Most general theory of 3d gravity: covariant phase space, dual diffeomorphisms, and more

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.

3d gravity in Bondi-Weyl gauge: charges, corners, and integrability

“…In this section we derive the charges associated with the internal symmetries of the triad. This method is specific to the three-dimensional case, and exploits the topological nature of the theory to trade the diffeomorphisms for internal symmetries [67,[72][73][74]. In addition to the Lorentz transformations, we consider the so-called "translations" acting as…”

“…• Finally, another possible generalization is to consider as the starting point the socalled Mielke-Baekler Lagrangian for three-dimensional gravity in triad variables [67,83,84]. This Lagrangian contains a Chern-Simons term and a torsion term, which allows to obtain two different central charges for the Virasoro algebras (or the two central charges of BMS 3 in the flat limit).…”

Chiral massive news: null boundary symmetries in topologically massive gravity

“…It would be interesting to compare the charges in the metric and dreibein formulation of Einstein-Λ or TMG. It would also be interesting to discuss the 3d dual charges defined in [101,102] and understand whether and/or how they are included in the maximal phase space discussed in this work.…”

Conservation and integrability in lower-dimensional gravity

“…For related works in other formalisms see e.g. [24,[67][68][69][70][71][72][73][74][75][76][77][78][79][80][81]. Note that it is expected that different formalisms lead to different symmetry algebras [77].…”