Evidence for Tadpole Cancellation in the Topological String

TL;DR

This paper demonstrates a tadpole cancellation condition in topological string theory on Calabi-Yau threefolds with orientifolds and D-branes, linking charge balance to decoupling of A- and B-models and providing new computational techniques.

Contribution

It introduces a topological tadpole cancellation condition, extending localization methods and the holomorphic anomaly equation to unoriented strings, with implications for enumerative geometry.

Findings

Topological string amplitudes require charge cancellation for BPS expansion.

Localization techniques adapted for orientifolds yield predictions for higher genus curves.

Extended holomorphic anomaly equation for unoriented strings is developed.

Abstract

We study the topological string on compact Calabi-Yau threefolds in the presence of orientifolds and D-branes. In examples, we find that the total topological string amplitude admits a BPS expansion only if the topological charge of the D-brane is equal to that of the orientifold plane. We interpret this as a manifestation of a general tadpole cancellation condition in the topological string that is necessary for decoupling of A- and B-model in loop amplitudes. Our calculations in the A-model involve an adapted version of existing localization techniques, and give predictions for the real enumerative geometry of higher genus curves in Calabi-Yau manifolds. In the B-model, we introduce an extension of the holomorphic anomaly equation to unoriented strings.