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Type IIA Orientifolds on General Supersymmetric Z_N Orbifolds

Ralph Blumenhagen, Joseph P. Conlon, Kerim Suruliz

TL;DR

The paper develops a general, crystallography-based framework for Type IIA Omega R orientifolds on non-factorisable Z_N orbifolds, enabling the calculation of one-loop amplitudes and tadpole conditions on lattices where T^6 does not factorise into T^2 × T^2 × T^2. It then applies these tools to a broad set of non-factorisable cases (Z7, Z8, Z8', Z12, Z12', Z4), deriving lattice sums, fixed-point data, and brane configurations that produce consistent, supersymmetric vacua with explicit gauge groups and massless spectra. The work extends known factorisable models, demonstrates sector-by-sector and global tadpole cancellation in non-trivial fixed-torus settings, and yields a suite of explicit orientifold models with D6-branes on O-planes ready for potential phenomenological applications and connections to M-theory on G2 manifolds. Overall, it broadens the landscape of calculable Type II orientifolds and provides practical methods for constructing chiral, supersymmetric vacua beyond factorisable geometries.

Abstract

We construct Type IIA orientifolds for general supersymmetric Z_N orbifolds. In particular, we provide the methods to deal with the non-factorisable six-dimensional tori for the cases Z7, Z8, Z8', Z12 and Z12'. As an application of these methods we explicitly construct many new orientifold models.

Type IIA Orientifolds on General Supersymmetric Z_N Orbifolds

TL;DR

The paper develops a general, crystallography-based framework for Type IIA Omega R orientifolds on non-factorisable Z_N orbifolds, enabling the calculation of one-loop amplitudes and tadpole conditions on lattices where T^6 does not factorise into T^2 × T^2 × T^2. It then applies these tools to a broad set of non-factorisable cases (Z7, Z8, Z8', Z12, Z12', Z4), deriving lattice sums, fixed-point data, and brane configurations that produce consistent, supersymmetric vacua with explicit gauge groups and massless spectra. The work extends known factorisable models, demonstrates sector-by-sector and global tadpole cancellation in non-trivial fixed-torus settings, and yields a suite of explicit orientifold models with D6-branes on O-planes ready for potential phenomenological applications and connections to M-theory on G2 manifolds. Overall, it broadens the landscape of calculable Type II orientifolds and provides practical methods for constructing chiral, supersymmetric vacua beyond factorisable geometries.

Abstract

We construct Type IIA orientifolds for general supersymmetric Z_N orbifolds. In particular, we provide the methods to deal with the non-factorisable six-dimensional tori for the cases Z7, Z8, Z8', Z12 and Z12'. As an application of these methods we explicitly construct many new orientifold models.

Paper Structure

This paper contains 18 sections, 112 equations, 7 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Type IIA orientifolds on non-factorisable orbifolds
  3. Definition of $\Omega R$ orientifolds
  4. Dealing with non-factorisable lattices
  5. The massless spectrum
  6. The closed string spectrum
  7. The open string spectrum
  8. Examples of factorisable orientifolds revisited
  9. Examples of non-factorisable orientifolds
  10. $\mathbb{Z}_7$ Model : $A_7$ with $v = \frac{1}{7}(1,2,-3)$
  11. $\mathbb{Z}_8^{'}$ Model: $B_2 \times B_4$ with $v = \frac{1}{8}(2,1,-3)$
  12. $\mathbb{Z}_8$ Model: $D_2 \times B_4$ with $v = \frac{1}{8}(-4,1,3)$
  13. $\mathbb{Z}_{12}$ Model: $A_2 \times F_4$ with $v = \frac{1}{12}(4,1,-5)$
  14. $\mathbb{Z}_{12}^{'}$ Model: $D_2 \times F_4$ with $v = \frac{1}{12}(-6,1,5)$
  15. $\mathbb{Z}_{4}$ Model: $A_3 \times A_3$ with $v = \frac{1}{4}(1,-2,1)$
  16. ...and 3 more sections

Figures (7)

  • Figure 1: The $\mathbb{Z}_{12}$ lattice vectors. The arrow marked ${\bf e}_i$ in each plane is not ${\bf e}_i$ itself; rather, it is the component of ${\bf e}_i$ in that plane.
  • Figure 2: The $\mathbb{Z}_7$ lattice
  • Figure 3: The $\mathbb{Z}_7$ branes
  • Figure 4: The $\mathbb{Z}_8^{'}$ lattice
  • Figure 5: The $\mathbb{Z}_8^{'}$ branes
  • ...and 2 more figures