Quantum Deformation of Lattice Gauge Theory

TL;DR

This paper introduces a quantum deformation of 3D lattice gauge theory using topological and algebraic methods, connecting it to known invariants and algebraic approaches in mathematical physics.

Contribution

It defines a new quantum deformation of lattice gauge theory via the Reshetikhin-Turaev functor and explores its topological and algebraic properties, including connections to existing invariants.

Findings

Partition function yields a 3-fold invariant matching Turaev-Viro in the topological limit.

Establishes a link between q-deformed gauge theory on Riemann surfaces and the Alekseev-Grosse-Schomerus algebraic approach.

Provides a framework for bounded manifolds and links within the quantum gauge theory context.

Abstract

A quantum deformation of 3-dimensional lattice gauge theory is defined by applying the Reshetikhin-Turaev functor to a Heegaard diagram associated to a given cell complex. In the root-of-unity case, the construction is carried out with a modular Hopf algebra. In the topological (weak-coupling) limit, the gauge theory partition function gives a 3-fold invariant, coinciding in the simplicial case with the Turaev-Viro one. We discuss bounded manifolds as well as links in manifolds. By a dimensional reduction, we obtain a q-deformed gauge theory on Riemann surfaces and find a connection with the algebraic Alekseev-Grosse-Schomerus approach.