Abstract
The attractor equations for an arbitrary one-parameter family of Calabi-Yau manifolds are studied in the large complex structure region. These equations are solved iteratively, generating what we term an N-expansion, which is a power series in the Gromov-Witten invariants of the manifold. The coefficients of this series are associated with integer partitions. In important cases we are able to find closed-form expressions for the general term of this expansion. To our knowledge, these are the first generic solutions to attractor equations that incorporate instanton contributions. In particular, we find a simple closed-form formula for the entropy associated to rank two attractor points, including those recently discovered. The applications of our solutions are briefly discussed. Most importantly, we are able to give an expression for the Wald entropy of black holes that includes all genus 0 instanton corrections.
Highlights
Ever since the discovery of the attractor mechanism [1], the study of supersymmetric black holes in N = 2 theories in four dimensions has been intimately linked to the study of attractor points on Calabi-Yau moduli spaces
While the attractor mechanism applies to any four-dimensional N = 2 supergravity theory coupled to vector multiplets, an especially interesting class of examples is provided by compactifying 10-dimensional type IIA and IIB supergravities on Calabi-Yau threefolds
In appendix B, we examine the role of the monodromy transformations around the large complex structure limit, showing that these can either be interpreted as a symmetry of the solutions or used to extend our solutions to cases where the D4 brane charge divides the D6 charge
Summary
In the case of Calabi-Yau compactifications, the attractor equations make it possible to find the complex or complexified Kähler structure of the internal manifold, and can be used to compute the central charge (1.12) of the supergravity theory, which in turn is related to the Bekenstein-Hawking entropy of the black hole associated to the attractor point. Solutions for the attractor equations in the approximation where the instanton corrections to the prepotential are ignored, but the genus 1 perturbative contribution is included, were found in [10, 23] and used to give an approximate formula for the Bekenstein-Hawking entropy in any Calabi-Yau compactification, where the internal manifold has a large complex structure. Identities like this were already anticipated by Moore [3]
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