String theory

Differential cohomology



An orientifold 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) 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) 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 model 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). See also (KMSV 16).

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 M-therory F-theory

Lifts of orientifold background from type II string theory to F-theory go back to (Sen 96, Sen 97a).

Lifts to D-type ADE singularities in M-theory goes back to (Sen 97b).


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

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

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

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

Lifts of type II orientifolds to F-theory were first discussed in

and to M-theory in

Last revised on April 13, 2018 at 05:49:59. See the history of this page for a list of all contributions to it.