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

TL;DR

This paper explores two non-linear realizations of the 4D conformal group, showing their geometric relation via AdS_5 coordinate changes, and maps conformal Galileons between these representations, revealing complex interrelations and S-matrix equivalence.

Contribution

It provides a geometric interpretation of the field redefinition relating two conformal group realizations and maps Galileons between these frameworks, highlighting their non-trivial correspondence.

Findings

The field redefinition is a coordinate change in AdS_5.

The DBI action maps to a linear combination of Galileons.

Dilaton S-matrix is equivalent in both representations.

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 \eta_{\mu\nu} e^{-2 \pi}; 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.