Combinatorial constructions of modules for infinite-dimensional Lie algebras, II. Parafermionic space

Abstract

The standard modules for an affine Lie algebra $\ga$ have natural subquotients called parafermionic spaces -- the underlying spaces for the so-called parafermionic conformal field theories associated with $\ga.$ We study the case $\ga = \widehat{sl}(n+1,\C)$ for any positive integral level $k\ge 2.$ Generalizing the $\mathcal{Z}$-algebra approach of Lepowsky, Wilson and Primc, we construct a combinatorial basis for the parafermionic spaces in terms of colored partitions. The parts of these partitions represent ''Fourier coefficients'' of generalized vertex operators (parafermionic currents) and can be interpreted as statistically interacting quasi-particles of color $i,1\le i\le n,$ and charge $s,1\le s\le k-1.$ From a combinatorial point of view, these bases are essentially identical with the bases for level $k-1$ principal subspaces constructed by the author in [GeI]. In the particular case of the vacuum module, the character (string function) associated with our basis is the formula of Kuniba, Nakanishi and Suzuki [KNS] conjectured in a Bethe Ansatz layout. New combinatorial characters are established for the whole standard vacuum $\ga$-modules.

34 pages, 2 tables, Latex

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• A quasi-particle description of the M(3,p) models
• Exclusion statistics in conformal field theory and the UCPF for WZW models
• Parafermionic derivation of Andrews-type multiple sums
• Integrability in anyonic quantum spin chains via a composite height model
• Paths and partitions: combinatorial descriptions of the parafermionic states
• Paths for Z_k parafermionic models
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• The Bailey lemma and Kostka polynomials
• Local height probabilities in a composite Andrews-Baxter-Forrester model