Magnetic monopoles without string in the Kähler–Clifford algebra bundle: A geometrical interpretation

Abstract

In substitution for Dirac monopoles with string (and for topological monopoles), ‘‘monopoles without string’’ have recently been introduced on the basis of a generalized potential, the sum of a vector A, and a pseudovector γ5 B potential. By making recourse to the Clifford bundle 𝒞(τM,g) [(TxM,g)=R1,3; 𝒞(TxM,g)=R1,3], which just allows adding together for each x∈M tensors of different ranks, in a previous paper a Lagrangian and Hamiltonian formalism was constructed for interacting monopoles and charges that can be regarded as satisfactory from various points of view. In the present article, after having ‘‘completed’’ the formalism, a purely geometrical interpretation of it is put forth within the Kähler–Clifford bundle 𝒦(τ*M,ĝ) of differential forms, essential ingredients being a generalized curvature and the Hodge decomposition theorem. Thus the way is paved for the extension of our ‘‘monopoles without string’’ to non-Abelian gauge groups. The analogy with supersymmetric theories is apparent.