Abstract
We extend a recent classification of three-dimensional spatially isotropic homogeneous spacetimes to Chern-Simons theories as three-dimensional gravity theories on these spacetimes. By this we find gravitational theories for all carrollian, galilean, and aristotelian counterparts of the lorentzian theories. In order to define a nondegenerate bilinear form for each of the theories, we introduce (not necessarily central) extensions of the original kinematical algebras. Using the structure of so-called double extensions, this can be done systematically. For homogeneous spaces that arise as a limit of (anti-)de Sitter spacetime, we show that it is possible to take the limit on the level of the action, after an appropriate extension. We extend our systematic construction of nondegenerate bilinear forms also to all higher-dimensional kinematical algebras.
Highlights
The crucial role played by symmetries in three-dimensional gravity becomes obvious from the intriguing possibility to write gravity in a gauge theory formulation
We extend a recent classification of three-dimensional spatially isotropic homogeneous spacetimes to Chern-Simons theories as three-dimensional gravity theories on these spacetimes
Since three-dimensional gravity does not allow for propagating degrees of freedom, the solutions of the respective theories are locally three-dimensional Minkowski, de Sitter, or anti-de Sitter (AdS), with the gauge algebras corresponding to the symmetries of the respective spacetime
Summary
The crucial role played by symmetries in three-dimensional gravity becomes obvious from the intriguing possibility to write gravity in a gauge theory formulation. Gravity on AdS3 in particular has received a great amount of attention as one finds that, imposing the right boundary conditions [3], the asymptotic symmetries of AdS3 yield the infinitedimensional symmetries of a two-dimensional conformal field theory (CFT). This set-up presents one of the best studied instances of the AdS/CFT duality [4,5,6]. In this work we want to define CS theories for all possible three-dimensional spatially isotropic homogeneous spacetimes. Ultrarelativistic gravity theories on the other hand could help to elucidate the structure of field theories with carrollian symmetries, which appear at event horizons [22, 23] and at null infinity (and arguably at spatial infinity [24,25,26]). Given the simplicity of gravity in three dimensions, it suggests itself as an interesting testing ground for the study of gravitational theories with non-lorentzian symmetries; see [30,31,32] for CS theories of the Carroll group and [33,34,35,36,37] for works on the (AdS-)Galilei group and supersymmetric extensions thereof
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