Extended Holomorphic Anomaly and Loop Amplitudes in Open Topological String

TL;DR

This paper extends the holomorphic anomaly equation to open topological strings on Calabi-Yau threefolds, linking D-brane boundary states to Hodge theory and solving for low-genus amplitudes.

Contribution

It introduces an extended anomaly equation for open strings, connecting D-brane charges, Hodge theory, and Feynman diagram expansions, with explicit solutions for low-genus cases.

Findings

Extended holomorphic anomaly equation for open topological strings.

Holomorphic ambiguity fixed for low-genus, few-boundary cases.

Boundary states described by Hodge-theoretic normal functions.

Abstract

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.