Counting Higher Genus Curves with Crosscaps in Calabi-Yau Orientifolds

TL;DR

This paper computes topological string amplitudes on Calabi-Yau orientifolds, counting non-orientable surfaces like Klein bottles and projective planes with handles, using advanced geometric and algebraic techniques.

Contribution

It introduces a comprehensive method to count higher genus non-orientable curves in Calabi-Yau orientifolds using geometric transitions and localization techniques.

Findings

Calculated all loop topological string amplitudes on orientifolds.

Counted Klein bottles and projective planes with handles.

Developed a unified approach combining geometric transitions and localization.

Abstract

We compute all loop topological string amplitudes on orientifolds of local Calabi-Yau manifolds, by using geometric transitions involving SO/Sp Chern-Simons theory, localization on the moduli space of holomorphic maps with involution, and the topological vertex. In particular we count Klein bottles and projective planes with any number of handles in some Calabi-Yau orientifolds.