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The Level 2 and 3 Modular Invariants for the Orthogonal Algebras

  Published:2000-06-01
 Printed: Jun 2000
  • Terry Gannon
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Abstract

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~$u(1)$.
Keywords: Kac-Moody algebra, conformal field theory, modular invariants Kac-Moody algebra, conformal field theory, modular invariants
MSC Classifications: 17B67, 81T40 show english descriptions Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Two-dimensional field theories, conformal field theories, etc.
17B67 - Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
81T40 - Two-dimensional field theories, conformal field theories, etc.
 

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