So, physicists like to attach a mysterious extra cohomology class in H^2(X;C^*) to a Kahler (or hyperkahler) manifold called a "B-field."  The only concrete thing I've seen this B-field do is change the Fukaya category/A-branes: when you have a B-field, you shouldn't take flat vector bundles on a Lagrangian subvariety, but rather ones whose curvature is the B-field.  How should I think about this gadget?