Axiomatic Holonomy Maps and Generalized Yang-Mills Moduli Space

TL;DR

This paper provides an alternative proof of the reconstruction theorem for connections using holonomy maps, generalizing the classical Yang-Mills configuration space to include all compatible maps and bundle classes.

Contribution

It introduces a new proof and construction of connection forms from holonomy maps, extending the classical Yang-Mills theory to a broader, axiomatic framework.

Findings

Established a one-to-one correspondence between holonomy maps and connection classes.

Generalized the classical Yang-Mills configuration space to include all compatible loop space maps.

Derived a formula for local connection coefficients from holonomy maps.

Abstract

This article is a follow-up of ``Holonomy and Path Structures in General Relativity and Yang-Mills Theory" by Barrett, J. W. (Int.J.Theor.Phys., vol.30, No.9, 1991). Its main goal is to provide an alternative proof of this part of the reconstruction theorem which concerns the existence of a connection. A construction of connection 1-form is presented. The formula expressing the local coefficients of 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 "On Loop Space Formulation of Gauge Theories" by Chan, H.-M., Scharbach, P. and Tsou S.T. (Ann.Phys., vol.167, 454-472, 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 from the loop space on $M$ to group $G$ fulfilling some axioms and all possible equivalence classes of $P(M,G)$ bundles with connection, where the equivalence relation is defined by bundle isomorphism in a natural way.