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 (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 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
Early accounts include
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 is in
and specifically in the context of intersecting D-brane models with an eye towards string phenomenology in
Exposition:
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
Pedram Hekmati, Michael Murray, Richard Szabo, Raymond Vozzo, Real bundle gerbes, orientifolds and twisted KR-homology (arXiv:1608.06466)
Pedram Hekmati, Michael Murray, Richard Szabo, Raymond Vozzo, Sign choices for orientifolds (arXiv:1905.06041)
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
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 Witten-Sakai-Sugimoto model for QCD on orientifolds:
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)
Philip C. Argyres, Ron Maimon, Sophie Pelland, The M theory lift of two O6 planes and four D6 branes, JHEP 0205 (2002) 008 (arXiv:hep-th/0204127)
following
Edward Witten, Solutions Of Four-Dimensional Field Theories Via M Theory, (arXiv:hep-th/9703166)
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 May 16, 2019 at 05:20:12. See the history of this page for a list of all contributions to it.