Black-Hole Attractors in N=1 Supergravity

TL;DR

This work extends the black-hole attractor mechanism to $N=1$ supergravity coupled to vector and chiral multiplets by deriving the attractor equations in terms of a moduli-dependent holomorphic gauge-kinetic matrix $f_{\Lambda\Sigma}$. It analyzes both general frameworks and concrete truncations from $N=2$ theories, including CY orientifold reductions, showing how attractor points satisfy $\partial_i V=0$ and, in special cases, reduce to pure-spinor conditions. The authors present explicit results for several $L(q,P,\dot P)$ homogeneous spaces, deriving attractor potentials, entropy formulas, and the role of duality invariants, illustrating when attractors exist and when they do not. The findings highlight a broad applicability of attractor behavior in $N=1$ theories with moduli-dependent gauge couplings and reveal a deep link to pure-spinor geometry and electric-magnetic duality structures, with implications for brane configurations in string compactifications. The work suggests future directions including Born–Infeld-type corrections and higher-curvature terms to assess robustness of attractors in more general settings.

Abstract

We study the attractor mechanism for N=1 supergravity coupled to vector and chiral multiplets and compute the attractor equations of these theories. These equations may have solutions depending on the choice of the holomorphic symmetric matrix f_{ΛΣ} which appears in the kinetic lagrangian of the vector sector. Models with non trivial electric-magnetic duality group which have or have not attractor behavior are exhibited. For a particular class of models, based on an N=1 reduction of homogeneous special geometries, the attractor equations are related to the theory of pure spinors.