Consistent nonlinear KK reduction of 11d supergravity on AdS7 × S4 and self-duality in odd dimensions

We show that there exists a consistent truncation of 11 dimensional supergravity to the ’massless’ fields of maximal (N=4) 7 dimensional gauged supergravity. We find the complete expressions for the nonlinear embedding of the 7 dimensional fields into the 11 dimensional fields, and check them by reproducing the d=7 susy transformation laws from the d=11 laws in various sectors. In particular we determine explicitly the matrix U which connects the Killing spinors to the gravitinos in the KK ansatz, and the dependence of the 4-index field strength on the scalars. This is the first time a complete nonlinear KK reduction of the original d=11 supergravity on a nontrivial compact space has been explicitly given. We need a first order formulation for the 3 index tensor field AΛΠΣ in d=11 to reproduce the 7 dimensional result. The concept of ’self-duality in odd dimensions’ is thus shown to originate from first order formalism in higher dimensions. For the AdS-CFT correspondence, our results imply that one can use 7d gauged supergravity (without further massive modes) to compute certain correlators in the d=6 (0,2) CFT at leading order in N. This eliminates an ambiguity in the formulation of the correspondence. The question whether in general a consistent Kaluza-Klein (KK) truncation exists at the nonlinear level is an old problem. For tori, the consistency is easy to prove, but for more complicated compact spaces little is known. In supergravity (sugra), the truncation of d=11 sugra on AdS4×S7 to maximal d = 4 gauged sugra was intensively studied 15 years ago [1], culminating in a series of papers by de Wit and Nicolai [2, 3]. The interest in those days was to find realistic 4 dimensional models from spontaneous compactification of maximal 11 dimensional sugra. Recent developments in the AdSCFT correspondence [4, 5, 6, 7, 8] have renewed interest in AdS compactifications Research supported by NSF Grant 9722101; E-mail addresses: [email protected], [email protected], [email protected]