Simple-Current Symmetries, Rank-Level Duality, and Linear Skein Relations for Chern-Simons Graphs

TL;DR

This paper improves algorithms for calculating Chern-Simons graph invariants by introducing linear equations, revealing symmetries, and establishing identities related to simple currents, rank-level duality, and quantum 6j-symbols.

Contribution

It introduces linear equations to enhance the calculation of tetrahedra in Chern-Simons graphs and uncovers symmetries and identities linking different gauge theories and quantum invariants.

Findings

Enhanced algorithm for tetrahedron calculations

Identities between CS theories at different levels and groups

Explicit signs for braid eigenvalues in all compact gauge groups

Abstract

A previously proposed two-step algorithm for calculating the expectation values of Chern-Simons graphs fails to determine certain crucial signs. The step which involves calculating tetrahedra by solving certain non- linear equations is repaired by introducing additional linear equations. As a first step towards a new algorithm for general graphs we find useful linear equations for those special graphs which support knots and links. Using the improved set of equations for tetrahedra we examine the symmetries between tetrahedra generated by arbitrary simple currents. Along the way we uncover the classical origin of simple-current charges. The improved skein relations also lead to exact identities between planar tetrahedra in level $K$ $G(N)$ and level $N$ $G(K)$ CS theories, where $G(N)$ denotes a classical group. These results are recast as identities for quantum $6j$-symbols and WZW braid matrices. We obtain the transformation properties of arbitrary graphs and links under simple current symmetries and rank-level duality. For links with knotted components this requires precise control of the braid eigenvalue permutation signs, which we obtain from plethysm and an explicit expression for the (multiplicity free) signs, valid for all compact gauge groups and all fusion products.