Formula omitted] supersymmetric black attractors in six and seven dimensions

Abstract

Using a quaternionic formulation of the moduli space M ( IIA / K 3 ) of 10D type IIA superstring on a generic K3 complex surface with volume V 0 , we study extremal N = 2 black attractors in 6D space–time and their uplifting to 7D. For the 6D theory, we exhibit the role played by 6D N = 1 hypermultiplets and the Z m central charges isotriplet of the 6D N = 2 superalgebra. We construct explicitly the special hyper-Kähler geometry of M ( IIA / K 3 ) and show that the SO ( 4 ) × SO ( 20 ) invariant hyper-Kähler potential is given by H = H 0 + Tr [ ln ( 1 − V 0 −1 S ) ] with Kähler leading term H 0 = Tr [ ln V 0 ] plus an extra term which can be expanded as a power series in V 0 −1 and the traceless and symmetric 3×3 matrix S . We also derive the holomorphic matrix prepotential G and the flux potential G BH of the 6D black objects induced by the topology of the RR field strengths F 2 = d A 1 and F 4 = d A 3 on the K3 surface and show that G BH reads as Q 0 + ∑ m = 1 3 q m Z m . Moreover, we reveal that Z m = ∑ I = 1 20 Q I ( ∫ C 2 I J m ) where the isotriplet J m is the hyper-Kähler 2-form on the K3 surface. It is found as well that the uplifting to seven dimensions is quite similar to 4D/5D correspondence for back hole potential considered in arXiv: 0707.0964 [hep-ph]. Then we study the N = 2 black object attractors in 6D and 7D obtained respectively from type IIA string and M-theory on K3.