Geometry of Orientifolds with NS-NS B-flux

TL;DR

<3-5 sentence high-level summary> This work analyzes the geometry of Type IIB orientifolds in the presence of NS-NS B-flux, identifying two distinct routes to gauge-group rank reduction: (i) D-branes wrapping a torus with half-integer $B$-flux yielding non-commuting Wilson lines, and (ii) D-branes transverse to such tori where the O-plane content alters tadpole cancellation. It further reveals that mixed D9/D5 sectors acquire multiplicities tied to discrete gauge symmetries, and that K3 orientifolds with B-flux exhibit subtleties and constraints that limit consistent CFT orbifold backgrounds. The paper then demonstrates how to construct consistent four-dimensional ${ m N}=2$ and ${ m N}=1$ models by isolating B-flux to specific factors and carefully arranging brane content, while highlighting persistent obstructions in several ${f Z}_M$ orbifold cases due to the 59-sector structure. Overall, the results illuminate how geometric data from $B$-flux shapes both the spectra and consistency conditions of higher-dimensional and 4D orientifolds, guiding viable compactifications with reduced supersymmetry.

Abstract

We discuss geometry underlying orientifolds with non-trivial NS-NS B-flux. If D-branes wrap a torus with B-flux the rank of the gauge group is reduced due to non-commuting Wilson lines whose presence is implied by the B-flux. In the case of D-branes transverse to a torus with B-flux the rank reduction is due to a smaller number of D-branes required by tadpole cancellation conditions in the presence of B-flux as some of the orientifold planes now have the opposite orientifold projection. We point out that T-duality in the presence of B-flux is more subtle than in the case with trivial B-flux, and it is precisely consistent with the qualitative difference between the aforementioned two setups. In the case where both types of branes are present, the states in the mixed (e.g., 59) open string sectors come with a non-trivial multiplicity, which we relate to a discrete gauge symmetry due to non-zero B-flux, and construct vertex operators for the the mixed sector states. Using these results we revisit K3 orientifolds with B-flux (where K3 is a T^4/Z_M orbifold) and point out various subtleties arising in some of these models. For instance, in the Z_2 case the conformal field theory orbifold does not appear to be the consistent background for the corresponding orientifolds with B-flux. This is related to the fact that non-zero B-flux requires the presence of both O5^- as well as O5^+ planes at various Z_2 orbifold fixed points, which appears to be inconsistent with the presence of the twisted B-flux in the conformal field theory orbifold. We also consider four dimensional N=2 and N=1 supersymmetric orientifolds. We construct consistent four dimensional models with B-flux which do not suffer from difficulties encountered in the K3 cases.