Abstract
In this paper we study the scalar geometries occurring in the dimensional reduction of minimal five-dimensional supergravity to three Euclidean dimensions, and find that these depend on whether one first reduces over space or over time. In both cases the scalar manifold of the reduced theory is described as an eight-dimensional Lie group L (the Iwasawa subgroup of G 2(2)) with a left-invariant para-quaternionic-Kähler structure. We show that depending on whether one reduces first over space or over time, the group L is mapped to two different open L-orbits on the pseudo-Riemannian symmetric space G 2(2) /(SL(2) · SL(2)). These two orbits are inequivalent in the sense that they are distinguished by the existence of integrable L-invariant complex or para-complex structures.
Highlights
In this paper we study the scalar geometries occurring in the dimensional reduction of minimal five-dimensional supergravity to three Euclidean dimensions, and find that these depend on whether one first reduces over space or over time
While in space-like reductions leading to Riemannian symmetric target spaces M = G/K of non-compact type one can rely on the Iwasawa decomposition G = KL, to provide a global parametrization of M using the transitive action of the solvable Iwasawa subgroup L ⊂ G, such a global parametrization is no longer possible for the pseudo-Riemannian symmetric spaces G/H appearing in time-like reductions
In [8] it was shown that solutions with regular event horizons correspond to complete geodesics which are contained within a ‘solv-patch’, i.e. an open orbit of the Iwasawa subgroup, whereas geodesics which are not fully contained in a single solv-patch lift to singular space-time geometries
Summary
We perform the dimensional reduction of pure five-dimensional supergravity to three dimensions. Following the procedure of [28] we have absorbed the Kaluza Klein scalar σ into h0 to obtain scalars fitting into four-dimensional vector multiplets In this formulation x and y are independent dynamical scalar fields, whereas σ is a dependent field which can be expressed in terms of y via eσ = 6−1/3y. Upon reduction to three dimensions, each gives rise to 2 scalars: p0 and p1 correspond to the four-dimensional components of the two vector fields,. It is known that the reduction over one space-like and one time-like dimension gives rise to a space which is locally isometric to the pseudo-Riemannian symmetric space G2(2)/(SL(2)·SL(2)), which is para-quaternionic-Kahler, as expected for three-dimensional Euclidean hypermultiplets [15].
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