The Level 2 and 3 Modular Invariants for the Orthogonal Algebras

Gannon, Terry

Canadian journal of mathematics, Tome 52 (2000) no. 3, pp. 503-521

Résumé

The ‘1-loop partition function’ of a rational conformal field theory is a sesquilinear combination of characters, invariant under a natural action of $\text{S}{{\text{L}}_{2}}(\mathbb{Z})$ , and obeying an integrality condition. Classifying these is a clearly defined mathematical problem, and at least for the affine Kac-Moody algebras tends to have interesting solutions. This paper finds for each affine algebra $B_{r}^{\left( 1 \right)}$ and $D_{r}^{(1)}$ all of these at level $k\le 3$ . Previously, only those at level 1 were classified. An extraordinary number of exceptionals appear at level 2—the $B_{r}^{(1)},D_{r}^{(1)}$ level 2 classification is easily the most anomalous one known and this uniqueness is the primary motivation for this paper. The only level 3 exceptionals occur for $B_{2}^{(1)}\cong C_{2}^{(1)}$ and $D_{7}^{(1)}$ . The ${{B}_{2,3}}$ and ${{D}_{7,3}}$ exceptionals are cousins of the ${{\varepsilon }_{6}}$ -exceptional and ${{\varepsilon }_{8}}$ -exceptional, respectively, in the $\text{A-D-E}$ classification for $A_{1}^{(1)}$ , while the level 2 exceptionals are related to the lattice invariants of affine $u(1)$ .

Détail

DOI : 10.4153/CJM-2000-023-2

Mots-clés : 17B67, 81T40, Kac-Moody algebra, conformal field theory, modular invariants

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