Calabi-Yau Black Holes

TL;DR

The paper addresses the macroscopic entropy of $N=2$ extremal black holes in Calabi–Yau moduli space with a cubic prepotential $F= d_{ABC}{X^A X^B X^C\over X^0}$, showing that the horizon entropy can be expressed in terms of charge–intersection-number combinations through stabilization equations. The main result is a general entropy formula $ {S\over \pi } = {1\over 3p^0} \sqrt{ {4\over 3}{(\Delta _A {\tilde{x}}^A)}^2 - 9\bigl(p^0(p\cdot q )- 2 D\bigr)^2 }$ with $\Delta_A = 3D_A - p^0 q_A$, $D_A = d_{ABC} p^B p^C$, $D = d_{ABC} p^A p^B p^C$, and $p\cdot q=p^0 q_0+p^A q_A$, where ${ (\Delta _A {\tilde{x}}^A) }^2$ is fixed by the quadratic system $d_{ABI}{\tilde{x}}^A {\tilde{x}}^B = \Delta _I$; special linearization occurs for $p^0=0$. The work provides explicit fixed-moduli solutions and multiple CY examples (including STU and other $F_{II}$ families) that yield closed-form entropy expressions and connect to known microscopic results. A topological correction via $F_1$ shifts charges to ${\tilde{q}}_\Sigma$, refining the entropy expressions. Overall, the paper offers a practical framework to compute $N=2$ CY black hole entropy across diverse CY geometries by solving a finite algebraic system tied to the intersection data $d_{ABC}$.

Abstract

We have found the entropy of N=2 extreme black holes associated with general Calabi-Yau moduli space. We show that for arbitrary d_{ABC} and black hole charges the entropy-area formula depends on combinations of these charges and parameters d_{ABC}. These combinations are the solutions of the simple system of algebraic equations. We gave a few examples of particular Calabi-Yau moduli space for which this system has an explicit solution. For special case when one of black hole charges is equal to zero (p^0=0) the solution always exists.