Abelian Vectors and Self-Dual Tensors in Six-Dimensional Supergravity

In this note we describe the most general coupling of {\it abelian} vector and tensor multiplets to six-dimensional $(1,0)$ supergravity. As was recently pointed out, it is of interest to consider more general Chern-Simons couplings to abelian vectors of the type $H^{r}=d B^{r}-1/2 c^{rab} A^{a}d A^{b}$, with $c^{r}$ matrices that may not be simultaneously diagonalized. We show that these couplings can be related to Green-Schwarz terms of the form $B^r c_r^{ab} F^a F^b$, and how the complete local Lagrangian, that embodies factorized gauge and supersymmetry anomalies (to be disposed of by fermion loops) is uniquely determined by Wess-Zumino consistency conditions, aside from an arbitrary quartic coupling for the gauginos.