Abstract
We extend the notion of super-Minkowski space-time to include Z 2 n -graded (Majorana) spinor coordinates. Our choice of the grading leads to spinor coordinates that are nilpotent but commute amongst themselves. The mathematical framework we employ is the recently developed category of Z 2 n -manifolds understood as locally ringed spaces. The formalism we present resembles N -extended superspace (in the presence of central charges), but with some subtle differences due to the exotic nature of the grading employed.
Highlights
Supersymmetry was independently discovered by three groups of authors: Gervais & Sakita [1], Gol’fand & Lichtman [2] and Volkov & Akulov [3] in the early 1970s
Supersymmetry can lead to milder divergences and even ‘non-renormalisation theorems’; offer a solution to the hierarchy problem in Grand Unified Theories; remove the tachyon from the spectrum of string theories and naturally leads to a theory of gravity when promoted to a local gauge theory
From the Batchelor–Gawȩdzki theorem for Z2n -manifolds [38] (Theorem 3.2) we know that any Z2n -manifold is noncanonically isomorphic to a Z2n \ {0}-graded vector bundle
Summary
Supersymmetry was independently discovered by three groups of authors: Gervais & Sakita [1], Gol’fand & Lichtman [2] and Volkov & Akulov [3] in the early 1970s. At the most basic level, one postulates that standard four-dimensional Minkowski space-time is extended by appending four anticommuting Majorana spinors. Extended supersymmetries are constructed by appending more and more anticommuting Majorana spinors. We will consider the very special instance of Majorana spinors that are Z2n -graded commutative We view this situation as a very mild form of ‘θ − θ’ non-anticommutativity. The constructions formally look very similar to the standard case of N -extended supersymmetry. In part, this is why the starting Z2n -Lie algebra was chosen as it is.
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