Chern-Simons terms in four-dimensional heterotic string theory

We consider a four-dimensional heterotic string theory whose massless states are the N=4 supergravity multiplet and the gauge multiplet of SO(44). The string is coupled to the gauge background maintaining the world-sheet supersymmetry. It is found that the non-Abelian gauge current conservation becomes anomalous at the quantum level. The anomalies, however, can be eliminated by requiring the antisymmetric tensor field to transform nontrivially under the gauge transformation. This, in turn, introduces a Chern-Simons term in the definition of the corresponding field strength for it to be gauge invariant.