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

TL;DR

This work investigates the existence and structure of non-supersymmetric black hole attractors in type IIB compactifications on a two-parameter Calabi–Yau mirror. By analyzing both the extremization of the black hole potential $V$ and Kallosh's nonsupersymmetric attractor equations, the authors uncover a dichotomy: away from the conifold locus, attractors exhibit connections to arithmetic elliptic curves via endomorphisms and Veronese-type projections, imposing a set of nonlinear constraints on fluxes and moduli; near the conifold locus, the attractor geometry aligns with an $A_1$-singularity, enforcing flux-switching conditions. The study combines period data, mass-matrix analysis, and attractor equations to reveal deep links between geometric, arithmetic, and flux-structure aspects of nonsupersymmetric attractors, offering a framework for numerical and further analytical exploration. These insights provide a structured path toward classifying nonsupersymmetric attractors and understanding flux vacua in multi-parameter Calabi–Yau compactifications.

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.