  We study a generalization of the Alexandrov-Kontsevich-Schwarz-Zaboronsky
(AKSZ) formulation of the A- and B-models which involves a doubling of
coordinates, and can be understood as a complexification of the Poisson
$\sigma$-model underlying these. In the flat space limit the construction
contains models obtained by twisting an N=2 supersymmetric $\sigma$-model on
Hull's doubled geometry. The curved space generalization involves a product of
two diffeomorphic Calabi-Yau manifolds, and the $O(d,d)$ metric can be
understood as a complexification of the CY metric. In addition, we consider
solutions that can not be obtained by twisting the above $\sigma$-model. For
these it is possible to interpolate between a model evaluated on holomorphic
maps and one evaluated on constant maps by different choices of gauge fixing
fermion. Finally, we discuss some intriguing similarities between aspects of
the doubled formulation and topological M-theory, and a possible relation with
results from the theory of Lie and Courant algebroids, where a doubled
formulation plays a role in relating two- and three-dimensional topological
theories.
