Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
An orientifold is a target spacetime for string sigma-models that combines aspects of -orbifolds with orientation reversal on the worldsheet (therefore the name).
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 -equivariant cohomology and typically in real-oriented cohomology theory. For instance the B-field which otherwise is a (twisted) coycles in (ordinary) ordinary 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 models for B-fields for the bosonic string on orientifolds (differential HZR-theory) given in (Schreiber-Schweigert-Waldorf 05) and examples are analyzed in this context in (Gawedzki-Suszek-Waldorf 08).
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 is (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).
Lift to F-theory
Lifts of orientifold background from type II string theory to F-theory go back to (Sen 96, Sen 97).
The original observation that D-brane charge for orientifolds should be in KR-theory is due to
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)
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
More details are in
Related lecture notes / slides include
Jacques Distler, Orientifolds and Twisted KR-Theory (2008) (pdf)
Daniel Freed, Dirac charge quantiation, K-theory, and orientifolds, talk at a workshop Mathematical methods in general relativity and quantum field theories, November, 2009 (pdf)
Greg Moore, The RR-charge of an orientifold (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
Sergei Gukov, K-Theory, Reality, and Orientifolds, Commun.Math.Phys. 210 (2000) 621-639 (arXiv:hep-th/9901042)
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 coming from the crossed module 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
Lifts of type II orientifolds to F-theory were first discussed in