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Abstract

In this paper we consider a Wilson loop in a 2+1 dimensional pure Yang-Mills theory with an SU(2) gauge group. The initial goal is to test a conjecture of A. M. Polyakov's which proposes that if one considers the field-strength, Fa„, and the gauge field, Aa, as independent, random variables, then a sum over surfaces spanning the Wilson loop will re-introduce the Bianchi Identity. We do this by introducing an additional functional integral over a sigma model variable which unravels the path-ordering of the loop variables. Then, via a non-Abelian Stokes' theorem, we express the Wilson loop as a surface integral with separate functional integrals over both Fa and Aa. At the semi-classical level, characterized by a large spin parameter, we find that the conjecture holds true - the Bianchi Identity arises as a natural constraint. Secondly, we find that this reformulation of the Wilson loop naturally allows for an arbitrary distribution of monopoles. We treat both the cases of a single monopole and a monopole gas. In the latter case case we demonstrate the confinement of quarks for states of half-odd-integer spin.