Abstract
We show that interpreting the inverse AdS_3 radius 1/l as a Grassmann variable results in a formal map from gravity in AdS_3 to gravity in flat space. The underlying reason for this is the fact that ISO(2,1) is the Inonu-Wigner contraction of SO(2,2). We show how this works for the Chern-Simons actions, demonstrate how the general (Banados) solution in AdS_3 maps to the general flat space solution, and how the Killing vectors, charges and the Virasoro algebra in the Brown-Henneaux case map to the corresponding quantities in the BMS_3 case. Our results straightforwardly generalize to the higher spin case: the recently constructed flat space higher spin theories emerge automatically in this approach from their AdS counterparts. We conclude with a discussion of singularity resolution in the BMS gauge as an application.
Highlights
Our simple observation is that this Inonu-Wigner contraction of the algebras can√be realized at the level of the theories, by taking the inverse AdS3 radius ≡ 1/l = λ to be a Grassmann parameter such that 2 = 0
We show that interpreting the inverse AdS3 radius 1/l as a Grassmann variable results in a formal map from gravity in AdS3 to gravity in flat space
The underlying reason for this is the fact that ISO(2, 1) is the Inonu-Wigner contraction of SO(2, 2). We show how this works for the Chern-Simons actions, demonstrate how the general (Banados) solution in AdS3 maps to the general flat space solution, and how the Killing vectors, charges and the Virasoro algebra in the Brown-Henneaux case map to the corresponding quantities in the BMS3 case
Summary
Witten [12] noticed that gravity in 2+1 dimensions can be written as a Chern-Simons gauge theory, with gauge group SO(2, 2) when the cosmological constant Λ is negative and gauge group ISO(2, 1) when it is zero. Is the Einstein-Hilbert action (with zero cosmological constant) in the first order formulation, if the generators satisfy the ISO(2, 1) algebra [Pa, Pb] = 0,. (where is a Grassmann parameter so that 2 = 0), in (2.18), the AdS EinsteinHilbert action (2.17) turns into the flat space Einstein-Hilbert action, but multiplied by an overall factor of. In other words we will see that the quantity multiplying the , after the above replacement, is the flat space gravitational action This makes sure that the Newton’s constant and Chern-Simons level after this replacement are related by (2.20). This generalizes very straightforwardly to higher spin theories as well, as we will briefly discuss later Another (non-Grassmann) way to think of the mapping from one theory to other is to think of it as the scaling limit where 1/l → 0 but with k/l is held fixed. This implies that we should take in the AdS case
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