Attractors with Large Complex Structure for One-Parameter Families of Calabi-Yau Manifolds

TL;DR

This paper develops a novel iterative method to solve attractor equations for one-parameter Calabi-Yau families in the large complex structure limit, incorporating instanton effects and providing explicit entropy formulas.

Contribution

It introduces the N-expansion solution for attractor equations, including instanton contributions, and derives closed-form expressions for entropy in specific cases.

Findings

First generic solutions with instanton effects for attractor equations.

Closed-form formula for entropy at rank two attractor points.

Inclusion of all genus 0 instanton corrections in black hole entropy.

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.