Extended Holomorphic Anomaly and Loop Amplitudes in Open Topological String

TL;DR

The paper extends BCOV's holomorphic anomaly to open topological strings by incorporating D-brane boundary data through normal functions and Griffiths' infinitesimal invariant, establishing extended anomaly equations that include boundary degenerations. It develops a B-model framework combining tt* geometry, normal functions, and disk/annulus contributions, and demonstrates the formalism on the real quintic by solving low-genus open+closed amplitudes and fixing holomorphic ambiguities via large-volume, orbifold, and conifold expansions. The work also connects disk two-point data to the infinitesimal invariant and discusses open-string integrality conjectures, offering a systematic method to compute open+closed amplitudes on compact Calabi-Yau manifolds with D-branes. Together, these results illuminate the role of D-branes as boundary data in the vacuum bundle and pave the way for a full open-sector extension of topological string theory.

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.

Figures (6)

• Figure 1: Left: The disk amplitude with $n+1$ boundary insertions captures the higher $A_\infty$ products \ref{['mn']}. Right: The disk amplitude with one bulk and one boundary insertion encodes the bulk to boundary obstruction map \ref{['obmap']}, and will also define the infinitesimal Abel-Jacobi map \ref{['infaj']}.
• Figure 2: The two degenerations that contribute to the RHS of the holomorphic anomaly for closed strings.
• Figure 3: The path-integral on the disk with a long strip attached defines a canonical open string vacuum.
• Figure 4: The disk amplitude with two bulk insertions.
• Figure 5: When adding boundaries, the two degenerations from Fig. \ref{['degenbcov']} remain unaffected.