Orientifolds of Gepner Models

TL;DR

The paper builds Type II orientifolds of Gepner models with ${ m N}=1$ in 4d by classifying parity symmetries, constructing crosscap/boundary states, and solving tadpole/ SUSY conditions. It develops a supersymmetric, worldsheet-based framework (in the NSR formalism) to enumerate D-brane configurations and spectra at the Gepner point, including both A- and B-type setups, and demonstrates that the B-type sector can yield large numbers of non-chiral or chiral-free vacua, with explicit counts in the quintic and a two-parameter model. The work establishes a precise match between Gepner-model data and large-volume geometric data, clarifying how moduli spaces, O-plane types, and RR charges interpolate across non-geometric phases, and highlights how Fayet–Iliopoulos terms and anomaly cancellation constraints shape the vacuum structure. It also proves a vanishing-index theorem for certain B-brane pairs in Type IIB orientifolds, constraining the occurrence of chiral, supersymmetric vacua within this class. Overall, the paper provides a comprehensive, exact-solution framework for orientifolds of Gepner models, linking worldsheet RCFT techniques to geometric and phenomenological aspects of Calabi–Yau compactifications.

Abstract

We systematically construct and study Type II Orientifolds based on Gepner models which have N=1 supersymmetry in 3+1 dimensions. We classify the parity symmetries and construct the crosscap states. We write down the conditions that a configuration of rational branes must satisfy for consistency (tadpole cancellation and rank constraints) and spacetime supersymmetry. For certain cases, including Type IIB orientifolds of the quintic and a two parameter model, one can find all solutions in this class. Depending on the parity, the number of vacua can be large, of the order of 10^{10}-10^{13}. For other models, it is hard to find all solutions but special solutions can be found -- some of them are chiral. We also make comparison with the large volume regime and obtain a perfect match. Through this study, we find a number of new features of Type II orientifolds, including the structure of moduli space and the change in the type of O-planes under navigation through non-geometric phases.

Figures (17)

• Figure 1: Kähler moduli space for a B-orientifold of the quintic
• Figure 2: Real section of the Kähler moduli space of the two parameter model $(k_i+2)=(8,8,4,4,4)$. The Gepner point $g$ is at the tip of a conical singularity. The left cone is the moduli space of the orientifold without dressing by quantum symmetry. The lines of singularity $C_1$ and $C_{\rm con}$ divide the moduli space into several perturbative regions. The right cone (shaded region), which reaches out all the way to the large volume regime, is the moduli space of the orientifold with dressing by quantum symmetry.
• Figure 3: To see the large volume limit in the compactified moduli space, we have to blowup the singular point $b=C_1\cap C_\infty$. We replace twice a small neighborhood of the origin with a Möbius strip, successively inserting the divisors $D_{(-1,-1)}$ and $D_{(0,-1)}$. The multiply stroked lines are identified. The shaded region can be reached smoothly from the Gepner point.
• Figure 4: Two possibilities of complexifying the real moduli space with a codimension one singularity. (a) One can go around the singularity and the two parts are smoothly connected. (b) The moduli space consists of branches. Any path from one region to the other must go through the singular point. Third possibility (not shown in Figure) would be that the two regions are disconnected.
• Figure 5: The O-plane corresponding to $|\Scr{C}_{2m-1,-1}(+)\rangle$. This is the example with $k+2=8$ and $m=2$.