On Spin Models, Modular Invariance, and Duality

A spin model is a triple (X,W+,W−), where W+ and W− are complex matrices with rows and columns indexed by X which satisfy certain equations (these equations allow the construction of a link invariant from (X,W+,W−)). We show that these equations imply the existence of a certain isomorphism 9 between two algebras M and H associated with (X,W+,W−). When M = H = A, A is the Bose-Mesner algebra of some association scheme, and 9 is a duality of A. These results had already been obtained in [15] when W+,W− are symmetric, and in [5] in the general case, but the present proof is simpler and directly leads to a clear reformulation of the modular invariance property for self-dual association schemes. This reformulation establishes a correspondence between the modular invariance property and the existence of “spin models at the algebraic level”. Moreover, for Abelian group schemes, spin models at the algebraic level and actual spin models coincide. We solve explicitly the modular invariance equations in this case, obtaining generalizations of the spin models of Bannai and Bannai [3]. We show that these spin models can be identified with those constructed by Kac and Wakimoto [20] using even rational lattices. Finally we give some examples of spin models at the algebraic level which are not actual spin models.