Axiomatic holonomy maps and generalized Yang-Mills moduli space

Abstract

This Letter is a follow-up of Barrett, J. W.,Internat. J. Theoret. Phys. 30(9), (1991). Its main goal is to provide an alternative proof of that part of the reconstruction theorem which concerns the existence of a connection. A construction of a connection 1-form is presented. The formula expressing the local coefficients of the connection in terms of the holonomy map is obtained as an immediate consequence of that construction. Thus, the derived formula coincides with that used in Chan, H.-M., Scharbach, P., and Tsou, S. T.,Ann. Physics 166, 396–421 (1986). The reconstruction and representation theorems form a generalization of the fact that the pointed configuration space of the classical Yang-Mills theory is equivalent to the set of all holonomy maps. The point of this generalization is that there is a one-to-one correspondence not only between the holonomy maps and the orbits in the space of connections, but also between all maps ΩM → G fulfilling some axioms and all possible equivalence classes ofP(M, G) bundles with connections, where the equivalence relation is defined by a bundle isomorphism in a natural way.