Contents

Idea

An orientifold is a background for string sigma-models that combines aspects of ${\mathbb{Z}}_2$-orbifolds with orientation reversal on the worldsheet (therefore the name): it consists of a bundle gerbe on a space with a ${\mathbb{Z}}_2$-action that satisfies a peculiar twisted equvariance condition with respect to this action.

Such orientifold gerbes with connection are the right structure for the definition of surface holonomy of unoriented surfaces. Therefore they serve for defining the gauge part of the action functional for unoriented strings.

More precisely, the gauge fields that constiture the background for a string $\sigma$-model, such as the Kalb-Ramond field and the RR-fields are modeled as cocycles in the differential cohomology of the target space, and an orientifold is the data given by an orbifold spacetime that involves the group ${\mathbb{Z}}_2$ and equipped with certain classes in its( twisted) differential cohomology that is suitably ${\mathbb{Z}}_2$-equivariant.

References

A definition and study of orientifold bundle gerbes, modeling the Kalb-Ramond field, is in

• Urs Schreiber, Christoph Schweigert, Konrad Waldorf, Unoriented WZW models and Holonomy of Bundle Gerbes (arXiv)

• Krzysztof Gawedzki, Rafal R. Suszek, Konrad Waldorf, Bundle Gerbes for Orientifold Sigma Models (arXiv)

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

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Precis (arXiv, slides)

A summary talk on this is

• Greg Moore, The RR-charge of an orientifold (ppt)

A formulation of some of the relevant aspects of orientifolds in terms of the differential nonabelian cohomology with coefficients in the 2-group $\mathrm{AUT}(U(1))$ coming from the crossed module $[U(1)\to {\mathbb{Z}}_2]$ is indicated in

• Urs Schreiber, talk Background fields in twisted differential nonabelian cohomology at Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology