Background Independence and the Open Topological String Wavefunction

TL;DR

This paper explores how the open topological string partition function encodes a background-independent state in the Hilbert space, emphasizing the role of the extended holomorphic anomaly equation and D-branes in this framework.

Contribution

It demonstrates the background independence of the open topological string wavefunction via the extended holomorphic anomaly equation and proposes that D-branes form a basis of the Hilbert space.

Findings

Partition function specifies a state in the Hilbert space.

Extended holomorphic anomaly equation encodes background independence.

D-branes potentially form a basis of the Hilbert space.

Abstract

The open topological string partition function in the background of a D-brane on a Calabi-Yau threefold specifies a state in the Hilbert space associated with the quantization of the underlying special geometry. This statement is a consequence of the extended holomorphic anomaly equation after an appropriate shift of the closed string variables, and can be viewed as the expression of background independence for the open-closed topological string. We also clarify various other aspects of the structure of the extended holomorphic anomaly equation. We conjecture that the collection of all D-branes furnishes a basis of the Hilbert space, and revisit the BPS interpretation of the open topological string wavefunction in this light.