Supersymmetric Black Attractors in Six and Seven Dimensions

Using a quaternionic formulation of the moduli space M (IIA/K3) 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 Zm central charges isotriplet of the 6D N = 2 superalgebra. We construct explicitly the special hyperKahler geometry of M (IIA/K3) and show that the SO (4)×SO (20) invariant hyperKahler potential is given by H = H0 + Tr [ ln ( 1− V −1 0 S )] with Kahler leading term H0 = Tr [lnV 0] plus an extra term which can be expanded as a power series in V −1 0 and the traceless and symmetric 3×3 matrix S. We also derive the holomorphic matrix prepotential G and the flux potential GBH of the 6D black objects induced by the topology of the RR field strengths F2 = dA1 and F4 = dA3 on the K3 surface and show that GBH reads as Q0 + ∑3 m=1 q mZm. Moreover, we reveal that Zm = ∑20 I=1 QI (∫ C 2 Jm ) where the isotriplet Jm is the hyperKahler 2form 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 [email protected] [email protected] [email protected] [email protected] considered in arXiv 0707.0964 [hep-th]. Then we study the N = 2 black object attractors in 6D and 7D obtained respectively from type IIA string and M-theory on K3.