N=4 twisted superspace from Dirac-Kähler twist and off-shell SUSY invariant actions in four dimensions

We propose a N=4 twisted superspace formalism in four dimensions by introducing a Dirac-Kähler twist. In addition to the BRST charge as a scalar counter part of the twisted supercharge we find vector and tensor twisted supercharges. By introducing twisted chiral superfields we explicitly construct an off-shell twisted N=4 SUSY invariant action. We propose a variety of supergauge invariant actions by introducing twisted vector superfields. We may, however, need to find further constraints in order to identify a twisted N=4 super-Yang-Mills action. We propose a superconnection formalism for the twisted superspace and constraints play a crucial role in the construction. It turns out that the N=4 superalgebras obtained by a Dirac-Kähler twist can be decomposed into N=2 sectors. We then construct twisted N=2 super-Yang-Mills actions using this formalism in two and four dimensions.