Mirror Fermat Calabi-Yau Threefolds and Landau-Ginzburg Black Hole Attractors

TL;DR

The paper analyzes extremal black hole attractors in 4D N=2 supergravity with a single complex-structure modulus, focusing on the mirror Fermat CY_3s near the Landau-Ginzburg point. It develops two complementary approaches—the attractor equations from the effective BH potential V_BH and the holomorphic SKG identities—to determine BH charge configurations that stabilize the LG point, uncovering a universal presence of a stable 1/2-BPS attractor and a stable non-BPS Z ≠ 0 attractor (when Z ≠ 0), with non-BPS Z = 0 attractors typically unstable. A detailed, case-by-case analysis for k = 5, 6, 8, 10 (mirror quintic, sextic, octic, dectic) reveals how the LG attractor structure depends on the Fermat parameter and the charge tuning, including explicit entropy expressions and charge configurations. The work connects LG-limit Yukawa couplings, PF equations, and SKG data, and contrasts the LG attractor landscape with the large-volume, cubic prepotential regime, highlighting both similarities and crucial differences in stability and attractor content. The results deepen understanding of attractor mechanisms in 1-modulus SK geometries and provide exact holomorphic Yukawa data for Fermat CY_3s in the LG limit, with implications for microscopic interpretations of BH entropy.

Abstract

We study black hole attractor equations for one-(complex structure)modulus Calabi-Yau spaces which are the mirror dual of Fermat Calabi-Yau threefolds (CY_{3}s). When exploring non-degenerate solutions near the Landau-Ginzburg point of the moduli space of such 4-dimensional compactifications, we always find two species of extremal black hole attractors, depending on the choice of the Sp(4,Z) symplectic charge vector, one 1/2-BPS (which is always stable, according to general results of special Kahler geometry) and one non-BPS. The latter turns out to be stable (local minimum of the ``effective black hole potential'' V_{BH}) for non-vanishing central charge, whereas it is unstable (saddle point of V_{BH}) for the case of vanishing central charge. This is to be compared to the large volume limit of one-modulus CY_{3}-compactifications (of Type II A superstrings), in which the homogeneous symmetric special Kahler geometry based on cubic prepotential admits (beside the 1/2-BPS ones) only non-BPS extremal black hole attractors with non-vanishing central charge, which are always stable.