N=4 Yang--Mills theory as a complexification of the N=2 theory

A complexification of the twisted $\N=2$ theory allows one to determine the N=4 Yang--Mills theory in its third twist formulation. The imaginary part of the gauge symmetry is used to eliminate two scalars fields and create gauge covariant longitudinal components for the imaginary part of the gauge field. The latter becomes the vector field of the thirdly twisted $\N=4$ theory. Eventually, one gets a one to one correspondence between the fields of both theories. Analogous complexifications can be done for topological 2d-gravity and topological sigma models.