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. Contents 1 Motivations 2 Extended phase spaces 3 Extended actions 4 Examples 4.1 Chern-Simons theory 4.1.1 Extended phase space 4.1.2 Boundary dynamics 4.1.3 Gluing of subregions 4.1.4 Entanglement entropy 4.2 Maxwell theory 4.3 Maxwell-Chern-Simons theory 4.3.1 Extended phase space 4.3.2 Boundary dynamics 4.3.3 Entanglement entropy 4.4 BF theory 5 Perspectives A Covariant phase space B Maxwell theory in radial gauge C Maxwell-Chern-Simons theory in temporal gauge D Boundary conditions as boundary sources E Hamiltonian quantization of the chiral bosons F Extended action and phase space for non-Abelian theories F.1 Chern-Simons theory F.2 Yang-Mills theory F.3 BF theory = −dc. (3.2)8 Although the result is unfortunately hidden in appendix B. 9 An important difference is that Harlow and Wu are not concerned with edge modes and extended phase spaces.

Boundary electromagnetic duality from homological edge modes

“…There are several reasons to be interested in the form of the action given in (37). First, we note that this is essentially the extended BF action discussed in [11], and thus on the basis of that analysis, we would expect a central charge to arise. Second, the term φ∧F that is needed to compensate for the failure of gauge invariance of Ã∧F (in the presence of a codimension 2 stratum) can be given the following interpretation: it is a 2-dimensional BF action term that can function as a duality wall operator.…”

“…In the next section, we discuss how such a stratum emerges from inserting electric Wilson lines into the boundary, and in Section VII we use these edge modes to write down a dressed action that leads to the desired non-commutation relation. We will see that this dressed action is equivalent to supplementing the action (11) with some codimension two data.…”

“…where we now see that we have transformed this action into the form of our earlier low-energy action (11) along with two additional terms that live on a codimension 2 stratum. Evidently, the third term has to be understood in the following way:…”

“…However, according to the covariant phase space formalism, we need dynamical input in order to define the boundary pre-symplectic form corresponding to such a φ: it should arise from a well-defined variational problem on the boundary. In [10] (see also [11]) we derive the boundary pre-symplectic form given in [2] as well as their bulk-to-boundary matching condition (indeed, we derive the symplectic structure of the entire BRST extension of the theory) from the following action:…”

“…In Section III, we lay the groundwork for a systematic construction of the relevant edge modes. In particular, we review our previous work [10], where we showed that the edge modes in [2] can be constructed by means of two ingredients: at the kinematical level, they arise when one implements topological boundary conditions by means of a homotopy pullback, and at the dynamical level, the relevant bulk-to-boundary matching conditions and boundary symplectic structure arise from a particular choice of boundary action (on the latter, see also [11]). [12] The subsequent sections then apply these tools to construct EM duality on the boundary and non-commuting electric and magnetic charges in codimension 2.…”