Abstract
The proper definition of subsystems in gauge theory and gravity requires an extension of the local phase space by including edge mode fields. Their role is on the one hand to restore gauge invariance with respect to gauge transformations supported on the boundary, and on the other hand to parametrize the largest set of boundary symmetries which can arise if both the gauge parameters and the dynamical fields are unconstrained at the boundary. In this work we construct the extended phase space for three-dimensional gravity in first order connection and triad variables. There, the edge mode fields consist of a choice of coordinate frame on the boundary and a choice of Lorentz frame on the bundle, which together constitute the Lorentz-diffeomorphism edge modes. After constructing the extended symplectic structure and proving its gauge invariance, we study the boundary symmetries and the integrability of their generators. We find that the infinite-dimensional algebra of boundary symmetries with first order variables is the same as that with metric variables, and explain how this can be traced back to the expressions for the diffeomorphism Noether charge in both formulations. This concludes the study of extended phase spaces and edge modes in three-dimensional gravity, which was done previously by the author in the BF and Chern-Simons formulations.
Highlights
JHEP02(2018)029 spirit of Stueckelberg fields [15]
This was answered in [16], where it was shown that the generators of the boundary symmetries on the extended phase space are a “dressed” version of the usual Hamiltonian boundary charges1 associated with gauge transformations
While the usual Hamiltonian boundary charges can be promoted to observables if the constraints are defined with smearing fields which are vanishing on the boundary, the extended phase space construction provides a way of relaxing this requirement, and of constructing boundary observables without imposing any boundary conditions on the gauge parameters or the dynamical fields
Summary
We are interested in first order three-dimensional gravity with a cosmological constant of arbitrary sign. From which one can see that the Lagrangian is not vanishing on-shell (unless l2 = ∞) This property is true in any dimension for gravity with a cosmological constant. Lξe = d(ξ e) + ξ (de), Lξω = d(ξ ω) + ξ (dω) These transformations are not independent, as diffeomorphisms can be realized on-shell as a combination of field dependent Lorentz transformations and translations [16, 18]. In order to describe the edge modes and the extended phase space, one has to first choose a set of independent gauge transformations. The goal of the present paper is to extend this construction to the case of Lorentz transformations and diffeomorphisms
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