A Grassmann Path From AdS_3 to Flat Space
Chethan Krishnan, Avinash Raju, Shubho Roy
TL;DR
This work presents a formal map from AdS$_3$ gravity to flat-space gravity by treating the inverse AdS radius $1/l$ as a Grassmann parameter with $\epsilon^2=0$, effectively realizing the Inonu–Wigner contraction $SO(2,2)\to ISO(2,1)$ at the level of the Chern–Simons action and its solutions. The authors show Banados’ AdS$_3$ solutions and the Brown–Henneaux Virasoro algebra contract to the flat-space BMS$_3$ structure, including the asymptotic charges and central extensions, and extend the construction to higher-spin theories, where flat-space higher-spin fields arise naturally from their AdS counterparts. This Grassmann approach provides an algebraic and tractable route to translate AdS results into flat space, with automatic provision of a nondegenerate trace and a clear readout for observables via holonomies. As an application, they demonstrate singularity resolution in a BMS-like gauge by embedding singular spin-2 configurations into a higher-spin framework and using holonomy constraints to obtain non-singular geometries with regular higher-spin fields.
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.