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

Hendryk Pfeiffer
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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.

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Hendryk Pfeiffer. 2001-09-06. Four-dimensional Lattice Gauge Theory with ribbon categories and the Crane-Yetter state sum. https://doi.org/10.1063/1.1398063

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