Yang-Mills gauge invariance on a space of Bose and Fermi coordinates

Abstract

A complete formalism is developed for imposing Yang-Mills gauge invariance induced by general coordinate transformations on superspace (i.e., a space containing both commuting and anticommuting coordinates). The appropriate group is the graded pseudo-Lie group of real, general linear transformations on superspace analogous to the role played by GL(4,$R$) in general relativity. The construction of derivatives which transform covariantly under this group forces the introduction of a connection. In the usual gauge theories the connection is just the vector potential, whereas here we expect it to be a function of all the dynamical fields. In this purely affine theory, field strengths and our proposed equations of motion for them result in a self-sourced theory involving only the connection. However, we find that there exist solutions which permit us to define a metric for which an inverse does not exist. These solutions are associated with a spontaneous symmetry breakdown of the vacuum which yields only the Lorentz metric and with no restriction on the internal-symmetry group. This spontaneous symmetry breaking introduces a parameter with the dimensions of ${(\mathrm{mass})}^{2}$.