Massive And Massless Gauge Fields Formed by Flat Connections

Abstract

The Yang - Mills type massive and massless gauge theories are interpreted in the geometrical frame of holomorphic principal bundles on a complex 2 - manifold. It is seen in this formalism that, the component (1,1) of the curvature of this connection appears because of flat connections generated by holomorphic structure although connection is flat. Thus it is possible to write a Lagrangian for a Yang - Mills theory including massive and massless gauge fields. However, the mass matrix of a massive gauge field on such a bundle isn’t nilpotent and this field is generated by a noncommutative flat connection on the same bundle, then the structure group of this bundle is non - Abelian complex Lie group. However, if the gauge field is massless, then this is generated by commutative flat connection, and so the structure group of the bundle is Abelian complex Lie group. Also one sees that the second Chern number or topological charge is proportional to the total volume of the base manifold for each massless and massive gauge theories and Abelian (massless) gauge theories are indeed the theories of the Kähler potential on the complex projective space $\mathbb {C}P^{2}$.