First Order Description of Black Holes in Moduli Space

TL;DR

The paper develops a unified first-order description for extremal black holes in moduli space by introducing a prepotential $W$ whose gradient drives the radial flow of scalars and the metric. It proves that the standard second-order field equations can be recast as a first-order system with $\dot U = e^U W$ and $\dot \Phi^r = 2 e^U g^{rs} \partial_s W$, and shows that the black-hole potential satisfies $V_{BH} = W^2 + 2 g^{rs} \partial_r W \partial_s W$, while horizon data correspond to extrema of $W$ and $V_{BH}$, and the horizon entropy equals $S_{BH} = W^2|_{hor}$. For extended supergravity theories with $N\ge 3$, explicit forms of $W$ are constructed in terms of U-duality invariants, yielding multiple inequivalent attractors (BPS and non-BPS) across $N=3,4,5,6,8$ cases and clarifying how different charge configurations map to distinct prepotentials. The work further discusses a non-extremal extension with explicit $\tau$-dependence and shows how a first-order description could persist in certain regimes, offering a path towards a fake-supergravity perspective. Overall, the framework unifies attractor phenomena across extended supergravities and links horizon data to duality-invariant charge structures with potential implications for c-functions in holographic contexts.

Abstract

We show that the second order field equations characterizing extremal solutions for spherically symmetric, stationary black holes are in fact implied by a system of first order equations given in terms of a prepotential W. This confirms and generalizes the results in [14]. Moreover we prove that the squared prepotential function shares the same properties of a c-function and that it interpolates between M^2_{ADM} and M^2_{BR}, the parameter of the near-horizon Bertotti-Robinson geometry. When the black holes are solutions of extended supergravities we are able to find an explicit expression for the prepotentials, valid at any radial distance from the horizon, which reproduces all the attractors of the four dimensional N>2 theories. Far from the horizon, however, for N-even our ansatz poses a constraint on one of the U-duality invariants for the non-BPS solutions with Z \neq 0. We discuss a possible extension of our considerations to the non extremal case.