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arXiv:hep-th/0307050 · Journal of Statistical Mechanics Theory and Experiment

The origin of multiplets of chiral fields inSU(2)kat rational level

Alexander Nichols

Published Sep 22, 2004• 5 citations
arXivHTMLPDF

Abstract

We study solutions of the Knizhnik–Zamolodchikov equation for discrete representations of SU (2) k at rational level k +2 = p / q using a regular basis in which the braid matrices are well defined for all spins. We show that at spin J = ( j +1) p −1 for there are always a subset of 2 j +1 solutions closed under the action of the braid matrices. For these fields have integer conformal dimension and all the 2 j +1 solutions are monodromy free. The action of the braid matrices on these can be consistently accounted for by the existence of a multiplet of chiral fields with extra SU (2) quantum numbers ( m = − j ,..., j ). In the quantum group SU q (2), with q = e −i π/( k +2) , there is an analogous structure and the related representations are trivial with respect to the standard generators but transform in a spin j representation of SU (2) under the extended centre.

DOI
10.1088/1742-5468/2004/09/p09006

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We study solutions of the Knizhnik–Zamolodchikov equation for discrete representations of SU (2) k at rational level k +2 = p / q using a regular basis in which the braid matrices are well defined for all spins. We show that at spin J = ( j +1) p −1 for there are always a subset of 2 j +1 solutions closed under the action of the braid matrices. For these fields have integer conformal dimension and all the 2 j +1 solutions are monodromy free. The action of the braid matrices on these can be consistently accounted for by the existence of a multiplet of chiral fields with extra SU (2) quantum numbers ( m = − j ,..., j ). In the quantum group SU q (2), with q = e −i π/( k +2) , there is an analogous structure and the related representations are trivial with respect to the standard generators but transform in a spin j representation of SU (2) under the extended centre.
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The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra U q ( 2 ) at q N = 1, a three-parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters, the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin-conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged centre Z of the algebra U q ( 2 ) at q N = 1. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a hypersurface Spec Z ⊂ ℂ 4 which remains invariant under the action of an infinite-dimensional group G of analytic transformations, called the quantum coadjoint action.
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