BPS Saturated Dyonic Black Holes of $N=8$ Supergravity Vacua

We summarize the results for four-dimensional Bogomol'nyi-Prasad- Sommerfield (BPS) saturated dyonic black hole solutions arising in the Kaluza-Klein sector and the three-form field sector of the eleven-dimensional supergravity on a seven-torus. These black hole solutions break $3\over 4$ of $N=8$ supersymmetry, fill out the multipletes of $U$-duality group, and the two classes of solutions are related to each other by a discrete symmetry transformation in $E_7$. Using the field redefinitions of the corresponding effective actions, we present these solutions in terms of fields describing classes of dyonic black holes carrying charges of $U(1)$ gauge fields in Neveu-Schwarz-Neveu-Schwarz and/or Ramond-Ramond sector(s) of the type-IIA superstring on a six-torus. We also summarize the dependence of their ADM masses on the asymptotic values of scalar fields, parameterizing the toroidal moduli space and the string coupling constant.