Abstract
Abstract We analyze some properties of the four dimensional supergravity theories which originate from five dimensions upon reduction. They generalize to N > 2 extended supersymmetries the d-geometries with cubic prepotentials, familiar from N = 2 special Kähler geometry. We emphasize the role of a suitable parametrization of the scalar fields and the corresponding triangular symplectic basis. We also consider applications to the first order flow equations for non-BPS extremal black holes.
Highlights
The allowed scalar manifolds for the N = 2 five-dimensional supergravity coupled to nV −1Abelian vector multiplets, parametrized by scalar fields φx (x = 1, . . . , nV − 1), can be described as the-dimensional cubic hypersurface 1 3! dijk λiλj λk =of an ambient space spanned by nV coordinates λi = λi(φx) (i = 1, . . . , nV ) [1]
When this theory is dimensionally reduced to four dimensions, it yields a particular class of N = 2 four-dimensional matter coupled models with special Kahler target space geometries, which were studied in [3] under the name “d-spaces”
There, the uplift between four and five dimensions was called “r-map”, since it associates real scalars to the N = 2 four dimensional complex scalar fields belonging to the nV D = 4 vector multiplets: zi = Xi/X0 = ai − i λi, with ai, λi real and with the index 0 pertaining to the D = 4 graviphoton
Summary
Our results have interesting applications to non-BPS extremal black holes, that we illustrate by making a precise and non trivial comparison between the methods of [22] and [26] in the computation of the fake superpotential [27] for non-BPS solutions and (p0, q0) charge configuration in the stu-truncation of N = 8 supergravity Beyond their interest in relation to supergravity structure and solutions, one may hope that these general properties of N ≥ 2 d-geometries and the corresponding triangular symplectic frame (with degree-4 nilpotent axionic translations) could play a role in understanding the symmetry structure of supergravity counterterms, in order to clarify the issue of ultraviolet finiteness of N = 8 and other extended supergravity theories in D = 4 space-time dimensions [28].
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