Attractors and first order formalism in five dimensions revisited

Abstract: The attractor mechanism in five dimensional Einstein-Maxwell Chern-Simons theory is studied. The expression of the five dimensional rotating black object potential depending on Taub-NUT, electric and magnetic charges as well as on all the scalar and gauge fields, is investigated. The first order formalism in d=5 is constructed and analyzed. We derive a general expression defining the fake superpotential which is valid for all charge configurations. An explicit expression for the fake superpotential is constructed, for all very special geometries, in the case of vanishing Taub-NUT charge. We carry out an analogous construction in the very special geometries corresponding to $t^3$ and $stu$ models, for the most general charge configurations. The attractor flows and horizon values of all fields are given.

• The paper’s main contribution is the explicit construction of a five-dimensional black object potential and an effective first order action, capturing both BPS and non-BPS attractor flows.
• It applies a fake superpotential approach to derive flow equations that accommodate nontrivial Taub-NUT charges and axion configurations within very special geometry.
• It clarifies the 4D/5D correspondence by mapping duality invariants and moduli space geometries, enhancing the analysis of extremal black objects and entropy computations.

Attractors and First Order Formalism in Five-Dimensional Supergravities

Introduction and Motivation

The paper addresses the attractor mechanism within five-dimensional Einstein-Maxwell Chern-Simons supergravity, focusing on the structure of extremal black objects and the development of a first order formalism. The analysis extends general understanding by including solutions with non-trivial Taub-NUT, electric, and magnetic charges and arbitrary axion configurations, thereby generalizing previously known results restricted to more special configurations or truncated field content. The interplay between the geometry of the moduli space—in particular "very special geometry"—and black hole physics is systematically explored, emphasizing the relation with the corresponding four-dimensional theories arising from dimensional reduction.

Black Object Potentials and Attractor Mechanism

A central contribution is the explicit construction of the five-dimensional black object potential, $V_5$, incorporating the complete set of charges and scalar and gauge degrees of freedom. The potential is expressed as a sum of squares of "real central charges," extending the potential-based framework standard in four dimensions to configurations with less supersymmetry and more intricate charge assignments. A key result is the establishment of a consistent one-dimensional effective action for stationary black objects, from which all equations of motion can be derived. The formalism naturally incorporates the effects of the Taub-NUT charge, which has nontrivial implications for near-horizon geometry and the degeneracy structure of the attractor points.

The attractor equations demonstrated in the paper cover both BPS and non-BPS configurations, and the algebraic characterization via invariants such as the cubic ($I_3$) and quartic ($I_4$) charge combinations is systematically elucidated. The paper reinforces that, for all physical black object solutions, attractor behaviour persists—i.e., horizon values of scalars depend only on the conserved charges—not on their asymptotic boundary values.

First Order (Fake Superpotential) Formalism

A significant technical outcome is the formulation of first order flow equations in five dimensions, paralleling the Hamilton-Jacobi approach in four-dimensional supergravity but adapted to the constraints of five-dimensional "very special" geometry. The black object potential is shown to be reducible to a function $W$ (the "fake superpotential"), where $V_5 = W^2$ plus gradient and geometric terms, for all charge configurations. The explicit form of $W$ is derived for a general class of symmetric and non-symmetric geometries, as well as for extremal non-BPS flows—unlike the standard central charge-based flow for BPS solutions.

In particular, for vanishing Taub-NUT charge, the authors obtain a closed formula for $W$ valid for all "very special" cubic geometries, encompassing cases previously treated in isolation or with restrictive assumptions. The non-uniqueness of $W$ in the non-BPS case is discussed, and the relation between these "fake superpotentials" and duality invariants is explicitly mapped out.

4D/5D Correspondence and Group Theoretic Structure

The relationship between the five-dimensional formalism and its four-dimensional counterpart is clarified, especially via Kaluza-Klein reduction. The scalar geometry in five dimensions, constrained by the cubic “very special” condition, is mapped to special Kähler geometry in four dimensions. The analysis includes explicit expressions for the metric, couplings, and potentials after dimensional reduction, and the transformations of topological (Taub-NUT) and dynamical charges under this map.

A detailed account of the embedding of the five-dimensional duality group into the four-dimensional one is provided, including the mapping of the fundamental representations and the relationship between cubic and quartic invariants. For $N=8$ and $N=2$ supergravities, this encompasses the embeddings $E_{6(6)} \to E_{7(7)}$, and the behavior of invariant polynomials under reduction (e.g., $I_4 \sim p^0 I_3$). The formalism is shown to preserve duality invariance properties and to provide a systematic procedure for constructing all relevant invariants using either "long" charges (including axion contributions) or central charge variables.

Explicit Models and Charge Configurations

The formalism is instantiated via several exemplary models. The $t^3$ model (pure supergravity) and the $stu$ model (with three vector multiplets) are analyzed in detail, with the explicit form of the black object potential, attractor flows, horizon data, and superpotentials, both for BPS and non-BPS branches. Strong algebraic results are obtained, e.g., explicit entropy formulas in terms of charge invariants:

• For the non-BPS branch in the $t^3$ and $stu$ models, the entropy is given in terms of critical points of the quartic invariant $I_4(p, q)$, and the analytic form for the attractor values of all scalars is provided.
• For cases with axion truncation, specific charge configurations (purely electric, purely magnetic, and D0-D6) are considered, with exact expressions for horizon quantities and entropies, highlighting when reduced potentials and superpotentials are sufficient to capture the attractor dynamics.

All analysis is performed for general values of the Taub-NUT charge, and the authors give careful attention to the physical distinctness of "topological" and "dynamical" charges in five-dimensional versus four-dimensional settings.

Implications and Future Directions

The paper establishes a unified formalism for the analysis of attractor flows in five-dimensional supergravity, applicable to all known classes of extremal black objects and robust against axion and charge deformations. It clarifies the role of superpotentials in encoding duality invariant information and enables computation of all attractor data from first principles for arbitrary charge configurations.

From a practical standpoint, the framework significantly streamlines the computation of black object entropies and moduli flows in five-dimensional supergravity, particularly in the presence of nontrivial Taub-NUT structures and rotations. The explicit connection with four-dimensional results supports the classification of higher-dimensional solutions via their four-dimensional progeny.

Theoretically, the work opens the possibility of systematic studies of multicenter and composite configurations, walls of marginal stability, and nontrivial topological backgrounds. The established formalism is expected to play a central role in the analysis of microstate geometries and the interplay between duality, string compactification, and black hole thermodynamics.

Conclusion

The paper presents a comprehensive analysis and extension of the attractor and first order formalism for five-dimensional Einstein-Maxwell Chern-Simons supergravity. Key achievements include the general construction of the black object potential for arbitrary charges and axionic backgrounds, formulation of the universal first order equations and superpotentials, explicit computation of attractor flows in $t^3$ and $stu$ models, and a clarified understanding of the 4D/5D correspondence under duality and dimensional reduction. The results both unify and generalize previous approaches, offering new tools for the analysis of extremal black objects and their moduli spaces in higher-dimensional gravity theories.

Reference: "Attractors and first order formalism in five dimensions revisited" (1010.3516).