The Real Topological String on a local Calabi-Yau

TL;DR

This paper investigates the real topological string on local P2 with orientifold planes and D-branes, employing three methods to compute invariants, reproduce amplitudes, and analyze the conifold expansion, revealing a new universality class.

Contribution

It introduces a refined localization technique for open and unoriented Gromov-Witten invariants, develops a real topological vertex formalism, and connects A- and B-model results through the extended holomorphic anomaly.

Findings

Computed open and unoriented Gromov-Witten invariants for toric Calabi-Yau with involution.

Reproduced topological string amplitudes using the real topological vertex.

Identified a new universality class in the conifold expansion.

Abstract

We study the topological string on local P2 with O-plane and D-brane at its real locus, using three complementary techniques. In the A-model, we refine localization on the moduli space of maps with respect to the torus action preserved by the anti-holomorphic involution. This leads to a computation of open and unoriented Gromov-Witten invariants that can be applied to any toric Calabi-Yau with involution. We then show that the full topological string amplitudes can be reproduced within the topological vertex formalism. We obtain the real topological vertex with trivial fixed leg. Finally, we verify that the same results derive in the B-model from the extended holomorphic anomaly equation, together with appropriate boundary conditions. The expansion at the conifold exhibits a gap structure that belongs to a so far unidentified universality class.