Heterotic Action in SUGRA-SYM Background

We consider the generalization of the heterotic action considered by Cherkis and Schwarz where the chiral bosons are introduced in a manifestly covariant way using an auxiliary field. In particular, we construct the kappa-symmetric heterotic action in ten-dimensional supergravity background coupled to super Yang-Mills theory and prove its kappa-symmetry. The usual Bianchi identity of Type I supergravity with super Yang-Mills $dH_3= -\tr F\wedge F$ is crucially used. For technical reason, the Yang-Mills field is restricted to be abelian.