  In this work, we discuss the interaction between anti-symmetric rank-two
tensor matter and topological Yang-Mills fields. The matter field considered
here is the rank-2 Avdeev-Chizhov tensor matter field in a suitably extended
$N_{T}=2$ SUSY. We start off from the $N_{T}=2$, D=4 superspace formulation and
we go over to Riemannian manifolds. The matter field is coupled to the
topological Yang-Mills field. We show that both actions are obtained as
$Q-$exact forms, which allows us to write the energy-momentum tensor as
$Q-$exact observables.
