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
G
-valued field to discretize the connection 1-form,
A
, we use an $\AG$-valued field
U
on the edges, which plays the role of the 1-form $\ad_A$, 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=v
v
¯
, defined with the Wilson action. We compute the Fourier transform,
v
^
, of this chiral Boltzmann weight on
G=S
U
3
and we obtain a finite sum of generalized hypergeometric functions. The dual model describes the dynamics of three spin fields :
λ
P
∈
G
^
and $m_P\in{\hat{Z(G)}}\simeq\Z_3$, on each oriented plaquette
P
, and $\epsilon_{ab}\in{\hat{\OG}}\simeq\Z_2$, 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
.