Crosscaps in Gepner Models and Type IIA Orientifolds

TL;DR

This work constructs crosscap states for Type IIA orientifolds of Gepner models by coupling Calabi–Yau orientifold data to RCFT methods, and identifies a dual M-theory picture on barely $G_2$ Joyce manifolds. It develops a comprehensive RCFT framework to realize antiholomorphic involutions at Gepner points, computes Klein bottle amplitudes for the $(k=1)^3$ toy model and the $(k=3)^5$ quintic, and demonstrates consistency with geometric and spacetime analyses. The treatment hinges on spectral-flow invariant orbits, simple-current techniques, and beta-projections, yielding explicit crosscaps and tadpole structures that motivate the inclusion of open-string sectors for RR-tadpole cancellation. The results validate the RCFT approach for orientifolds of Gepner models and set the stage for exploring even levels, B-type crosscaps, and D-brane spectra in future work, with potential insights for M-theory duals and phenomenology.

Abstract

As a first step to a detailed study of orientifolds of Gepner models associated with Calabi-Yau manifolds, we construct crosscap states associated with anti-holomorphic involutions (with fixed points) of Calabi-Yau manifolds. We argue that these orientifolds are dual to M-theory compactifications on (singular) seven-manifolds with $G_2$ holonomy. Using the spacetime picture as well as the M-theory dual, we discuss aspects of the orientifold that should be obtained in the Gepner model. This is illustrated for the case of the quintic.