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.
在c=-1玻色子鬼系统和c=2复标量理论之间建立了一个“正则映射”，在c=-2费米子鬼系统和c=1狄拉克理论之间建立了一个类似的映射。这种映射的存在是由各自理论的特征的同一性所暗示的。各个c <0 and c>0理论共享相同的状态空间，而共形场的空间是不同的。在c<0对应物的这种映射上，（c>0）复标量和狄拉克理论继承了隐藏的非定域sl（2）对称性。
