  Open topological string amplitudes on compact Calabi-Yau threefolds are shown
to satisfy an extension of the holomorphic anomaly equation of Bershadsky,
Cecotti, Ooguri and Vafa. The total topological charge of the D-brane
configuration must vanish in order to satisfy tadpole cancellation. The
boundary state of such D-branes is holomorphically captured by a Hodge
theoretic normal function. Its Griffiths' infinitesimal invariant is the
analogue of the closed string Yukawa coupling and plays the role of the
terminator in a Feynman diagram expansion for the topological string with
D-branes. The holomorphic anomaly equation is solved and the holomorphic
ambiguity is fixed for some representative worldsheets of low genus and with
few boundaries on the real quintic.
