SU(N) grand unified models in the Grassmannian scheme

Abstract

Non-supersymmetric gauge theories may be formulated in the Grassmannian scheme by appending an internal Grassmannian manifold to the usual four dimensional spacetime, similar to the approach taken in the Kaluza-Klein Theory. SU(N) gauge theories for example, are constructed by appending an internal N-dimensional complex Grassmannian space to the spacetime. Superfields describing the particles are then defined as Taylor expansions in the Grassmannian space
Ψ(θ) = ∑_r,s=0^N (θ)^r(θ̅ )^sψ(r,s),
where θ and θ̅ are the Grassmann coordinates while ψ(r,s) are the multiplets representing the particles. The Lagrangian densities are obtained by considering Berezin integrals involving the superfields. For example, the free matter Lagrangian is given by
∫ d^Nθ̅ d^Nθ Ψ(θ) iγ ⋅ ∇Ψ̅(θ).
Ψ̅ here is the dual of Ψ in the θ-space.