An orientifold (Dai-Lin-Polchinski 89, p. 12) is a target spacetime for string sigma-models that combines aspects of $\mathbb{Z}_2$-orbifolds with orientation reversal on the worldsheet, whence the name.
In type II string theory orientifold backgrounds (inducing type I string theory) with $\mathbb{Z}_2$-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 $\mathbb{Z}_2$-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 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
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
Exposition:
Lifts of orientifolds to M-theory and F-theory are discussed in
Ashoke Sen, F-theory and Orientifolds (arXiv:hep-th/9605150)
Ashoke Sen, Orientifold Limit of F-theory Vacua (arXiv:hep-th/9702165)
Ashoke Sen, A Note on Enhanced Gauge Symmetries in M- and String Theory, JHEP 9709:001,1997 (arXiv:hep-th/9707123)
Kentaro Hori, Consistency Conditions for Fivebrane in M Theory on $\mathbb{R}^5/\mathbb{Z}_2$ Orbifold, Nucl.Phys.B539:35-78, 1999 (arXiv:hep-th/9805141)
Amihay Hanany, Barak Kol, section 4 of On Orientifolds, Discrete Torsion, Branes and M Theory, JHEP 0006 (2000) 013 (arXiv:hep-th/0003025)
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
Urs Schreiber, Christoph Schweigert, Konrad Waldorf, Unoriented WZW models and Holonomy of Bundle Gerbes, Communications in Mathematical Physics August 2007, Volume 274, Issue 1, pp 31-64 (arXiv)
Krzysztof Gawedzki, Rafal R. Suszek, Konrad Waldorf, Bundle Gerbes for Orientifold Sigma Models Adv. Theor. Math. Phys. 15(3), 621-688 (2011) (arXiv:0809.5125)
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
Daniel Freed, Dirac charge quantiation, K-theory, and orientifolds, talk at a workshop Mathematical methods in general relativity and quantum field theories, Paris, November 2009 (pdf, pdf)
Greg Moore, The RR-charge of an orientifold, Oberwolfach talk 2010 (pdf, pdf, ppt)
Daniel Freed, Lectures on twisted K-theory and orientifolds, lecures at K-Theory and Quantum Fields, ESI 2012 (pdf)
A detailed list of examples of KR-theory of orientifolds and their T-duality is in
Charles Doran, Stefan Mendez-Diez, Jonathan Rosenberg, T-duality For Orientifolds and Twisted KR-theory (arXiv:1306.1779)
Charles Doran, Stefan Mendez-Diez, Jonathan Rosenberg, String theory on elliptic curve orientifolds and KR-theory (arXiv:1402.4885)
A formulation of some of the relevant aspects of (bosonic) orientifolds in terms of the differential nonabelian cohomology with coefficients in the 2-group $AUT(U(1))$ coming from the crossed module $[U(1) \to \mathbb{Z}_2]$ 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:
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