Orientifolds of the 3-sphere

TL;DR

This paper explores the geometry of orientifolds in the SU(2) WZW model, linking geometric involutions to algebraic results and revealing new constraints on Cardy states related to discrete B-fluxes.

Contribution

It provides a geometric interpretation of orientifolds in the SU(2) WZW model and uncovers novel constraints on Cardy states based on flux considerations.

Findings

Geometric involutions correspond to algebraic Klein bottle and Möbius amplitudes.

Selection rules and signs in crosscap couplings are derived semiclassically.

Only integer- or half-integer-spin Cardy states can coexist in O0 orientifolds.

Abstract

We study the geometry of orientifolds in the SU(2) WZW model. They correspond to the two inequivalent, orientation-reversing involutions of $S^3$, whose fixed-point sets are: the north and south poles (O0), or the equator two-sphere (O2). We show how the geometric action of these involutions leads unambiguously to the previously obtained algebraic results for the Klein bottle and Moebius amplitudes. We give a semiclassical derivation of the selection rules and signs in the crosscap couplings, paying particular attention to discrete B-fluxes. A novel observation, which does not follow from consistency of the one-loop vacuum diagrams, is that in the case of the O0 orientifolds only integer- or only half-integer-spin Cardy states may coexist.