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(Chalmers University of Technology)
The conjectured AGT correspondence, which relates a class of four-dimensional superconformal gauge theories to two-dimensional Liouville theory, has been suggested to have its origins in a certain twisted version of the mysterious six-dimensional (2,0) theory. Using this as motivation, we consider a topologically twisted version of the abelian $(2,0)$ theory on 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.