Homotopy and duality in non-Abelian 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 $G$-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, $w=v\bar{v}$, defined with the Wilson action. We compute the Fourier transform, $\hat{v}$, 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 : ${\lambda }_P\in \hat{G}$ and $m_P\in \widehat{Z(G)}\simeq {\mathbb{Z}}_3$, on each oriented plaquette $P$, and $ε_{ab}\in{\hat{\OG}}\simeq\Z_2$, 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 $G$.

References in corpus (3)

• Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants
• Differential Geometry of Gerbes
• Combinatorics of Non-Abelian Gerbes with Connection and Curvature