Evidence for tadpole cancelation in the topological string

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 Bogomol'nyiPrasad-Sommerfield (BPS) expansion only if the topological charge of the D-brane configuration 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 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.

Real mirror symmetry for one-parameter hypersurfaces

“…As observed in [14], it appears that in order to obtain a satisfactory BPS interpretation for open topological string amplitudes on compact Calabi-Yau manifold, one has to consider an orientifold model and choose a D-brane configuration that cancels the tadpoles. In our models, we naturally choose the orientifold action that we used to define the D-branes, and put exactly one D-brane on top of the orientifold plane.…”

“…As explained in [13,14], the domainwall tensions that we obtained as solutions of the inhomogeneous Picard-Fuchs equation in the previous sections constitute tree-level for the three models X (6) , X (8) , and X (10) .…”

“…For the one-loop amplitudes however, we have a complete proposal [14], generalizing the result of [15]. We can therefore just plug in the tree-level data into this formula, and extract [12,28,14] one-loop BPS invariants for our three one-parameter hypersurfaces.…”

“…We can therefore just plug in the tree-level data into this formula, and extract [12,28,14] one-loop BPS invariants for our three one-parameter hypersurfaces.…”

“…These computations, the details of which we leave for the future, use the extension [13,14] of the BCOV holomorphic anomaly equations [15,16], and yield further numerical predictions as well as additional consistency checks. Again, the most interesting example is X (8) , where the disconnected fixed point set (as orientifold plane)…”