COMPLEXIFICATION OF GAUGE-THEORIES

For the case of a first-class constrained system with equivariant mome ntum map, we study the conditions under which the double process of re ducing to the constraint surface and dividing out by the group of gaug e transformations G is equivalent to the single process of dividing ou t the initial phase space by the complexification G(C) of G. For the p articular case of a phase space action that is the lift of a configura tion space action, conditions are found under which, in finite dimensi ons, the physical phase space of a gauge system with first-class const raints is diffeomorphic to a manifold imbedded in the physical configu ration space of the complexified gauge system. Similar conditions are shown to hold for the infinite-dimensional example of Yang-Mills theor ies. As a physical application we discuss the adequateness of using ho lomorphic Wilson loop variables as (generalized) global coordinates on the physical phase space of Yang-Mills theory.