Non-linear representations of the conformal group and mapping of galileons

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 in [1] 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.