Three-dimensional spin-3 theories based on general kinematical algebras
Jan Rosseel, Stefan Prohazka, Eric Bergshoeff, Daniel Grumiller
We initiate the study of non-and ultra-relativistic higher spin theories. For sake of simplicity we focus on the spin-3 case in three dimensions. We classify all kinematical algebras that can be obtained by all possible Inönü-Wigner contraction procedures of the kinematical algebra of spin-3 theory in three dimensional (anti-) de Sitter space-time. We demonstrate how to construct associated actions of Chern-Simons type, directly in the ultra-relativistic case and by suitable algebraic
... in the non-relativistic case. We show how to give these kinematical algebras an infinite-dimensional lift by imposing suitable boundary conditions in a theory we call "Carroll Gravity", whose asymptotic symmetry algebra turns out to be an infinite-dimensional extension of the Carroll algebra. to but different from BMS, while the near horizon boundary conditions in    lead to infinite copies of the Heisenberg algebra, in terms of which BMS (or related symmetry algebras) are composite. 2 See however [56-62] for attempts to consider higher spin theories in non-AdS backgrounds with nonrelativistic CFT duals. 3 See however [68-70] for recent progress concerning higher spin theories in four dimensional flat space. -2 - JHEP01(2017)114 we will start from the observation made in  that all kinematical algebras can be obtained by taking sequential Inönü-Wigner (IW) contraction procedures 4 of the (A)dS algebras. We will then classify all possible IW contraction procedures of the kinematical algebra of spin-3 theory in (A)dS 3 , as well as all possible kinematical algebras that can be obtained by sequential contraction procedures. Some of the kinematical algebras that are obtained in this way can be interpreted as spin-3 extensions of the Galilei and Carroll algebras. We will show that one can construct Chern-Simons theories for (suitable extensions of) these algebras. These can then be interpreted as non-and ultra-relativistic three-dimensional spin-3 theories. We will in particular argue that these theories can be viewed as higher spin generalizations of Extended Bargmann gravity [13,    and Carroll gravity  , two examples of non-and ultra-relativistic gravity theories that have been considered in the literature recently. The kinematical algebras of spin-3 theories that we obtain in this paper are finitedimensional. Relativistic three-dimensional kinematical algebras have infinite-dimensional extensions that are obtained as asymptotic symmetry algebras upon imposing suitable boundary conditions on metric and higher spin fields, such as the Virasoro algebra (for the AdS algebra) , the BMS algebra (for the Poincaré algebra) [77, 78] or W -algebras (for their higher spin generalizations) [79, 80] . It is interesting to ask whether the non-and ultra-relativistic algebras found in this paper also have infinite-dimensional extensions that correspond to asymptotic symmetry algebras of their corresponding higher spin gravity theories. We will not attempt to address this question in full generality in this paper. We will, however, show that the spin-2 Carroll algebra allows for an infinite-dimensional extension. In particular, we will show that there exist suitable boundary conditions in three-dimensional Carroll gravity, such that the resulting asymptotic symmetry algebra is an infinite-dimensional extension of the Carroll algebra. This suggests that a similar result should also hold for the non-and ultra-relativistic spin-3 theories constructed in this paper as well as for the other spin-2 theories that have not been investigated in detail yet. The organization of this paper is as follows. In section 2, we classify all IW contraction procedures of the kinematical algebra of spin-3 theory in (A)dS 3 . We then classify all kinematical algebras that can be obtained by combining these various contraction procedures. In section 3, we restrict ourselves to the algebras that can be interpreted as non-and ultrarelativistic ones, for zero cosmological constant. We argue that in the ultra-relativistic cases, a Chern-Simons theory can be constructed in a straightforward manner. This is not true for the non-relativistic cases. However, we demonstrate that the non-relativistic kinematical algebras can be suitably extended in such a way that a Chern-Simons action can be written down. We then show via a linearized analysis that the non-and ultra-relativistic spin-3 Chern-Simons theories thus obtained can be viewed as spin-3 generalizations of Extended Bargmann gravity and Carroll gravity, respectively. In section 4 we discuss boundary conditions for Carroll gravity that lead to an infinite-dimensional extension of the Carroll algebra. This section does not depend on the results of the previous sections 4 The terminology 'IW contraction procedures' might perhaps sound a little unconventional at this point. We refer to section 2.2 for a more precise discussion about the difference in our use of the terms 'contraction' and 'contraction procedures' and why this is relevant for our work.