Geometry of WZW Orientifolds

TL;DR

This paper provides a geometric analysis of unoriented WZW models, identifying orientifold planes with conjugacy classes and relating boundary states to D-branes, with explicit formulas connecting algebraic and geometric data.

Contribution

It offers a geometric interpretation of crosscap states and boundary conjugation in WZW models, and derives explicit formulas linking modular matrices to group characters.

Findings

Orientifold planes are localized on conjugacy classes of the group manifold.

Boundary conjugation corresponds to D-branes and their images.

Explicit relations between P-matrix, S-matrix, and group characters.

Abstract

We analyze unoriented Wess-Zumino-Witten models from a geometrical point of view. We show that the geometric interpretation of simple current crosscap states is as centre orientifold planes localized on conjugacy classes of the group manifold. We determine the locations and dimensions of these planes for arbitrary simply-connected groups and orbifolds thereof. The dimensions of the O-planes turn out to be given by the dimensions of symmetric coset manifolds based on regular embeddings. Furthermore, we give a geometrical interpretation of boundary conjugation in open unoriented WZW models; it yields D-branes together with their images under the orientifold projection. To find the agreement between O-planes and crosscap states, we find explicit answers for lattice extensions of Gaussian sums. These results allow us to express the modular P-matrix, which is directly related to the crosscap coefficient, in terms of characters of the horizontal subgroup of the affine Lie algebra. A corollary of this relation is that there exists a formal linear relation between the modular P- and the modular S-matrix.