Graded gauge theories over supersymmetric space

Abstract

We generalize the usual gauge theories, as well as the supergauge theories, in the following way. We construct a graded group associated with a compact semisimple Lie group G. This graded group contains G and the linear space of anticommuting G-spinors on which G acts through a highly reducible representation. The graded group generalizes the notion of the super-Poincaré group. Next we construct a fiber bundle the basis of which is the superspace, the structural group being the graded group. Then we introduce the connection, curvature, and calculate the corresponding Yang–Mills Lagrangian. The nontrivial content of such a theory is put forward if we impose the Grassmann parity condition on our connection and curvature; we supposed here that both Grassmann parities (i.e., the one in the superspace and that in the graded group) add up to define the Grassmann parity of the corresponding field components. Together with the Hermiticity condition this supergauge leaves almost no room for arbitrariness in the expansion of the superconnection; it contains only the usual gauge field, the adjoint Higgs multiplet, and the spinor multiplet belonging to the spinorial representation of G. The conformal symmetry of the Lagrangian is broken, and the mass terms appear for the Higgs scalar and the spinor multiplet. The Yukawa and current–current interactions are also obtained, together with the Fermi four-point interaction term. The theory yields the ratio of the Higgs scalar mass versus the bare spinor mass equal to 27/40; the strengths of other couplings depend on the group via the decomposition of the spinor multiplet into the irreducible representations.