Lorentz-diffeomorphism edge modes in 3d gravity

Abstract

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.

Kinematical gravitational charge algebra

“…A. Boundary pre-symplectic 2-form In order to allow for boundary field variations, we can extend the bulk phase space by adding a boundary term to the pre-symplectic 2-form (6). In the present work, we focus on electric gauge transformations and translations, and we postpone the detailed study of magnetic gauge transformations to future investigation.…”

“…This will lead us to show that the constraints G α and P ϕ form an isu(2) Poincaré Lie algebra. 6 δ d ϕ is equivalent on-shell to the action of the covariant Lie derivative. More precisely we have that…”

“…There is growing understanding that edge modes must play a key role in quantum gravity. They are central in the quantization of 2d Yang-Mills and 2d gravity [2][3][4], They play a key role in 3d quantum gravity [5][6][7][8]. They are essential to the understanding of boundary dynamics and to the construction of defects operators in Chern-Simons theory [9][10][11][12].…”

Dual2+1Dloop quantum gravity on the edge

“…When discretizing gauge theories, and gravity in particular, a major problem is preserving gauge invariance despite the discreteness of the resulting theory. The presence of boundaries can be shown to introduce new degrees of freedom, called edge modes [25,26,27] 2 , which may be used to dress observables and make them gauge-invariant. These edge modes are associated to new boundary symmetries, which transform them and control the gluing map between subsystems.…”

Novel edge states in self-dual gravity

“…We are deeply grateful to Giorgio Immirzi for very helpful suggestions and correcting errors. After submitting this article to the arXiv, our attention was drawn to the reference [16] where edge states in the context of three dimensional gravity is discussed.…”