Extended actions, dynamics of edge modes, and entanglement entropy

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.

Edge modes of gravity. Part I. Corner potentials and charges

“…showing how the boundary Lagrangian affects the form of the corner symplectic potential, in agreement with [44,49,50]. As expected, this variation is naturally of the form δL = E L + dθ L , with E L the equations of motion and θ L the corner potential.…”

“…Proceeding with the systematic study of the corner symplectic potentials gives us an organizing principle for understanding the corner symmetries. Moreover, this systematic treatment requires to acknowledge that boundary Lagrangians also posses their own symplectic potentials, which naturally live at corners 4 [44,49,50] (see also [23] in the asymptotic context).…”

“…Having established that different formulations of gravity have different corner symmetry algebras, and that this latter is controlled by the corner symplectic potential, we can then apply the edge mode formalism introduced in [51] and pushed further in [50,[57][58][59][60]. This consists in restoring the gauge-invariance broken by the presence of a boundary by adding edge mode fields.…”

Edge modes of gravity. Part II. Corner metric and Lorentz charges

“…A more modern take on this, introduced in [118] and developed in [157][158][159][160][161][162], consists in explicitly introducing new corner degrees of freedom, called edge modes, in order to restore gauge invariance and decouple the notion of corner symmetry from that of gauge. The advantage is that we can, in this case, construct two sets of canonical charges, which for the example of Lorentz transformations with parameter α we denote C JHEP11(2020)027…”

“…They are gauge-invariant, while ϕ allows us to specify the choice of the corner gauge frame. With these new corner edge mode fields we can now introduce the notion of extended symplectic potential [118,123,158,162].…”