Attractors and the Holomorphic Anomaly

TL;DR

This paper clarifies how attractor equations interface with the holomorphic anomaly by casting the topological string partition function as a wave function on H^3(M,R) and showing that a background-independent real polarization removes the anomaly. It develops complex and real quantizations (Kähler and real polarization), identifies Psi_top as a background-independent state, and derives a generalized, background-independent relation between black-hole state counting and topological strings. The results offer a non-perturbative perspective on topological strings, suggesting a density-matrix formulation that integrates both topological and anti-topological sectors and remains robust under moduli deformations. The framework provides a concrete route to compute microscopic black hole degeneracies across backgrounds and clarifies the non-perturbative structure of the OSV conjecture.

Abstract

Motivated by the recently proposed connection between N=2 BPS black holes and topological strings, I study the attractor equations and their interplay with the holomorphic anomaly equation. The topological string partition function is interpreted as a wave-function obtained by quantizing the real cohomology of the Calabi-Yau. In this interpretation the apparent background dependence due to the holomorphic anomaly is caused by the choice of complex polarization. The black hole attractor equations express the moduli in terms of the electric and magnetic charges, and lead to a real polarization in which the background dependence disappears. Our analysis results in a generalized formula for the relation between the microscopic density of black hole states and topological strings valid for all backgrounds.