The Dynamics of Supertranslations and Superrotations in 2+1 Dimensions

TL;DR

The paper analyzes the dynamics of asymptotic symmetries in 2+1 dimensional, asymptotically flat gravity. By evaluating the boundary variation of the Einstein–Hilbert action, it derives an induced boundary action on future null infinity that comprises a Liouville-like bulk term and Schwarzian-like corner terms, encoding Goldstone-like modes for supertranslations and superrotations. This boundary action is shown to be equivalent to a chiral Liouville theory with a central charge c = 24π/κ^2 and is closely linked to Virasoro coadjoint orbits, with Hill’s equation surfacing in the analysis. The work connects 2D conformal dynamics at infinity to the BMS3 algebra and sets the stage for extending these ideas to higher dimensions and to quantum aspects of flat-space holography.

Abstract

Supertranslations, and at least in 2+1 dimensions superrotations, are asymptotic symmetries of the metric in asymptotically flat spacetimes. They are not, however, symmetries of the boundary term of the Einstein-Hilbert action, which therefore induces an action for the Goldstone-like fields that parametrize these symmetries. I show that in 2+1 dimensions, this action is closely related to a chiral Liouville action, as well as the "Schwarzian" action that appears in two-dimensional near-AdS physics.

Figures (1)

• Figure 1: A (2+1)-dimensional spacetime foliated by null cones $u=\hbox{const.}$