Consistent Truncation to Three Dimensional (Super-)gravity

TL;DR

The work develops a bulk procedure to consistently truncate general three-dimensional (super-)gravity theories with arbitrary matter and higher-derivative corrections to a pure gravity theory whose gravitational sector comprises only the Einstein-Hilbert term, a cosmological constant Λ, and a gravitational Chern-Simons term. It provides a concrete algorithm to determine the effective Λ(φ) by evaluating the action on an AdS_3 background and extremizing with respect to the scale, then setting φ to its extremum to complete the truncation; fermionic higher-derivative terms are shown to be removable via field redefinitions under unbroken SUSY. The authors demonstrate the method with a dimensional reduction of five-dimensional minimal supergravity on S^2 with curvature-squared terms, obtaining a three-dimensional pure supergravity plus a CS term whose central charges agree with AdS_3/CFT expectations and black hole entropy calculations. Overall, the paper offers a practical, bulk-side construction for consistent truncation in AdS_3 contexts and clarifies how the boundary CFT data emerges from the bulk action.

Abstract

For a general three dimensional theory of (super-)gravity coupled to arbitrary matter fields with arbitrary set of higher derivative terms in the effective action, we give an algorithm for consistently truncating the theory to a theory of pure (super-)gravity with the gravitational sector containing only Einstein-Hilbert, cosmological constant and Chern-Simons terms. We also outline the procedure for finding the parameters of the truncated theory. As an example we consider dimensional reduction on S^2 of the 5-dimensional minimal supergravity with curvature squared terms and obtain the truncated theory without any curvature squared terms. This truncated theory reproduces correctly the exact central charge of the boundary CFT.