  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.
