dorsal/arxiv
View SchemaCombinatorial constructions of modules for infinite-dimensional Lie algebras, II. Parafermionic space
| Authors | Galin Georgiev |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9504024 |
| URL | https://arxiv.org/abs/q-alg/9504024 |
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 $k \geq 2.$ Generalizing the $\cal Z$-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 $i,\;1\leq i \leq n,$ and charge $s,\; 1\leq s \leq k-1.$ 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.
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"abstract": "The standard modules for an affine Lie algebra $\\ga$ have natural\nsubquotients called parafermionic spaces -- the underlying spaces for the\nso-called parafermionic conformal field theories associated with $\\ga.$\n We study the case $\\ga = \\widehat{sl}(n+1,\\C)$ for any positive integral\nlevel $k \\geq 2.$ Generalizing the $\\cal Z$-algebra approach of Lepowsky,\nWilson and Primc, we construct a combinatorial basis for the parafermionic\nspaces in terms of colored partitions. The parts of these partitions represent\n\u0027\u0027Fourier coefficients\u0027\u0027 of generalized vertex operators (parafermionic\ncurrents) and can be interpreted as statistically interacting quasi-particles\nof color $i,\\;1\\leq i \\leq n,$ and charge $s,\\; 1\\leq s \\leq k-1.$ From a\ncombinatorial point of view, these bases are essentially identical with the\nbases for level $k-1$ principal subspaces constructed by the author in [GeI].\nIn the particular case of the vacuum module, the character (string function)\nassociated with our basis is the formula of Kuniba, Nakanishi and Suzuki [KNS]\nconjectured in a Bethe Ansatz layout.\n New combinatorial characters are established for the whole standard vacuum\n$\\ga$-modules.",
"arxiv_id": "q-alg/9504024",
"authors": [
"Galin Georgiev"
],
"categories": [
"q-alg",
"hep-th",
"math.QA"
],
"title": "Combinatorial constructions of modules for infinite-dimensional Lie algebras, II. Parafermionic space",
"url": "https://arxiv.org/abs/q-alg/9504024"
},
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