The level 2 and 3 modular invariants for the orthogonal algebras

This paper finds for all orthogonal algebras (i.e. the B and D series) all modular invariant 1-loop partition functions at levels 1,2,3. Previously, only those at level 1 were classified. An extraordinary number of exceptionals appear at level 2 -- indeed this is the primary motivation for this paper -- and we find infinitely many new ones there. The only level 3 exceptionals occur for $B_2$ and $D_7$, and the latter appear to be new. The $B_{2,3}$ and $D_{7,3}$ exceptionals are cousins of the "$E_6$" and "$E_8$" exceptionals in the A-D-E classification for affine $A_1$, while the level 2 exceptionals are related to the lattice invariants of affine u(1).