On Black Hole Effective Potential in 6D/7D N=2 Supergravity

Using the harmonic superspace method and the duality between real and complex representations of hypermultiplets, we compute the explicit scalar field expression of the quaternionic metric $G_{mn}$ of the moduli space SO(1,1)x[SO(4,k)/(SO(4)xSO(k))] of 6D N=2 supergravity with generic k Maxwell supermultiplets. The obtained metric includes the particular case k=20 associated with 10D type IIA superstring on K3. Uplifting to 7D N=2 supergravity is described and aspects of 6D/7D black attractor effective potentials are studied.