Non-linear Representations of the Conformal Group and Mapping of Galileons

TL;DR

This paper shows that two non-linear realizations of the 4D conformal group—the Weyl (dilaton) representation and the DBI (brane in AdS$_5$) representation—are related by a geometric AdS$_5$ coordinate change. It derives explicit forward and inverse mappings between the Weyl and DBI variables via coset construction and demonstrates that conformal Galileons map into linear combinations across representations, preserving second-order equations of motion. The authors verify S-matrix equivalence for dilaton scattering and analyze how non-trivial backgrounds alter light-cone propagation, revealing that Minkowski-luminal behavior in one frame can become subluminal in the other. These results illuminate how dual IR descriptions encode the same physics and raise subtle questions about causality and EFTs in curved-space duals.

Abstract

There are two common non-linear realizations of the 4D conformal group: in the first, the dilaton is the conformal factor of the effective metric η_{μν} e^{-2 π}; in the second it describes the fluctuations of a brane in AdS_5. The two are related by a complicated field redefinition, found by Bellucci, Ivanov and Krivonos (2002) to all orders in derivatives. We show that this field redefinition can be understood geometrically as a change of coordinates in AdS_5. In one gauge the brane is rigid at a fixed radial coordinate with a conformal factor on the AdS_5 boundary, while in the other one the brane bends in an unperturbed AdS_5. This geometrical picture illuminates some aspects of the mapping between the two representations. We show that the conformal Galileons in the two representations are mapped into each other in a quite non-trivial way: the DBI action, for example, is mapped into a complete linear combination of all the five Galileons in the other representation. We also verify the equivalence of the dilaton S-matrix in the two representations and point out that the aperture of the dilaton light-cone around non-trivial backgrounds is not the same in the two representations.