Non(anti)Commutative Superspace, Baker-Campbell-Hausdorff Closed Forms, and Dirac-K\'ahler Twisted Supersymmetry

Starting from an elementary calculation of super Lie group elements associating with non(anti)-commutative Grassmann parameters, we derive several closed expressions of Baker-Campbell-Hausdorff (BCH) formula which represent multiplication properties of super Lie group elements in the corresponding superspace. We then show that parametrization of superspace in general may become infinite dimensional due to the presence of non(anti)commutativity. We show that a Dirac-K\"ahler Twisted SUSY Algebra (also referred to as Marcus B-type Twisted SUSY Algebra or Geometric Langlands Twisted SUSY Algebra) with a certain type of deformation, which we call an exponential deformation, may circumvent this problem. We also provide, in terms of gauge covariantization of the SUSY algebra, a geometric understanding of the exponential deformation, and see that the framework constructed in this paper may serve as a non(anti)commutative superspace framework providing the gauge covariant link formulation of twisted super Yang-Mills on a lattice.
Comment: 44 pages, 2 figures