[Paper Review] Exact Solution and Attractor Mechanism for Non-BPS Black Holes: the D2-D6 System

Rong-Gen Cai, Da-Wei Pang|arXiv (Cornell University)|Dec 3, 2007

Black Holes and Theoretical Physics3 citations

TL;DR

This paper presents an exact, spherically symmetric solution for extremal non-BPS black holes in 4D supergravity carrying D2-D6 charges with general complex moduli. By assuming a solution form and verifying it satisfies the equations of motion and constraints, the authors derive a configuration manifestly dual to a previously known solution, establishing a new realization of the attractor mechanism in non-BPS systems.

ABSTRACT

The extremal, spherically symmetric black hole solution to 4D supergravity is obtained, which is a non-BPS configuration carrying D2-D6 charges with general complex moduli. Rather than solving the equations of motion directly, we assume the form of the solutions and then find that the assumption satisfies the equations of motion and the constraint. Our solution is manifestly dual to the solution presented in arXiv:0710.4967.

Motivation & Objective

• To construct an exact solution for extremal, spherically symmetric non-BPS black holes in 4D supergravity with D2-D6 charges.
• To incorporate general complex moduli in the black hole solution, extending beyond special cases.
• To demonstrate that the solution satisfies the equations of motion and constraints without direct solving.
• To establish manifest duality between the new solution and the one in arXiv:0710.4967.
• To explore the attractor mechanism in non-BPS black hole configurations with non-trivial moduli.

Proposed method

• Assumes a specific functional form for the metric and gauge field configurations based on spherical symmetry and charge structure.
• Imposes the ansatz on the equations of motion and constraints of 4D supergravity to verify consistency.
• Uses the structure of the D2-D6 charge system to fix the dependence of fields on radial coordinates.
• Relies on the fact that the ansatz satisfies all dynamical and constraint equations, confirming its validity.
• Demonstrates that the solution is manifestly dual to the one in arXiv:0710.4967 through field redefinitions and symmetry transformations.
• Analyzes the attractor mechanism by examining the fixed-point behavior of moduli at the horizon.

Experimental results

Research questions

• RQ1 Can an exact, extremal non-BPS black hole solution with D2-D6 charges and general complex moduli be constructed in 4D supergravity?
• RQ2 Does the assumed solution form satisfy the equations of motion and constraints of the theory?
• RQ3 How does the attractor mechanism manifest in this non-BPS configuration with non-trivial moduli?
• RQ4 What is the duality relation between this solution and the one presented in arXiv:0710.4967?
• RQ5 Can the attractor mechanism be realized in non-BPS black holes without requiring special moduli values?

Key findings

• The proposed ansatz yields a consistent solution that satisfies all equations of motion and constraints in 4D supergravity.
• The solution describes an extremal, spherically symmetric non-BPS black hole with D2-D6 charges and arbitrary complex moduli.
• The horizon moduli are fixed by the attractor mechanism, even in the non-BPS case with general complex structure moduli.
• The solution is manifestly dual to the configuration in arXiv:0710.4967, confirming its consistency within the duality web.
• The attractor mechanism operates in the non-BPS sector, fixing moduli at the horizon despite the absence of supersymmetry.
• The solution provides a concrete realization of non-BPS attractors in the D2-D6 system with full moduli dependence.