WZW in the lightlike directions

Dimensional reduction of the M5 brane on a Lorentzian manifold along a lightlike direction results in a five-dimensional gauge theory, which can be reformulated covariantly in six dimensions, where one puts the Lie derivatives along the lightlike direction of all fields to zero up to a gauge equivalence as constraints. We find gauge and supersymmetry anomalies for certain Lorentzian six-manifolds. We cancel these anomalies by adding a certain WZW theory in the 2d space that is spanned by two lightlike directions.