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

TL;DR

This paper develops a non(anti)commutative superspace framework by deriving BCH formulas, addressing infinite-dimensional parametrization issues, and introducing an exponential deformation to facilitate gauge covariant twisted super Yang-Mills theories on lattices.

Contribution

It introduces closed BCH expressions for super Lie groups, proposes an exponential deformation to manage infinite-dimensional superspaces, and connects these to gauge covariant twisted supersymmetry on lattices.

Findings

Derived explicit BCH formulas for super Lie groups.

Identified exponential deformation as a solution to infinite-dimensional parametrization.

Proposed a gauge covariant superspace framework for lattice twisted super Yang-Mills.

Abstract

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.