The representations of Temperley-Lieb-Jones algebras

Abstract

Representations of braid group obtained from rational conformal field theories can be used to obtain explicit representations of Temperley-Lieb-Jones algebras. The method is described in detail for SU(2)${}_k$ Wess - Zumino conformal field theories and its generalization to an arbitrary rational conformal field theory outlined. Explicit definition of an associated linear trace operation in terms of a certain matrix element in the space of conformal blocks of such a conformal theory is presented. Further for every primary field of a rational conformal field theory, there is a subfactor of hyperfinite II${}_1$ factor with trivial relative commutant. The index of the subfactor is given in terms of identity - identity element of certain duality matrix for conformal blocks of four-point correlators. Jones formula for index ( $<$ 4 ) for subfactors corresponds to spin $\frac{1}{2}$ representation of SU(2)${}_k$ Wess-Zumino conformal field theory. Definition of the trace operation also provides a method of obtaining link invariants explicitly.

24 pages

References in corpus (4)

• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links III : Compact Semi-simple Group
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links
• Chern-Simons Theory, Colored-Oriented Braids and Link invariants
• Complete Solution of SU(2) Chern-Simons Theory

Cited by in corpus (3)

• Knot invariants from rational conformal field theories
• Three-Manifold Invariants from Chern-Simons Field Theory with Arbitrary Semi-Simple Gauge Groups
• Three-partite vertex model and knot invariants