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 Gmn(φ) of the moduli space SO(4,k)×SO(1,1)SO(4)×SO(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.