N=2 Supersymmetric Black Attractors in Six and Seven Dimensions

Using a quaternionic formulation of the moduli space $\boldsymbol{M}({\small IIA/K3})$ of 10D type IIA superstring on a generic K3 complex surface with volume $\boldsymbol{V}_{0}$, we study extremal $\mathcal{N}=2$ black attractors in 6D space-time and their uplifting to 7D. For the 6D theory, we exhibit the role played by 6D $\mathcal{N}=1$ hypermultiplets and the $Z^{m}$ central charges isotriplet of the 6D $\mathcal{N}=2$ superalgebra. We construct explicitly the special hyperKahler geometry of $$ and show that the $SO(4) \times SO(20)$ invariant hyperKahler potential is given by $\mathcal{H}=\mathcal{H}_{0}+\mathrm{Tr}[ \ln (1-$ with Kahler leading term $\mathcal{H}_{0}=\mathrm{Tr}[ \ln \boldsymbol{V}_{0}]$ plus an extra term which can be expanded as a power series in $\boldsymbol{V}$ and the traceless and symmetric 3$\times$3 matrix $\boldsymbol{S}$. We also derive the holomorphic matrix prepotential $\mathcal{G}$ and the flux potential $\mathcal{G}_{BH}$ of the 6D black objects induced by the topology of the RR field strengths $\mathcal{F}_{2}=d\mathcal{A}_{1}$ and $$ on the K3 surface and show that \mathcal{G% }_{BH} reads as $Q_{0}+\sum_{m=1}^{3}q^{m}Z^{m}$. Moreover, we reveal that $$ where the isotriplet $J^{m}$ is the hyperKahler 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-th].