Homotopy and duality in nonAbelian lattice gauge theory
Abstract
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 Gvalued field to discretize the connection 1form, A, we use an Aut( G)valued field U on the edges, which discretizes the 1form ad _{A}, and a Gvalued field V on the plaquettes, which corresponds to the Faraday tensor, F. The 1connection, U, and the 2connection, V, are then supposed to have a 2curvature which vanishes. This constraint determines V as a function of U up to a phase in Z( G), the center of G. The 3curvature 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=v v̄, defined with the Wilson action. We compute the Fourier transform, v̂, of this chiral Boltzmann weight on G= SU_{3} and we obtain a finite sum of generalized hypergeometric functions. The dual model describes the dynamics of three spin fields: λ _{P}∈ Ĝ and m _{P}∈ overlineZ(G)≃ Z_{3}, on each oriented plaquette P, and ∊ _{ab}∈ overlineOut(G) ≃ Z_{2}, 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.
 Publication:

Nuclear Physics B
 Pub Date:
 April 2004
 DOI:
 10.1016/j.nuclphysb.2004.01.025
 arXiv:
 arXiv:hepth/0308100
 Bibcode:
 2004NuPhB.684..369A
 Keywords:

 Lattice gauge theory;
 KramersWannier duality;
 Homotopy;
 Fibered categories;
 High Energy Physics  Theory
 EPrint:
 Nucl.Phys. B684 (2004) 369383