Analytic derivation of dual gluons and monopoles from SU(2) lattice Yang-Mills theory. I. BF Yang-Mills representation

In this series of three papers, we generalize the derivation of dual photons and monopoles by Polyakov, and Banks, Myerson and Kogut, to obtain approximative models of SU(2) lattice gauge theory. The papers take three different representations as their starting points: the representation as a BF Yang-Mills theory, the spin foam representation and the plaquette representation. The derivations are based on stationary phase approximations. In this first article, we cast 3- and 4-dimensional SU(2) lattice gauge theory in the form of a lattice BF Yang-Mills theory. In several steps, the expectation value of a Wilson loop is transformed into a path integral over a dual gluon field and monopole-like degrees of freedom. The action contains the tree-level Coulomb interaction and a nonlinear coupling between dual gluons, monopoles and current.