The Level 2 and 3 Modular Invariants for the Orthogonal Algebras

Abstract

AbstractThe ‘1-loop partition function’ of a rational conformal field theory is a sesquilinear combination of characters, invariant under a natural action of SL2(), 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 Br(1) and Dr(1) all of these at level k ≤ 3. Previously, only those at level 1 were classified. An extraordinary number of exceptionals appear at level 2—the Br(1), Dr(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 B2(1) ≅ C2(1) and D7(1). The B2,3 and D7,3 exceptionals are cousins of the ε6-exceptional and ε8-exceptional, respectively, in the A-D-E classification for A1(1), while the level 2 exceptionals are related to the lattice invariants of affine u(1).