The Level 2 and 3 Modular Invariants for the Orthogonal Algebras

http://dx.doi.org/10.4153/CJM-2000-023-2
Canad. J. Math. 52(2000), 503-521

  Published:2000-06-01
 Printed: Jun 2000

The `1-loop partition function' of a rational conformal field theory is a sesquilinear combination of characters, invariant under a natural action of $\SL_2(\bbZ)$, 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 $B_r^{(1)}$ 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 ${\cal E}_6$-exceptional and $\E_8$-exceptional, respectively, in the A-D-E classification for $A_1^{(1)}$, while the level 2 exceptionals are related to the lattice invariants of affine~$u(1)$.