Quantum holonomy and link invariants

Abstract

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.

Flash Technology: Challenges and Opportunities

“…Recently, it was shown that for any quantum gauge theory, the quantum holonomy (L x ) is initial-point independent and G-invariant (Lee and Zhu 1991). Hence for irreducible π , π( (L x )) must equal to an x-independent eigenvalue P π (L) times 1 π .…”

“…That is, the triplet { [L x ], P π [L], W π [L]} to three-dimensional CST is what {V[L x ], Q π [L], V A π [L]} of section 3.3 and section 4.5 is to a quantum algebra. Thus, for example, for the gauge group G = SL(2), P π (in the fundamental representation) is the Jones polynomial (Witten 1989) and W π = (q −1 + q)P π ; for G = SL(M|N), M = N, P π is the HOMFLY polynomial (Freyd et al 1985, Horne 1990) and W π = (M − N)P π ; for G = SL(M|M), P π is the Alexander-Conway polynomial (Lee 1989, Lee andZhu 1991) but W π ≡ 0. That is, in quantum field theory the Alexander-Conway polynomial cannot be obtained from a Wilson line as in Witten's original approach; it must be obtained as the eigenvalue of a quantum holonomy.…”

Universal tangle invariant and commutants of quantum algebras

“…W π [L] is essentially the link polynomial V A π [L] of section 3.1 valued on the π representation of U q (g), where g is the Lie algebra of the (Lie) group G, and q, linearly related to the coupling constant of the three-dimensional CST, is restricted to being a root of unity, exp(i2π/(k+h)), whereh is the Coxeter number of G and k is the level of the representation of the affine algebra associated with g (Witten 1989, Horne 1990). Recently, it was shown that for any quantum gauge theory, the quantum holonomy (L x ) is initial-point independent and G-invariant (Lee and Zhu 1991). Hence for irreducible π , π( (L x )) must equal to an x-independent eigenvalue P π (L) times 1 π .…”

“…Thus, for example, for the gauge group G = SL(2), P π (in the fundamental representation) is the Jones polynomial (Witten 1989) and W π = (q −1 + q)P π ; for G = SL(M|N), M = N, P π is the HOMFLY polynomial (Freyd et al 1985, Horne 1990) and W π = (M − N)P π ; for G = SL(M|M), P π is the Alexander-Conway polynomial (Lee 1989, Lee andZhu 1991) but W π ≡ 0. That is, in quantum field theory the Alexander-Conway polynomial cannot be obtained from a Wilson line as in Witten's original approach; it must be obtained as the eigenvalue of a quantum holonomy.…”

Quantum holonomy in three-dimensional general covariant field theory and the link invariant

“…We show that quantum holonomy retains metric independence after explicit gauge fixing and hence possesses the topological property of a link invariant. We examine the generalized quantum holonomy defined on a multi-component link and discuss its relation to a polynomial for the link.PACS number(s): 11.15.Tk, 03.65.Fd, 02.40.Pc Some time ago it was shown that quantum holonomy in a three-dimensional general covariant non-Abelian gauge field theory possesses topological information of the link on which the holonomy operator is defined [1]. The quantum holonomy operator was shown to be a central element of the gauge group so that, in a given representation of the gauge group, * Also at ICSC-World Laboratory, Switzerland…”