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 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
Sunil Mukhi, Orientifolds: The Unique Personality Of Each Spacetime Dimension, Workshop on Frontiers of Field Theory, Quantum Gravity and String Theory, Puri, India, 12 - 21 Dec 1996 (arXiv:hep-th/9710004, cern:335233)
Jan de Boer, Robbert Dijkgraaf, Kentaro Hori, Arjan Keurentjes, John Morgan, David Morrison, Savdeep Sethi, section 3 of Triples, Fluxes, and Strings, Adv.Theor.Math.Phys. 4 (2002) 995-1186 (arXiv:hep-th/0103170)
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
Specifically K3 orientifolds ($\mathbb{T}^4/G_{ADE}$) in type IIB string theory, hence for D9-branes and D5-branes:
Eric G. Gimon, Joseph Polchinski, Section 3.2 of: Consistency Conditions for Orientifolds and D-Manifolds, Phys. Rev. D54: 1667-1676, 1996 (arXiv:hep-th/9601038)
Eric Gimon, Clifford Johnson, K3 Orientifolds, Nucl. Phys. B477: 715-745, 1996 (arXiv:hep-th/9604129)
Alex Buchel, Gary Shiu, S.-H. Henry Tye, Anomaly Cancelations in Orientifolds with Quantized B Flux, Nucl.Phys. B569 (2000) 329-361 (arXiv:hep-th/9907203)
P. Anastasopoulos, A. B. Hammou, A Classification of Toroidal Orientifold Models, Nucl. Phys. B729:49-78, 2005 (arXiv:hep-th/0503044)
Specifically K3 orientifolds ($\mathbb{T}^4/G_{ADE}$) in type IIA string theory, hence for D8-branes and D4-branes:
J. Park, Angel Uranga, A Note on Superconformal N=2 theories and Orientifolds, Nucl. Phys. B542:139-156, 1999 (arXiv:hep-th/9808161)
G. Aldazabal, S. Franco, Luis Ibanez, R. Rabadan, Angel Uranga, D=4 Chiral String Compactifications from Intersecting Branes, J. Math. Phys. 42:3103-3126, 2001 (arXiv:hep-th/0011073)
G. Aldazabal, S. Franco, Luis Ibanez, R. Rabadan, Angel Uranga, Intersecting Brane Worlds, JHEP 0102:047, 2001 (arXiv:hep-ph/0011132)
H. Kataoka, M. Shimojo, $SU(3) \times SU(2) \times U(1)$ Chiral Models from Intersecting D4-/D5-branes, Progress of Theoretical Physics, Volume 107, Issue 6, June 2002, Pages 1291–1296 (arXiv:hep-th/0112247, doi:10.1143/PTP.107.1291)
The $\mathbb{Z}_N$ action with even $N$ 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 $N$ of $\mathbb{Z}_N$ is odd. (p. 4)
Gabriele Honecker, Non-supersymmetric Orientifolds with D-branes at Angles, Fortsch.Phys. 50 (2002) 896-902 (arXiv:hep-th/0112174)
Gabriele Honecker, Intersecting brane world models from D8-branes on $(T^2 \times T^4/\mathbb{Z}_3)/\Omega\mathcal{R}_1$ type IIA orientifolds, JHEP 0201 (2002) 025 (arXiv:hep-th/0201037)
Gabriele Honecker, Non-supersymmetric orientifolds and chiral fermions from intersecting D6- and D8-branes, thesis 2002 (pdf)
The Witten-Sakai-Sugimoto model on D4-D8-brane bound states for QCD with orthogonal gauge groups on O-planes:
Toshiya Imoto, Tadakatsu Sakai, Shigeki Sugimoto, $O(N)$ and $USp(N)$ QCD from String Theory, Prog.Theor.Phys.122:1433-1453, 2010 (arXiv:0907.2968)
Hee-Cheol Kim, Sung-Soo Kim, Kimyeong Lee, 5-dim Superconformal Index with Enhanced $E_n$ Global Symmetry, JHEP 1210 (2012) 142 (arXiv:1206.6781)
Specifically D5 brane models T-dual to D6/D8 models:
Angel Uranga, A New Orientifold of $\mathbb{C}^2/\mathbb{Z}_N$ and Six-dimensional RG Fixed Points, Nucl. Phys. B577:73-87, 2000 (arXiv:hep-th/9910155)
Bo Feng, Yang-Hui He, Andreas Karch, Angel Uranga, Orientifold dual for stuck NS5 branes, JHEP 0106:065, 2001 (arXiv:hep-th/0103177)
Specifically for D6-branes:
S. Ishihara, H. Kataoka, Hikaru Sato, $D=4$, $N=1$, 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:
Dongfeng Gao, Kentaro Hori, Section 7.3 of: On The Structure Of The Chan-Paton Factors For D-Branes In Type II Orientifolds (arXiv:1004.3972)
Charles Doran, Stefan Mendez-Diez, Jonathan Rosenberg, String theory on elliptic curve orientifolds and KR-theory (arXiv:1402.4885)
Various:
Orientifolds of Gepner models
Specifically string phenomenology and the landscape of string theory vacua of Gepner model orientifold compactifications:
T.P.T. Dijkstra, L. R. Huiszoon, Bert Schellekens, Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models, Phys.Lett. B609 (2005) 408-417 (arXiv:hep-th/0403196)
T.P.T. Dijkstra, L. R. Huiszoon, Bert Schellekens, Supersymmetric Standard Model Spectra from RCFT orientifolds, Nucl.Phys.B710:3-57,2005 (arXiv:hep-th/0411129)
Lifts of orientifolds to M-theory (MO5, MO9) 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)
Eric Gimon, On the M-theory Interpretation of Orientifold Planes (arXiv:hep-th/9806226, spire:472499)
Changhyun Ahn, Hoil Kim, Hyun Seok Yang, $SO(2N)$ $(0,2)$ SCFT and M Theory on $AdS_7 \times \mathbb{R}P^4$, Phys.Rev. D59 (1999) 106002 (arXiv:hep-th/9808182)
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
Keshav Dasgupta, Sunil Mukhi, Orbifolds of M-theory, Nucl. Phys. B465 (1996) 399-412 (arXiv:hep-th/9512196)
Edward Witten, Five-branes And M-Theory On An Orbifold, Nucl. Phys. B463:383-397, 1996 (arXiv:hep-th/9512219)
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
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