Quantum holonomy and link invariants

It is shown that, in a non-Abelian quantum field theory without an anomaly and broken symmetry, the set of all matrix-valued quantum holonomies $\Psi \left[\gamma \right]\equiv 〈P\exp \left(i\oint \int {\gamma }^{}\mathrm{Adx}\right)〉$ for closed contours $\gamma$ form a commutative semigroup, whereas $〈P\exp \left(i\int {\alpha }^{}\mathrm{Adx}\right)〉=0\mathrm{}$ for every open path $\alpha$. The eigenvalues $\Phi \left[\gamma \right]$ of $\Psi \left[\gamma \right]$ are classified according to the irreducible representations of the gauge group. In an irreducible representation $\rho$, $\mathrm{Tr}\left(\Psi \right[\gamma \left]\right)=\Phi \left[\gamma \right]\mathrm{Tr}\left(1_{\rho }\right)$ is a Wilson loop. This equation solves a puzzle in the relation between link invariants and Wilson loops in the Chern-Simons theory in three dimensions when the gauge group is $\mathrm{SU}\mathrm{}\left(N\right|N)$, and provides useful insight in understanding nonperturbative quantum chromodynamics as a string theory.