Black Holes and U-Duality

TL;DR

This work constructs the most general charged rotating black holes in $4\le D\le 9$ from toroidally compactified Type II/String/M-theory by starting from a minimal generating solution and acting with the classical duality group, thereby generating all charge configurations without changing the Einstein-frame metric. It shows that the full set of solutions can be obtained via orbits of the maximal compact subgroups ($C_U$) of the duality groups, yielding explicit charge counting commensurate with the dimension of the charge lattice. For BPS-saturated static black holes, the horizon area and ADM mass can be expressed in duality-invariant forms: in $D=4$ via the quartic invariant $J_4$ of $E_{7(7)}$, in $D=5$ via the cubic invariant $J_3$ of $E_{6(6)}$, while in $D\ge 6$ such invariants vanish and the horizon area is zero. The results illuminate how microscopic entropy should depend only on duality-invariant combinations of bare charges, and they provide a procedure to obtain manifestly $U$-duality invariant mass formulas across dimensions, with explicit connections to known attractor-type behavior for BPS black holes.

Abstract

We find the general charged rotating black hole solutions of the maximal supergravities in dimensions $4\le D\le 9$ arising from toroidally compactified Type II string or M-theories. In each dimension, these are obtained by acting on a generating solution with classical duality symmetries. In D=4, D=5 and $6\le D \le 9$ the generating solution is specified by the ADM mass, $[{D-1}/2]$-angular momentum components and five, three and two charges, respectively. We discuss the BPS-saturated (static) black holes and derive the U-duality invariant form of the area of the horizon. We also comment on the U-duality invariant form of the BPS mass formulae.