Homotopy and duality in non-Abelian lattice gauge theory Original Research Article

We propose an approach of lattice gauge theory based on a homotopic interpretation of its degrees of freedom. The basic idea is to dress the plaquettes of the lattice to view them as elementary homotopies between nearby paths. Instead of using a unique G-valued field to discretize the connection 1-form, A, we use an Aut(G)-valued field U on the edges, which discretizes the 1-form adA, and a G-valued field V on the plaquettes, which corresponds to the Faraday tensor, F. The 1-connection, U, and the 2-connection, V, are then supposed to have a 2-curvature which vanishes. This constraint determines V as a function of U up to a phase in Z(G), the center of G. The 3-curvature around a cube is then Abelian and is interpreted as the magnetic charge contained inside this cube. Promoting the plaquettes to elementary homotopies induces a chiral splitting of their usual Boltzmann weight, w=vv̄, defined with the Wilson action. We compute the Fourier transform, v̂, of this chiral Boltzmann weight on G=SU3 and we obtain a finite sum of generalized hypergeometric functions. The dual model describes the dynamics of three spin fields: λP∈Ĝ and mP∈Z(G)≃Z3, on each oriented plaquette P, and εab∈Out(G)≃Z2, on each oriented edge (ab). Finally, we sketch a geometric interpretation of this spin system in a fibered category modeled on the category of representations of G.