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
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.
