WZW Orientifolds and Finite Group Cohomology

Abstract

The simplest orientifolds of the WZW models are obtained by gauging a ${\mathbb{Z}_2}$ symmetry group generated by a combined involution of the target Lie group G and of the worldsheet. The action of the involution on the target is by a twisted inversion ${g\mapsto(\zeta g)^{-1}}$, where ζ is an element of the center of G. It reverses the sign of the Kalb-Ramond torsion field H given by a bi-invariant closed 3-form on G. The action on the worldsheet reverses its orientation. An unambiguous definition of Feynman amplitudes of the orientifold theory requires a choice of a gerbe with curvature H on the target group G, together with a so-called Jandl structure introduced in [31]. More generally, one may gauge orientifold symmetry groups ${\Gamma=\mathbb{Z}_2\times Z}$ that combine the ${\mathbb{Z}_2}$-action described above with the target symmetry induced by a subgroup Z of the center of G. To define the orientifold theory in such a situation, one needs a gerbe on G with a Z-equivariant Jandl structure. We reduce the study of the existence of such structures and of their inequivalent choices to a problem in group-Γ cohomology that we solve for all simple simply connected compact Lie groups G and all orientifold groups ${\Gamma=\mathbb{Z} _2\times Z}$.