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 ~[ y l = (~e x p ( i $ ,~d x ) ) for closed contours y form a commutative semigroup, whereas (~e x p ( i J ,~d x ) ) =O for every open path a. The eigenvalues @[yl of ~[ y l are classified according to the irreducible representations of the gauge group. In an irreducible representation p, Tr(Y[yI) =@[ylTr(l,) 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 SU(NIN), and provides useful insight in understanding nonperturbative quantum chromodynamics as a string theory.
The spontaneous breaking of topological symmetry induced by a vanishingly small symmetry breaking term is investigated. It is shown that, in the presence of Gribov zero modes, topological theories without smearing terms, which are inequivalent to theories with smearing terms, permit spontaneous symmetry breaking. The relation with reducible configurations is briefly discussed. PACS numbers: 1 UO.Qc, 02.40.+m One of the motivations for developing topological field theories [1] is the hope that these models can describe the highly symmetrical phases of some realistic, less symmetric, field theories. The main obstacle to this approach is the difficulty in finding a way to break topological symmetry [1]. Hitherto, the models that have been investigated on this issue all contain gauge "smearing" terms that smear the gauge-fixing 8 function (these models correspond to the Feynman gauge in gauge theories), and no spontaneous breaking of topological symmetries has yet been found [2]. (An exception might be the symmetry breaking attributed to an instanton in a noncompact base manifold [3]; we will only consider compact manifolds in which such instantons do not exist.) Meanwhile, on other issues, models without smearing terms (corresponding to the Landau gauge), owing to their simplicity, have been extensively studied as substitutes to models with smearing terms [2-6]. The presumed interchangeability of the two classes of models is based on the argument that the only difference between them-a gauge smearing term which is BRST (Becchi-Rouet-Stora-Tyutin) exact -does not affect any of the properties of the theories [1], However, this is not unconditionally true. In fact, if Gribov zero modes [7] occur, a model in the Landau gauge may well not be connected to a model in the Feynman gauge by a smooth gauge transformation [7,8]. To see this [8], let F(X) be the gauge-fixing function in theLandau gauge. Its presence in the action picks out a configuration X that is a solution of FiX) =0. We now try to transfer to the Feynman gauge by replacing the F(X) =0 by F(X) = P, where P is arbitrary and depends locally on the coordinates of the base space in which X lives. If the path integration does not depend on P, the Feynman gauge is arrived at by averaging over the gauge function with weight exp( -aP 2 ). However, such P independence is contingent upon the existence, for every pair (P,8P), of an associated infinitesimal gauge transformation v that will transform the solution to F(X)=P to the solution to F{X)=P + SF SX SX 8v V = P + SP.
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