The trouble with twisting (2,0) theory

We consider a twisted version of the abelian (2, 0) theory placed upon a Lorentzian six-manifold with a product structure, M6 = C × M4. 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 M4 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 M4. 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 M4 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.