Abstract
The conformal extension of the BMS_{3} algebra is constructed. Apart from an infinite number of "superdilatations," in order to incorporate superspecial conformal transformations, the commutator of the latter with supertranslations strictly requires the presence of nonlinear terms in the remaining generators. The algebra appears to be very rigid, in the sense that its central extensions as well as the coefficients of the nonlinear terms become determined by the central charge of the Virasoro subalgebra. The wedge algebra corresponds to the conformal group in three spacetime dimensions SO(3,2), so that the full algebra can also be interpreted as an infinite-dimensional nonlinear extension of the AdS_{4} algebra with nontrivial central charges. Moreover, since the Lorentz subalgebra [sl(2,R)] is nonprincipally embedded within the conformal (wedge) algebra, according to the conformal weight of the generators, the conformal extension of BMS_{3} can be further regarded as a W_{(2,2,2,1)} algebra. An explicit canonical realization of the conformal extension of BMS_{3} is then shown to emerge from the asymptotic structure of conformal gravity in three dimensions, endowed with a new set of boundary conditions. The supersymmetric extension is also briefly addressed.
Highlights
Introduction.—The symmetries of special relativity are embodied through the Poincarealgebra
The wedge algebra corresponds to the conformal group in three spacetime dimensions SO(3,2), so that the full algebra can be interpreted as an infinite-dimensional nonlinear extension of the AdS4 algebra with nontrivial central charges
An explicit canonical realization of the conformal extension of BMS3 is shown to emerge from the asymptotic structure of conformal gravity in three dimensions, endowed with a new set of boundary conditions
Summary
Firmly established by the superconformal algebra ospð1j4Þ [40]. Interestingly, in contradistinction to the four-dimensional case, the BMS3 algebra [41,42] is known to admit a fully fledged supersymmetric extension, in the sense that supertranslations possess suitable fermionic square roots, spanned by an infinite number of fermionic canonical generators [43]. The commutator of supertranslations with special conformal transformations strictly requires the presence of nonlinear terms in the remaining generators, which become well defined provided that the BMS3-Weyl subalgebra is endowed with nonvanishing central extensions. It is worth pointing out that, as it occurs for classical W algebras, the conformal BMS3 algebra is well defined provided that the Virasoro central charge does not vanish; since otherwise, the coefficients that give support to the nonlinear terms would blow up This is not necessarily the case for the quantum algebra because these coefficients as well as the central extensions generically acquire corrections.
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