Consistent SO(6) reduction of type IIB supergravity on S5

Type IIB supergravity can be consistently truncated to the metric and the self-dual 5-form. We obtain the complete non-linear Kaluza–Klein S5 reduction ansatz for this theory, giving rise to gravity coupled to the fifteen Yang–Mills gauge fields of SO(6) and the twenty scalars of the coset SL(6,R)/SO(6) . This provides a consistent embedding of this subsector of N=8 , D=5 gauged supergravity in type IIB in D=10 . We demonstrate that the self-duality of the 5-form plays a crucial role in the consistency of the reduction. We also discuss certain necessary conditions for a theory of gravity and an antisymmetric tensor in an arbitrary dimension D to admit a consistent sphere reduction, keeping all the massless fields. We find that it is only possible for D=11 , with a 4-form field, and D=10 , with a 5-form. Furthermore, in D=11 the full bosonic structure of eleven-dimensional supergravity is required, while in D=10 the 5-form must be self-dual. It is remarkable that just from the consistency requirement alone one would discover D=11 and type IIB supergravities, and that D=11 is an upper bound on the dimension.