WZW Fusion Research Articles

Positivity and periodicity of Q-systems in the WZW fusion ring

We study properties of solutions of Q-systems in the WZW fusion ring obtained by the Kirillov–Reshetikhin modules. We make a conjecture about their positivity and periodicity and give a proof of it in some cases. We also construct a positive solution of the level k restricted Q-system of classical types in the fusion rings. As an application, we prove some conjectures of Kirillov and Kuniba–Nakanishi–Suzuki on the level k restricted Q-systems.

PRESENTATIONS OF WESS–ZUMINO–WITTEN FUSION RINGS

The fusion rings of the Wess–Zumino–Witten models are re-examined. Attention is drawn to the difference between fusion rings over ℤ (which are often of greater importance in applications) and fusion algebras over ℂ. Complete proofs are given by characterizing the fusion algebras (over ℂ) of the SU (r+1) and Sp (2r) models in terms of the fusion potentials, and it is shown that the analagous potentials cannot describe the fusion algebras of the other models. This explains why no other representation-theoretic fusion potentials have been found. Instead, explicit generators are then constructed for general WZW fusion rings (over ℤ). The Jacobi–Trudy identity and its Sp (2r) analogue are used to derive the known fusion potentials. This formalism is then extended to the WZW models over the spin groups of odd rank, and explicit presentations of the corresponding fusion rings are given. The analogues of the Jacobi–Trudy identity for the spinor representations (for all ranks) are derived for this purpose, and may be of independent interest.

An algorithm for twisted fusion rules

We present an algorithm for an efficient calculation of the fusion rules of twisted representations of untwisted affine Lie algebras. These fusion rules appear in WZW orbifold theories and as annulus coefficients in boundary WZW theories; they provide NIM-reps of the WZW fusion rules.

Higher level WZW sectors from free fermions

We introduce a gauge group of internal symmetries of an ambient algebra as a new tool for investigating the superselection structure of WZW theories and the representation theory of the corresponding affine Lie algebras. The relevant ambient algebra arises from the description of these conformal field theories in terms of free fermions. As an illustration we analyze in detail the so(N) WZW theories at level two, which allows us in particular to construct explicit bases for the level-two irreducible highest weight modules. In this case there is actually a homomorphism from the representation ring of the gauge group to the subring of the WZW fusion ring that corresponds to the Neveu–Schwarz sector, even though the level-two observable algebra is smaller than the gauge invariant subalgebra of the field algebra.

WZW Fusion Rings in the Limit of Infinite Level

We show that the WZW fusion rings at finite levels form a projective system with respect to the partial ordering provided by divisibility of the height, i.e. the level shifted by a constant. From this projective system we obtain WZW fusion rings in the limit of infinite level. This projective limit constitutes a mathematically well-defined prescription for the `classical limit' of WZW theories which replaces the naive idea of `sending the level to infinity'. The projective limit can be endowed with a natural topology, which plays an important role for studying its structure. The representation theory of the limit can be worked out by considering the associated fusion algebra; this way we obtain in particular an analogue of the Verlinde formula.

WZW fusion rules for the classical Lie algebras

Fusion rules for WZW models based on the Lie algebras su(n), sp(2n) and so(n) are considered. Previous work has shown how, in the cases of su(n) and sp(2n), fusion rules may be computed using Young diagram methods and applying fusion modification rules similar to the rank modification rules required for tensor products. In this note we extend these results to include the so(n) case.

Simple WZW superselection sectors

The superselection structure of simple currents of chiral Wess-Zumino-Witten theories, at arbitrary valuek ∈ ℕ of the corresponding affine Lie algebra, is described in terms of explicit localizable automorphisms of the affine algebra. These automorphisms are induced by certain Dynkin diagram automorphisms; under composition, they form an Abelian group isomorphic to the center of the relevant simply connected simple Lie group and, hence, reproduce the WZW fusion rules.

Level 1 WZW superselection sectors

The superselection structure of the Wess-Zumino-Witten theory based on the affine Lie algebra\(\widehat{so}(N)\) at level one is investigated for arbitraryN. By making use of the free fermion representation of the affine algebra, the endomorphisms which represent the superselection sectors on the observable algebra can be constructed as endomorphisms of the underlying Majorana algebra. These endomorphisms do not close on the chiral algebra of the theory, but we are able to obtain a larger algebra on which the endomorphisms close. The composition of equivalence classes of the endomorphisms reproduces the WZW fusion rules.

Group-level duality of WZW fusion coefficients and Chern-Simons link observables

We present the duality of the fusion rule multiplicities, modular tranformation matrices and conformal weights of the Wess-Zumino-Witten models based on any of the classical affine Lie algebras SU(N) K,Sp(N) Kand SO(N) K under the exchange of N and K. We also exhibit the transformation properties of these quantities under the symmetries of the extended Dynkin diagrams, which play a central role in this group-level duality. We interpret these results in Chern-Simons theory, and derive identities between link observables. Several surprising results involving spinor representations of SO(N) are proven.

WZW fusion rules, quantum groups, and the modular matrix S

A simple algorithm, due to Walton, for calculating the fusion rules of WZW theories is proved and applied to several nontrivial examples. The results are used to gain new insight in the structure of non-fully reducible modules of the quantum groups U q g , and to find the explicit form of the modular transformation matrix S for some cases where it had not yet been worked out before.