String theory

Differential cohomology



An orientifold (Dai-Lin-Polchinski 89, p. 12) is a target spacetime for string sigma-models that combines aspects of 2\mathbb{Z}_2-orbifolds with orientation reversal on the worldsheet, whence the name.

In type II string theory orientifold backgrounds (inducing type I string theory) with 2\mathbb{Z}_2-fixed points – called O-planes (see there for more) – are required for RR-field tadpole cancellation. This is a key consistency condition in particular for intersecting D-brane models used in string phenomenology.

Where generally (higher gauge) fields in physics/string theory are cocycles in (differential) cohomology theory and typically in complex oriented cohomology theory, fields on orientifolds are cocycles in genuinely 2\mathbb{Z}_2-equivariant cohomology and typically in real-oriented cohomology theory. For instance, the B-field, which otherwise is a (twisted) cocycle in (ordinary) differential cohomology, over an orientifold is a cocycle in (twisted) HZR-theory, and the RR-fields, which usually are cocycles in (twisted differential) K-theory, over an orientifold are cocycles in KR-theory (Witten 98).

An explicit model for B-fields for the bosonic string on orientifolds (differential HZR-theory) is given in (Schreiber-Schweigert-Waldorf 05) and examples are analyzed in this context in (Gawedzki-Suszek-Waldorf 08). See also (HMSV 16, HMSV 19).

The claim that for the superstring the B-field is more generally a cocycle with coefficients in the Picard infinity-group of complex K-theory (super line 2-bundles) and a detailed discussion of the orientifold version of this can be found in (Distler-Freed-Moore 09, Distler-Freed-Moore 10) with details in (Freed 12).

The quadratic pairing entering the 11d Chern-Simons theory that governs the RR-field here as a self-dual higher gauge field is given in (DFM 10, def. 6).


Lifts to M-theory F-theory

Lifts of orientifold background from type II string theory to F-theory go back to (Sen 96, Sen 97a).

Lifts of type IIA string theory orientifolds of D6-branes to D-type ADE singularities in M-theory (through the duality between M-theory and type IIA string theory) goes back to (Sen 97b). See at heterotic M-theory on ADE-orbifolds.

A more general scan of possible lifts of type IIA orientifolds to M-theory is indicated in (Hanany-Kol 00, around (3.2)), see (Huerta-Sati-Schreiber 18, Prop. 4.7) for details.

For instance the O4-plane lifts to the MO5-plane.




The concept originates around

Early accounts include

Traditional lecture notes include

Textbook discussion is in

and specifically in the context of intersecting D-brane models with an eye towards string phenomenology in


  • Marcus Berg, Introduction to Orientifolds (pdf, pdf)

The original observation that D-brane charge for orientifolds should be in KR-theory is due to

and was re-amplified in

In terms of KO-theory

Discussion of orbi-orienti-folds in terms of equivariant KO-theory is in

  • N. Quiroz, Bogdan Stefanski, Dirichlet Branes on Orientifolds, Phys.Rev. D66 (2002) 026002 (arXiv:hep-th/0110041)

  • Volker Braun, Bogdan Stefanski, Orientifolds and K-theory, Braun, Volker. “Orientifolds and K-theory.” Progress in String, Field and Particle Theory. Springer, Dordrecht, 2003. 369-372 (arXiv:hep-th/0206158)

  • H. Garcia-Compean, W. Herrera-Suarez, B. A. Itza-Ortiz, O. Loaiza-Brito, D-Branes in Orientifolds and Orbifolds and Kasparov KK-Theory, JHEP 0812:007, 2008 (arXiv:0809.4238)

A definition and study of orientifold bundle gerbes, modeling the B-field background for the bosonic string (differential HZR-theory), is in

see also

An elaborate formalization in terms of differential cohomology in general and twisted differential K-theory in particular that also takes the spinorial degrees of freedom into account is briefly sketched out in

based on stuff like

Details on the computation of string scattering amplitudes in such a background:

Related lecture notes / slides include

A detailed list of examples of KR-theory of orientifolds and their T-duality is in

A formulation of some of the relevant aspects of (bosonic) orientifolds in terms of the differential nonabelian cohomology with coefficients in the 2-group AUT(U(1))AUT(U(1)) coming from the crossed module [U(1) 2][U(1) \to \mathbb{Z}_2] is indicated in

More on this in section 3.3.10 of

Examples and Models

Specifically K3 orientifolds (𝕋 4/G ADE\mathbb{T}^4/G_{ADE}) in type IIB string theory, hence for D9-branes and D5-branes:

Specifically K3 orientifolds (𝕋 4/G ADE\mathbb{T}^4/G_{ADE}) in type IIA string theory, hence for D8-branes and D4-branes:

The N\mathbb{Z}_N action with even NN contains an order 2 element [...][ ...] Then there will be D8-branes in the type IIA D4-brane theory. Since the concept of intersecting D-branesinvolves use of the same dimensional D-branes, we restrict ourselves to the case that the order NN of N\mathbb{Z}_N is odd. (p. 4)

The Witten-Sakai-Sugimoto model on D4-D8-brane bound states for QCD with orthogonal gauge groups on O-planes:

Specifically D5 brane models T-dual to D6/D8 models:

Specifically for D6-branes:

  • S. Ishihara, H. Kataoka, Hikaru Sato, D=4D=4, N=1N=1, Type IIA Orientifolds, Phys. Rev. D60 (1999) 126005 (arXiv:hep-th/9908017)

  • Mirjam Cvetic, Paul Langacker, Tianjun Li, Tao Liu, D6-brane Splitting on Type IIA Orientifolds, Nucl. Phys. B709:241-266, 2005 (arXiv:hep-th/0407178)

Specifically for D3-branes/D7-branes:

Specifically 2d toroidal orientifolds:

2d toroidal orientifolds:


  • Dieter Lüst, S. Reffert, E. Scheidegger, S. Stieberger, Resolved Toroidal Orbifolds and their Orientifolds, Adv.Theor.Math.Phys.12:67-183, 2008 (arXiv:hep-th/0609014)

Orientifold Gepner models

Orientifolds of Gepner models

  • Brandon Bates, Charles Doran, Koenraad Schalm, Crosscaps in Gepner Models and the Moduli space of T 2T^2 Orientifolds, Advances in Theoretical and Mathematical Physics, Volume 11, Number 5, 839-912, 2007 (arXiv:hep-th/0612228)

Specifically string phenomenology and the landscape of string theory vacua of Gepner model orientifold compactifications:

Lift to M-theory

Lifts of orientifolds to M-theory (MO5, MO9) and F-theory are discussed in

The MO5 is originally discussed in

The classification in Hanany-Kol 00 (3.2) also appears, with more details, in Prop. 4.7 of

The “higher orientifold” appearing in Horava-Witten theory with circle 2-bundles replaced by the circle 3-bundles of the supergravity C-field is discussed towards the end of

Last revised on September 16, 2019 at 03:45:39. See the history of this page for a list of all contributions to it.