Attractors and the Holomorphic Anomaly

TL;DR

This paper explores the relationship between black hole attractor equations, topological string theory, and the holomorphic anomaly, proposing a new interpretation of the topological string partition function as a wave-function in a specific polarization.

Contribution

It introduces a novel interpretation of the topological string partition function as a wave-function and derives a generalized formula linking black hole microstates to topological strings across backgrounds.

Findings

Background dependence due to the holomorphic anomaly is explained by polarization choice.

A real polarization removes background dependence in attractor equations.

A generalized formula relates black hole microstates to topological strings for all backgrounds.

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.