We construct the most general supersymmetric configuration of $D2\mathrm{}$-branes and $D6\mathrm{}$-branes on a 6-torus. It contains arbitrary numbers of branes at relative $U\left(3\right)\mathrm{}$ angles. The corresponding supergravity solutions are constructed and expressed in a remarkably simple form, using the complex geometry of the compact space. The spacetime supersymmetry of the configuration is verified explicitly, by solution of the Killing spinor equations, and the equations of motion are verified too. Our configurations can be interpreted as a 16-parameter family of regular extremal black holes in four dimensions. Their entropy is interpreted microscopically by counting the degeneracy of bound states of $D$-branes. Our result agrees in detail with the prediction for the degeneracy of BPS states in terms of the quartic invariant of the E(7,7) duality group.

• Received 12 May 1997

DOI:https://doi.org/10.1103/PhysRevD.57.3509