Type IIB Orientifolds, F-theory, Type I Strings on Orbifolds and Type I - Heterotic Duality

TL;DR

The work analyzes Type IIB orientifolds on orbifolded K3 and Calabi–Yau three-folds in 6D and 4D, assessing when perturbative world-sheet methods suffice and when non-perturbative sectors are essential. It employs the webs of dualities—F-theory, Type I–heterotic duality, and explicit M/F-theory geometry—to map orientifold vacua to non-perturbative regimes and identify when D-branes on collapsed cycles generate missing states. The main results show a narrow set of 4D models admitting a purely perturbative description, while many others require non-perturbative content that resolves tadpole puzzles and anomalous spectra; these can be understood via Voisin–Borcea orbifolds and their F-theory duals. The findings illuminate how non-perturbative heterotic vacua arise as Type I/II orientifolds duals and demonstrate the crucial role of F-theory in capturing states invisible to world-sheet techniques, with implications for constructing consistent chiral ${\cal N}=1$ vacua in four dimensions.

Abstract

We consider six and four dimensional ${\cal N}=1$ supersymmetric orientifolds of Type IIB compactified on orbifolds. We give the conditions under which the perturbative world-sheet orientifold approach is adequate, and list the four dimensional ${\cal N}=1$ orientifolds (which are rather constrained) that satisfy these conditions. We argue that in most cases orientifolds contain non-perturbative sectors that are missing in the world-sheet approach. These non-perturbative sectors can be thought of as arising from D-branes wrapping various collapsed 2-cycles in the orbifold. Using these observations, we explain certain ``puzzles'' in the literature on four dimensional orientifolds. In particular, in some four dimensional orientifolds the ``naive'' tadpole cancellation conditions have no solution. However, these tadpole cancellation conditions are derived using the world-sheet approach which we argue to be inadequate in these cases due to appearance of additional non-perturbative sectors. The main tools in our analyses are the map between F-theory and orientifold vacua and Type I-heterotic duality. Utilizing the consistency conditions we have found in this paper, we discuss consistent four dimensional chiral ${\cal N}=1$ Type I vacua which are non-perturbative from the heterotic viewpoint.

Figures (2)

• Figure 1: The relations between Type IIB orientifolds, Type I, heterotic and F-theory.
• Figure 2: Open circles and dots represent the original Voisin--Borcea orbifolds. The line of $\otimes$'s corresponds to the extension discussed in section VIII.