Centrally extended BMS4 Lie algebroid

Abstract

We explicitly show how the field dependent 2-cocycle that arises in the current algebra of 4 dimensional asymptotically flat spacetimes can be used as a central extension to turn the BMS4 Lie algebra, or more precisely, the BMS4 action Lie algebroid, into a genuine Lie algebroid with field dependent structure functions. Both a BRST formulation, where the extension appears as a ghost number 2 cocyle, and a formulation in terms of vertex operator algebras are introduced. The mapping of the celestial sphere to the cylinder then implies zero mode shifts of the asymptotic part of the shear and of the news tensor.

Double-soft graviton amplitudes and the extended BMS charge algebra

“…It might be interesting to pursue these issues further for the case at hand, and to understand whether these poles can have nontrivial physical consequences. It may be the case, however, that they indicate that the interpretation of the charges is in fact more subtle, and that the integrals should be considered as formal objects in order to drop the boundary terms -such subtleties can arise for instance in the case of a vertex operator algebra [30].…”

“…As also discussed in [2], this term comes from the failure of the commutator of two asymptotic transformations to satisfy the gauge fixing conditions on the boundary as well as in the bulk. The extra boundary term is sometimes referred to as a central charge, or more precisely as a field-dependent central extension [1,29,30] since the corresponding bulk part of the charge is trivial. While in the Chern-Simons example in [3] the extended terms can be thought of as a purely boundary effect, in gravity the situation is a little different, since here the current is a total derivative and there is no unambiguous definition of bulk and boundary terms 21 .…”

“…This generalizes the work of [6,7,12] to amplitudes where more than one graviton is taken to be soft, and a particular combination of soft limits corresponds to the commutator of the BMS charge algebra. The general structure of the BMS algebra and the corresponding Dirac bracket in three and four dimensions was analyzed by studying the form of the classical symmetry transformations and charges in [29] (see also [30]), and it was found that while the global subalgebra in 4d has no central charges, there is a nontrivial extension of the classical algebra by a generalized 2-cocycle when the BMS algebra is promoted to include local (singular) superrotations. The extension term breaks the symmetry, similar to the breaking of conformal invariance by a nonzero central charge.…”

BMS4 algebra, its stability and deformations

“…in which M is a g-module, C p is the space of p-cochains and E's are related to each other by the differential operator d p,q n : E p,q n −→ E p+n,q−n+1 n [41,66]. In some specific cases one finds the 10 Of course, recalling that the global part of the supertranslations T 00 , T 01 , T 10 , T 11 are in the (2, 2) representation of the Lorentz group su(2) L × su(2) R , it is also natural to choose the indices to be half-integer valued, as suggested in [14]. differential function becomes trivial for n ≥ n 0 (for certain n 0 ) and E p,q n , ∀n ≥ n 0 are isomorphic to each other and therefore, E p,q n ∼ = E p,q ∞ .…”

Asymptotic Symmetries in the Gauge fixing Approach and the BMS group

“…Let us mention that a Lie algebroid structure is showing up at this stage [40,40,77]. The base manifold is given by the solution space, the field-dependent Lie algebra is the Lie algebra of asymptotic symmetry generators introduced above and the anchor is ab = 0.…”