Abstract
We present higher-spin algebras containing a Poincaré subalgebra and with the same set of generators as the Lie algebras that are relevant to Vasiliev’s equations in any space-time dimension D ≥ 3. Given these properties, they can be considered either as candidate rigid symmetries for higher-spin gauge theories in Minkowski space or as Carrollian conformal higher-spin symmetries in one less dimension. We build these Lie algebras as quotients of the universal enveloping algebra of mathfrak{iso}left(1,D-1right) and we show how to recover them as Inönü-Wigner contractions of the rigid symmetries of higher-spin gauge theories in Anti de Sitter space or, equivalently, of relativistic conformal higher-spin symmetries. We use the same techniques to also define higher-spin algebras with the same set of generators and containing a Galilean conformal subalgebra, to be interpreted as non-relativistic limits of the conformal symmetries of a free scalar field. We begin by showing that the known flat-space higher-spin algebras in three dimensions can be obtained as quotients of the universal enveloping algebra of mathfrak{iso}left(1,2right) and then we extend the analysis along the same lines to a generic number of space-time dimensions. We also discuss the peculiarities that emerge for D = 5.
Highlights
The Poincaré and (Anti) de Sitter algebras describe the isometries of the vacuum in gravitational theories
They can be considered either as candidate rigid symmetries for higher-spin gauge theories in Minkowski space or as Carrollian conformal higher-spin symmetries in one less dimension. We build these Lie algebras as quotients of the universal enveloping algebra of iso(1, D − 1) and we show how to recover them as Inönü-Wigner contractions of the rigid symmetries of higher-spin gauge theories in Anti de Sitter space or, equivalently, of relativistic conformal higherspin symmetries
We focus on the global symmetries of gauge theories involving only Fronsdal’s fields in AdSD and on their construction as quotients of the universal enveloping algebra (UEA) of so(2, D − 1)
Summary
The Poincaré and (Anti) de Sitter algebras describe the isometries of the vacuum in gravitational theories. The situation is instead radically different in three-dimensional Minkowski space, where global higher-spin symmetries can be naturally obtained as Inönü-Wigner contractions of the AdS ones [16, 48–51] In this context, one can build interacting gauge theories using Chern-Simons actions as in AdS3, while specific interactions with matter were introduced following Vasiliev’s approach [52]. We start filling the gap by presenting bosonic higher-spin extensions of the Poincaré algebra in any space-time dimensions We reach this goal by classifying the quotients of the universal enveloping algebra of iso(1, D − 1) that have the same set of generators as the Lie algebras entering Vasiliev’s equations.
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