Multi-Centered First Order Formalism

TL;DR

This paper introduces a unified first order formalism for multi-centered black holes in N=2 supergravity, extending previous models to include non-axisymmetric, rotating, and multi-centered solutions with a comprehensive mathematical framework.

Contribution

It develops a D=3 Cartesian formalism for multi-centered black holes, generalizes first order flow equations, and unifies various known solutions within a broader theoretical context.

Findings

Derived general first order flow equations for non-BPS and almost BPS black holes.

Unified description of static, rotating, single, and multi-centered black hole solutions.

Identified the structure of the almost BPS class as two distinct branches.

Abstract

We propose a first order formalism for multi-centered black holes with flat tree-dimensional base-space, within the stu model of N=2, D=4 ungauged Maxwell-Einstein supergravity. This provides a unified description of first order flows of this universal sector of all models with a symmetric scalar manifold which can be obtained by dimensional reduction from five dimensions. We develop a D=3 Cartesian formalism which suitably extends the definition of central and matter charges, as well as of black hole effective potential and first order "fake" superpotential, in order to deal with not necessarily axisimmetric solutions, and thus with multi-centered and/or (under-)rotating extremal black holes. We derive general first order flow equations for composite non-BPS and almost BPS classes, and we analyze some of their solutions, retrieving various single-centered (static or under-rotating) and multi-centered known systems. As in the t^3 model, the almost BPS class turns out to split into two general branches, and the well known almost BPS system is shown to be a particular solution of the second branch.