The representations of Temperley-Lieb-Jones algebras

TL;DR

This paper explores how representations from rational conformal field theories can be used to explicitly construct and analyze representations of Temperley-Lieb-Jones algebras, linking conformal blocks, subfactors, and link invariants.

Contribution

It provides a detailed method for constructing Temperley-Lieb-Jones algebra representations from SU(2)$_k$ Wess-Zumino theories and generalizes this to arbitrary rational conformal field theories.

Findings

Explicit definition of a trace operation in conformal blocks

Connection between subfactors and primary fields in conformal theories

Jones index formula for subfactors with index less than 4

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.