Timelike Hopf duality and type IIA* string solutions
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abstract
The usual T-duality that relates the type IIA and IIB theories compactified on circles of inversely related radii does not operate if the dimensional reduction is performed on the time direction rather than a spatial one. This observation led to the recent proposal that there might exist two further ten-dimensional theories, namely type IIA* and IIB*, related to type IIB and IIA, respectively, by a timelike dimensional reduction. In this paper we explore such dimensional reductions in cases where time is the coordinate of a non-trivial U(1) fibre bundle. We focus, in particular, on situations where there is an odd-dimensional anti-de Sitter spacetime AdS2n+l, which can be described as a U (1) bundle over CPn, a non-compact version of CPn corresponding to the coset manifold SU(n, l)/U(n). In particular, we study the AdS5 x S5 and AdS7 x S4 solutions of type IIB supergravity and 11-dimensional supergravity. Applying a timelike Hopf T-duality transformation to the former provides a new solution of the type IIA* theory, of the form CP2 x S1 x S5. We show how the Hopf-reduced solutions provide further examples of 'supersymmetry without supersymmetry'. We also present a detailed discussion of the geometrical structure of the Hopf-fibred metric on AdS2n+l, and its relation to the horospherical metric that arises in the AdS/CFT correspondence.
