Non(anti)Commutative Superspace, Baker-Campbell-Hausdorff Closed Forms, and Dirac-Kähler Twisted Supersymmetry
Kazuhiro Nagata
TL;DR
This work analyzes non(anti)commutative Grassmann parameters in the vector sector of superspace, where $\\\\\\\\\\\\\\lacklight\\{\\theta_A,\\theta_B\\}=a_{AB}$ and the SUSY algebra satisfies $\\{Q_A, Q_B\\}=P_{AB}$. It derives exact BCH closed forms for the multiplication of supergroup elements and demonstrates that generic nonzero $a_{AB}$ yields an infinite-dimensional superspace, complicating SUSY realizations. To address this, the paper introduces an exponential deformation of the Dirac-Kähler twisted SUSY algebra (via shift operators $g_\pm(P_{AB})$) that satisfies lattice Leibniz-rule-type conditions, yielding a finite, lattice-friendly framework in several dimensions ($N=D=2$, $N=4\!\ D=3$, $N=D=4$, and $N=4\!\ D=5$). It shows how, under carefully chosen ratio parameters $r_a, r_b, r_c$, these deformed algebras support exact or near-exact BCH composition and connect to gauge-covariant link formulations of twisted super Yang-Mills on a lattice, providing a practical non(anti)commutative superspace foundation for lattice SUSY and related dualities.
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-Kähler 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.