Black Hole Attractors and Pure Spinors

TL;DR

The paper develops a ten-dimensional pure spinor formulation of black hole attractors for a broad class of Type II compactifications preserving ${\cal N}=2$, including non-Kähler and flux backgrounds. By translating the SUSY variations into generalized complex geometry, it derives attractor flow equations for internal pure spinors and shows that at the horizon a generalized stabilization condition $\sum_k f_k = 2\mathrm{Im}(\bar{C}\,\Phi)$ fixes the moduli in terms of charges, with Hitchin's functional controlling solvability and entropy. The formalism reproduces standard Calabi–Yau results and extends to explicit examples like ${\rm IIB}$ on $T^6$, highlighting how pure spinors encode the dynamics of vector and hypermultiplet moduli along the radial flow. The results offer a unified, higher-dimensional perspective on attractors, connected to generalized complex geometry and potential links to topological string theory beyond Calabi–Yau geometries.

Abstract

We construct black hole attractor solutions for a wide class of N=2 compactifications. The analysis is carried out in ten dimensions and makes crucial use of pure spinor techniques. This formalism can accommodate non-Kaehler manifolds as well as compactifications with flux, in addition to the usual Calabi-Yau case. At the attractor point, the charges fix the moduli according to sum_k f_k = Im(C Phi), where Phi is a pure spinor of odd (even) chirality in IIB (A). For IIB on a Calabi-Yau, Phi=Omega and the equation reduces to the usual one. Methods in generalized complex geometry can be used to study solutions to the attractor equation.