  We consider a twisted version of the abelian $(2,0)$ theory placed upon a
Lorenzian six-manifold with a product structure, $M_6=C \times M_4 $. This is
done by an investigation of the free tensor multiplet on the level of equations
of motion, where the problem of its formulation in Euclidean signature is
circumvented by letting the time-like direction lie in the two-manifold $C$ and
performing a topological twist along $M_4$ alone. A compactification on $C$ is
shown to be necessary to enable the possibility of finding a topological field
theory. The hypothetical twist along a Euclidean $C$ is argued to amount to the
correct choice of linear combination of the two supercharges scalar on $M_4$.
This procedure is expected and conjectured to result in a topological field
theory, but we arrive at the surprising conclusion that this twisted theory
contains no $Q$-exact and covariantly conserved stress tensor unless $M_4$ has
vanishing curvature. This is to our knowledge a phenomenon which has not been
observed before in topological field theories. In the literature, the setup of
the twisting used here has been suggested as the origin of the conjectured
AGT-correspondence, and our hope is that this work may somehow contribute to
the understanding of it.
