On the Existence of Non-Supersymmetric Black Hole Attractors for Two-Parameter Calabi-Yau's and Attractor Equations

TL;DR

This paper investigates the existence and nature of non-supersymmetric black hole attractor solutions in type II string compactifications on a specific Calabi-Yau threefold, revealing geometric and algebraic structures influencing attractor behavior.

Contribution

It identifies geometric structures associated with non-supersymmetric attractors and analyzes attractor equations near and away from the conifold locus in a detailed Calabi-Yau setting.

Findings

Attractors linked to elliptic curves away from conifold locus.

Near conifold locus, attractors relate to A_1-singularity structures.

Attractor equations impose constraints on moduli and flux components.

Abstract

We look for possible nonsupersymmetric black hole attractor solutions for type II compactification on (the mirror of) CY_3(2,128) expressed as a degree-12 hypersurface in WCP^4[1,1,2,2,6]. In the process, (a) for points away from the conifold locus, we show that the attractors could be connected to an elliptic curve fibered over C^8 which may also be "arithmetic" (in some cases, it is possible to interpret the extremization conditions as an endomorphism involving complex multiplication of an arithmetic elliptic curve), and (b) for points near the conifold locus, we show that the attractors correspond to a version of A_1-singularity in the space Image(Z^6-->R^2/Z_2(embedded in R^3)) fibered over the complex structure moduli space. The potential can be thought of as a real (integer) projection in a suitable coordinate patch of the Veronese map: CP^5-->CP^{20}, fibered over the complex structure moduli space. We also discuss application of the equivalent Kallosh's attractor equations for nonsupersymmetric attractors and show that (a) for points away from the conifold locus, the attractor equations demand that the attractor solutions be independent of one of the two complex structure moduli, and (b) for points near the conifold locus, the attractor equations imply switching off of one of the six components of the fluxes. Both these features are more obvious using the atractor equations than the extremization of the black hole potential.