Spin Networks, Wilson Loops and 3nj Wigner Identities

TL;DR

This paper derives new identities among Wigner 3nj coefficients using spin network properties of SU(2) lattice gauge theory and topological states, with potential extensions to higher dimensions and larger groups.

Contribution

It introduces a general method to obtain Wigner 3nj identities from gauge invariant eigenvalue equations, applicable to various spin networks and SU(N) groups.

Findings

Derived new Wigner 3nj identities from SU(2) lattice gauge theory.

Connected topological ground states of SU(2) toric code to Wigner identities with phases.

Method is broadly applicable to higher-dimensional spin networks and larger gauge groups.

Abstract

We exploit the spin network properties of the magnetic eigenstates of SU(2) Hamiltonian lattice gauge theory and use the Wilson loop operators to obtain a wide class of new identities amongst 3nj Wigner coefficients. We also show that the topological ground states of the SU(2) toric code Hamiltonian lead to Wigner 3nj identities with non-trivial phases. The method is very general and involves only the eigenvalue equations of any gauge invariant operator and their solutions. Therefore, it can be extended to any higher dimensional spin networks as well as larger SU(N) groups.