We investigate the topological theory obtained by twisting the N=(2,2) supersymmetric nonlinear sigma model with target a bihermitian space with torsion. For the special case in which the two complex structures commute, we show that the action is a Q
-exact term plus a quasi-topological term. The quasi-topological term is locally given by a closed two-form which corresponds to a flat gerbe-connection and generalises the usual topological term of the A-model. Exponentiating it gives a Wilson surface, which can be regarded as a generalization of a Wilson line. This makes the quantum theory globally well-defined.
We discuss two dimensional N-extended supersymmetry in Euclidean signature and its R-symmetry. For N=2, the R-symmetry is SO(2)times SO(1,1), so that only an A-twist is possible. To formulate a B-twist, or to construct Euclidean N=2 models with H-flu
x so that the target geometry is generalised Kahler, it is necessary to work with a complexification of the sigma models. These issues are related to the obstructions to the existence of non-trivial twisted chiral superfields in Euclidean superspace.
