Abstract
We show that the phase space of three-dimensional gravity contains two layers of dualities: between diffeomorphisms and a notion of “dual diffeomorphisms” on the one hand, and between first order curvature and torsion on the other hand. This is most elegantly revealed and understood when studying the most general Lorentz-invariant first order theory in connection and triad variables, described by the so-called Mielke-Baekler Lagrangian. By analyzing the quasi-local symmetries of this theory in the covariant phase space formalism, we show that in each sector of the torsion/curvature duality there exists a well-defined notion of dual diffeomorphism, which furthermore follows uniquely from the Sugawara construction. Together with the usual diffeomorphisms, these duals form at finite distance, without any boundary conditions, and for any sign of the cosmological constant, a centreless double Virasoro algebra which in the flat case reduces to the BMS3 algebra. These algebras can then be centrally-extended via the twisted Sugawara construction. This shows that the celebrated results about asymptotic symmetry algebras are actually generic features of three-dimensional gravity at any finite distance. They are however only revealed when working in first order connection and triad variables, and a priori inaccessible from Chern-Simons theory. As a bonus, we study the second order equations of motion of the Mielke-Baekler model, as well as the on-shell Lagrangian. This reveals the duality between Riemannian metric and teleparallel gravity, and a new candidate theory for three-dimensional massive gravity which we call teleparallel topologically massive gravity.
Highlights
In spite of the apparent simplicity owing to its topological nature, three-dimensional gravity is a very rich theory [1]
We show that the phase space of three-dimensional gravity contains two layers of dualities: between diffeomorphisms and a notion of “dual diffeomorphisms” on the one hand, and between first order curvature and torsion on the other hand
By analyzing the quasi-local symmetries of this theory in the covariant phase space formalism, we show that in each sector of the torsion/curvature duality there exists a well-defined notion of dual diffeomorphism, which follows uniquely from the Sugawara construction
Summary
In spite of the apparent simplicity owing to its topological nature, three-dimensional gravity is a very rich theory [1]. The interest in studying these dual charges and their algebra with the usual diffeomorphisms, the Lorentz transformations, and the translations, comes from the following question: what are the boundary symmetries of a given formulation of gravity? In a first order theory like the MB model (or even BF theory), this seemingly simple question has no clear answer Even though they are equivalent as symmetries of the Lagrangian, it is not clear at the level of the charge algebra whether one should consider the diffeomorphisms as independent from the Lorentz transformations and the translations. After having studied the finite distance covariant phase space of the MB model and introduced the dual charges, we conclude this work with a study of the metric and teleparallel formulations of the theory, both at the level of the equations of motion and of the Lagrangian itself.
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