Two dimensional lattice gauge theory based on a quantum group
E. Buffenoir Ph. Roche
TL;DR
This work develops a two-dimensional lattice gauge theory whose gauge symmetry is encoded by a quantum group, realized as a coaction of a quantum gauge group on a noncommutative lattice-algebra of gauge fields. Wilson loops are constructed via a quantum trace and satisfy gauge and cyclic invariance, with non-crossing loops commuting, yielding a rich algebraic structure governed by the $R$-matrix of $U_q(\mathcal{G})$. The theory defines a q-deformed Yang–Mills measure and Boltzmann weights, producing partition functions and correlators that are triangulation-independent and exhibit a topological (or quasitopological) character; for $t_{\alpha}=-1$ these reduce to expressions related to Turaev–Viro invariants and connect to Chern–Simons theory in three dimensions. The results establish a concrete bridge between quantum-group gauge theory in 2D and topological quantum field theories, while outlining future directions, including representations of the gauge-field algebra and higher-dimensional generalizations.
Abstract
In this article we analyze a two dimensional lattice gauge theory based on a quantum group.The algebra generated by gauge fields is the lattice algebra introduced recently by A.Yu.Alekseev,H.Grosse and V.Schomerus we define and study wilson loops and compute explicitely the partition function on any Riemann surface. This theory appears to be related to Chern-Simons Theory.