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 (KMSV 16).

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 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).

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.


The concept originates around

  • Jin Dai, R.G. Leigh, Joseph Polchinski, p. 12 of New Connections Between String Theories, Mod.Phys.Lett. A4 (1989) 2073-2083 (spire:25758)

Traditional lecture notes include

  • Atish Dabholkar, Lectures on Orientifolds and Duality (arXiv:hep-th/9804208)

  • Carlo Angelantonj, Augusto Sagnotti, Open Strings, Phys.Rept.371:1-150,2002; Erratum-ibid.376:339-405, 2003 (arXiv:hep-th/0204089)

Textbook discussion in the context of intersecting D-brane models with an eye towards string phenomenology is in


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

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

and was re-amplified in

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

A more encompassing 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 being announced in

based on stuff like

More details are in

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

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

The Witten-Sakai-Sugimoto model for QCD on orientifolds:

Last revised on April 8, 2019 at 10:46:50. See the history of this page for a list of all contributions to it.