Four-dimensional Lattice Gauge Theory with ribbon categories and the Crane-Yetter state sum

TL;DR

This paper extends 4D lattice gauge theory by incorporating ribbon categories, enabling a quantum group framework and unifying Yang-Mills theory with the Crane-Yetter invariant through a generalized Spin Foam Model.

Contribution

It introduces a novel framework replacing traditional gauge groups with ribbon categories, generalizing Spin Foam Models and defining a new partition function for 4-manifolds.

Findings

Unified formulation of 4D Yang-Mills and Crane-Yetter invariants

Well-defined Spin Foam Models using ribbon categories

Extension of lattice gauge theory to quantum group settings

Abstract

Lattice Gauge Theory in 4-dimensional Euclidean space-time is generalized to ribbon categories which replace the category of representations of the gauge group. This provides a framework in which the gauge group becomes a quantum group while space-time is still given by the `classical' lattice. At the technical level, this construction generalizes the Spin Foam Model dual to Lattice Gauge Theory and defines the partition function for a given triangulation of a closed and oriented piecewise-linear 4-manifold. This definition encompasses both the standard formulation of d=4 pure Yang-Mills theory on a lattice and the Crane-Yetter invariant of 4-manifolds. The construction also implies that a certain class of Spin Foam Models formulated using ribbon categories are well-defined even if they do not correspond to a Topological Quantum Field Theory.