Branes at Angles and Black Holes

TL;DR

The article develops the most general supersymmetric bound states of D2- and D6-branes on a 6-torus, with an arbitrary number of branes at relative U(3) angles, and derives the corresponding four-dimensional black-hole solutions. By expressing the solutions in terms of the complex geometry of the compact space and harmonic functions, the authors obtain a remarkably simple classical description that unifies D2/D6 (and, via T-duality, D4/D0) configurations. They prove spacetime supersymmetry through explicit Killing spinor analysis and compute the black-hole entropy, showing it matches the microscopic degeneracy predicted by the quartic invariant of the $E(7,7)$ duality group. The work implies a broad geometric structure for BPS bound states, with potential extensions to Calabi–Yau compactifications and fluxes, and provides a consistent microscopic account of black-hole entropy in this rich brane setup.

Abstract

We construct the most general supersymmetric configuration of D2-branes and D6-branes on a 6-torus. It contains arbitrary numbers of branes at relative U(3) 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. 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.