Quantum holonomy and link invariants.

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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 $\ensuremath{\Psi}[\ensuremath{\gamma}]\ensuremath{\equiv}〈P\mathrm{exp}(i\ensuremath{\oint}\ensuremath{\int}{\ensuremath{\gamma}}^{}\mathrm{Adx})〉$ for closed contours $\ensuremath{\gamma}$ form a commutative semigroup, whereas $〈P\mathrm{exp}(i\ensuremath{\int}{\ensuremath{\alpha}}^{}\mathrm{Adx})〉=0$ for every open path $\ensuremath{\alpha}$. The eigenvalues $\ensuremath{\Phi}[\ensuremath{\gamma}]$ of $\ensuremath{\Psi}[\ensuremath{\gamma}]$ are classified according to the irreducible representations of the gauge group. In an irreducible representation $\ensuremath{\rho}$, $\mathrm{Tr}(\ensuremath{\Psi}[\ensuremath{\gamma}])=\ensuremath{\Phi}[\ensuremath{\gamma}]\mathrm{Tr}({1}_{\ensuremath{\rho}})$ 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 $\mathrm{SU}(N|N)$, and provides useful insight in understanding nonperturbative quantum chromodynamics as a string theory.