WZW orientifolds and finite group cohomology

TL;DR

This paper investigates the mathematical structures needed to define orientifold models in WZW theories, linking gerbes with Jandl structures to group cohomology, and provides a comprehensive classification for simple Lie groups.

Contribution

It reduces the existence and classification problem of Z-equivariant Jandl structures on gerbes to a group cohomology problem, solving it for all simple simply-connected compact Lie groups.

Findings

Classification of Z_2-equivariant gerbes with Jandl structures for all simple Lie groups.

Reduction of orientifold consistency conditions to group cohomology calculations.

Explicit solutions for the existence of orientifold structures in WZW models.

Abstract

The simplest orientifolds of the WZW models are obtained by gauging a 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 \zeta 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 hep-th/0512283. More generally, one may gauge orientifold symmetry groups \Gamma = Z_2 \ltimes Z that combine the 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-\Gamma cohomology that we solve for all simple simply-connected compact Lie groups G and all orientifold groups \Gamma = Z_2 \ltimes Z.