An Sl(2, Z) multiplet of nine-dimensional type II supergravity theories

Abstract We show that only by performing generalized dimensional reductions all possible brane configurations are taken into account and one gets the complete lower-dimensional theory. We apply this idea to the reduction of type IIB supergravity in an SL (2, Z)-covariant way and establish T -duality for the type II superstring effective action in the context of generalized dimensional reduction giving the corresponding generalized Buscher's T -duality rules. The full (generalized) dimensional reduction involves all the S -duals of D7-branes: Q7-branes and a sort of composite 7-branes. The three species constitute an SL (2, Z) triplet. Their presence induces the appearance of the triplet of masses of the 9-dimensional theory. The T -duals, including a “KK-8A-brane”, which must have a compact transverse dimension have to be considered in the type IIA side. Compactification of 11-dimensional KK-9M-branes (a.k.a. M9-branes) on the compact transverse dimension give D8-branes while compactification on a world-volume dimension gives KK-8A-branes. The presence of these KK-monopole-type objects breaks translation invariance and two of them given rise to an SL (2, Z)-covariant massive 11-dimensional supergravity whose reduction gives the massive 9-dimensional type II theories.