# nLab orientifold

### Context

#### Differential cohomology

differential cohomology

# Contents

## Idea

An orientifold is a target spacetime for string sigma-models that combines aspects of $\mathbb{Z}_2$-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 $\mathbb{Z}_2$-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 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).

## Properties

### Lift to F-theory

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

## References

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

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 $AUT(U(1))$ 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

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

Revised on May 19, 2014 04:52:15 by Urs Schreiber (217.39.7.253)