dorsal/arxiv
View SchemaOn Fusion Algebras and Modular Matrices
| Authors | T. Gannon, M. A. Walton |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9709039 |
| URL | https://arxiv.org/abs/q-alg/9709039 |
| DOI | 10.1007/s002200050695 |
| Journal | Commun.Math.Phys.206:1-22,1999 |
Abstract
We consider the fusion algebras arising in e.g. Wess-Zumino-Witten conformal field theories, affine Kac-Moody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix $S$, we find small sets of primary fields (equivalently, sets of highest weights) which can be identified with the variables of a polynomial realization of the $A_r$ fusion algebra at level $k$. We prove that for many choices of rank $r$ and level $k$, the number of these variables is the minimum possible, and we conjecture that it is in fact minimal for most $r$ and $k$. We also find new, systematic sources of zeros in the modular matrix $S$. In addition, we obtain a formula relating the entries of $S$ at fixed points, to entries of $S$ at smaller ranks and levels. Finally, we identify the number fields generated over the rationals by the entries of $S$, and by the fusion (Verlinde) eigenvalues.
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"abstract": "We consider the fusion algebras arising in e.g. Wess-Zumino-Witten conformal\nfield theories, affine Kac-Moody algebras at positive integer level, and\nquantum groups at roots of unity. Using properties of the modular matrix $S$,\nwe find small sets of primary fields (equivalently, sets of highest weights)\nwhich can be identified with the variables of a polynomial realization of the\n$A_r$ fusion algebra at level $k$. We prove that for many choices of rank $r$\nand level $k$, the number of these variables is the minimum possible, and we\nconjecture that it is in fact minimal for most $r$ and $k$. We also find new,\nsystematic sources of zeros in the modular matrix $S$. In addition, we obtain a\nformula relating the entries of $S$ at fixed points, to entries of $S$ at\nsmaller ranks and levels. Finally, we identify the number fields generated over\nthe rationals by the entries of $S$, and by the fusion (Verlinde) eigenvalues.",
"arxiv_id": "q-alg/9709039",
"authors": [
"T. Gannon",
"M. A. Walton"
],
"categories": [
"q-alg",
"hep-th",
"math.QA"
],
"doi": "10.1007/s002200050695",
"journal_ref": "Commun.Math.Phys.206:1-22,1999",
"title": "On Fusion Algebras and Modular Matrices",
"url": "https://arxiv.org/abs/q-alg/9709039"
},
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