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Comments on Orientifolds without Vector Structure

Constantin Bachas, Massimo Bianchi, Ralph Blumenhagen, Dieter Lust, Timo Weigand

TL;DR

The paper analyzes Type I and heterotic compactifications with Spin(32)/${\bb Z}_2$ bundles lacking vector structure, showing the obstruction is governed by a mod-2 class ${\bf b}\in H^2(X,{\mathbb Z}_2)$ tied to a discrete NS-NS $B$-field and the generalized Stiefel-Whitney class ${\tilde w_2(E)}$. It clarifies the T-duality to Type IIA orientifolds and revisits the 3-generation magnetized D-brane model C, demonstrating consistency when the discrete $B$-flux is included and that the unbroken gauge rank can be preserved. The framework is extended to genuine Calabi-Yau manifolds, with a mirror Type IIA description in which discrete complex-structure data correspond to $B$-flux choices, and the quintic is worked out as a key example. Together these results expand the landscape of consistent string vacua with vector-structure obstructions and open new directions for model building via discrete flux moduli and D-brane instanton effects.

Abstract

We revisit type I compactifications with a Spin(32)/Z2 gauge bundle that admits no vector structure. We elucidate the relation of this Z2 obstruction to discrete B-field flux and to 't Hooft flux and clarify some subtleties in the T-duality transformation to type IIA intersecting D-brane models. We reexamine the earliest 3-generation GUT model on magnetized D-branes and show its consistency when a discrete B-flux is switched on. We further generalize partially known results for toroidal models to type I compactifications without vector structure and their mirror dual type IIA orientifolds on genuine Calabi-Yau manifolds. We illustrate this by working out the example of the quintic in some detail.

Comments on Orientifolds without Vector Structure

TL;DR

The paper analyzes Type I and heterotic compactifications with Spin(32)/ bundles lacking vector structure, showing the obstruction is governed by a mod-2 class tied to a discrete NS-NS -field and the generalized Stiefel-Whitney class . It clarifies the T-duality to Type IIA orientifolds and revisits the 3-generation magnetized D-brane model C, demonstrating consistency when the discrete -flux is included and that the unbroken gauge rank can be preserved. The framework is extended to genuine Calabi-Yau manifolds, with a mirror Type IIA description in which discrete complex-structure data correspond to -flux choices, and the quintic is worked out as a key example. Together these results expand the landscape of consistent string vacua with vector-structure obstructions and open new directions for model building via discrete flux moduli and D-brane instanton effects.

Abstract

We revisit type I compactifications with a Spin(32)/Z2 gauge bundle that admits no vector structure. We elucidate the relation of this Z2 obstruction to discrete B-field flux and to 't Hooft flux and clarify some subtleties in the T-duality transformation to type IIA intersecting D-brane models. We reexamine the earliest 3-generation GUT model on magnetized D-branes and show its consistency when a discrete B-flux is switched on. We further generalize partially known results for toroidal models to type I compactifications without vector structure and their mirror dual type IIA orientifolds on genuine Calabi-Yau manifolds. We illustrate this by working out the example of the quintic in some detail.

Paper Structure

This paper contains 15 sections, 67 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Gauge bundles and orientifolds on $T^2$
  3. Long strings and 't Hooft flux
  4. Relation to the $B$-flux
  5. Including the orientifold
  6. Toroidal 4d orientifolds and the model C
  7. Consistency conditions for $T^6/\Omega{\cal R}$ orientifolds
  8. Three generations and the model C
  9. Euclidean D1-brane instantons
  10. Calabi-Yau compactifications of $Spin(32)/{\mathbb Z}_2$ bundles with or without vector structure
  11. $Spin(32)/{\mathbb Z}_2$ gauge bundles
  12. Smooth Calabi-Yau Type IIA orientifolds
  13. Example: The Quintic
  14. Outlook
  15. Toroidal Example

Figures (3)

  • Figure 1: The left figure shows the T-dual of an abelian bundle $(n,m)=(1,7)$ and the right image the T-dual of a 't Hooft bundle $(n,m)=(5,1)$.
  • Figure 2: Two configurations of D8-branes canceling the tadpole with the branes indicated by the blue arrows.
  • Figure 3: Complex structure moduli space for the mirror quintic ${\cal W}$ 5 in the $\psi$- and the $U$-plane. The blue lines indicate the two discrete branches after the orientifold projection, related to ${\cal B}=0,1/2$ in the mirror dual Type I model.