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Orientifolds with branes at angles

Stefan Forste, Gabriele Honecker, Ralph Schreyer

TL;DR

This work constructs four-dimensional, non-supersymmetric Type IIA orientifolds on $T^2\times T^4/{\mathbb Z}_N$ with D6-branes at angles, achieving RR tadpole cancellation while generating chiral open-string spectra. By explicitly computing Klein bottle, annulus, and Möbius strip amplitudes and performing the modular transform to the tree channel, the authors derive tadpole cancellation conditions for ${\mathbb Z}_2$, ${\mathbb Z}_3$, ${\mathbb Z}_4$, and ${\mathbb Z}_6$ orbifolds; these ensure the absence of non-abelian gauge anomalies and determine anomaly-free $U(1)$ factors. They present concrete examples: a ${\mathbb Z}_3$ model with gauge group effectively $SU(3)\times U(1)$ and a non-anomalous $U(1)$ combination, and a ${\mathbb Z}_2$ model yielding $SU(3)^4\times SU(2)^2\times U(1)^{10}$ with several non-anomalous $U(1)$s, though neither realizes a full three-generation Standard Model nor explains the hierarchy. The paper discusses limitations (tachyons, NSNS tadpoles, lack of hierarchy) and prospects for improvement via generalized orientifolds or background fields and backreaction effects.

Abstract

We present supersymmetry breaking four dimensional orientifolds of type IIA strings. The compact space is a torus times a four dimensional orbifold. The orientifold group reflects one direction in each torus. RR tadpoles are cancelled by D6-branes intersecting at angles in the torus and in the orbifold. The angles are chosen such that supersymmetry is broken. The resulting four dimensional theories contain chiral fermions. The tadpole cancellation conditions imply that there are no non-abelian gauge anomalies. The models also contain anomaly-free U(1) factors.

Orientifolds with branes at angles

TL;DR

This work constructs four-dimensional, non-supersymmetric Type IIA orientifolds on with D6-branes at angles, achieving RR tadpole cancellation while generating chiral open-string spectra. By explicitly computing Klein bottle, annulus, and Möbius strip amplitudes and performing the modular transform to the tree channel, the authors derive tadpole cancellation conditions for , , , and orbifolds; these ensure the absence of non-abelian gauge anomalies and determine anomaly-free factors. They present concrete examples: a model with gauge group effectively and a non-anomalous combination, and a model yielding with several non-anomalous s, though neither realizes a full three-generation Standard Model nor explains the hierarchy. The paper discusses limitations (tachyons, NSNS tadpoles, lack of hierarchy) and prospects for improvement via generalized orientifolds or background fields and backreaction effects.

Abstract

We present supersymmetry breaking four dimensional orientifolds of type IIA strings. The compact space is a torus times a four dimensional orbifold. The orientifold group reflects one direction in each torus. RR tadpoles are cancelled by D6-branes intersecting at angles in the torus and in the orbifold. The angles are chosen such that supersymmetry is broken. The resulting four dimensional theories contain chiral fermions. The tadpole cancellation conditions imply that there are no non-abelian gauge anomalies. The models also contain anomaly-free U(1) factors.

Paper Structure

This paper contains 23 sections, 52 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. General Setup
  3. Details of the construction
  4. Amplitudes and tadpole cancellation
  5. The spectrum
  6. The ${\mathbb Z}_3$ and ${\mathbb Z}_2$ case
  7. Models on $T^2\times T^4/{\mathbb Z}_3$
  8. Example 1, ${\mathbb Z}_3$
  9. Models on $T^2\times T^4/{\mathbb Z}_2$
  10. Example 2, ${\mathbb Z}_2$
  11. Conclusions
  12. Computation of one-loop diagrams
  13. Lattice contributions
  14. Computation of tree channel diagrams
  15. Construction of crosscap states
  16. ...and 8 more sections

Figures (5)

  • Figure 1: Orientifold fixed planes for ${\mathbb Z}_4$
  • Figure 2: Possible torus lattices on $T_1$.
  • Figure 3: Possible torus lattices on $T_{2,3}$ for ${\mathbb Z}_4$. The black circles are the fixed points under $\Theta^{1,3}$ and the white circles are the additional fixed points under $\Theta^2$.
  • Figure 4: One stack of $N_a$ D6$_a$-branes with wrapping number $(n_a,m_a) = (3,1)$ intersecting another stack of $N_b$ D6$_b$-branes with wrapping number $(n_b,m_b) = (1,2)$ on $T_1$. The black circles denote the multiple intersection points on the torus.
  • Figure 5: Brane configurations for the examples