Theta Functions for Lattices of SU(3) Hyper-Roots

Abstract

We recall the definition of the hyper-roots that can be associated to modules-categories over fusion categories defined by the choice of a simple Lie group G together with a positive integer k. This definition was proposed in 2000, using another language, by Adrian Ocneanu. If G = SU(2), the obtained hyper-roots coincide with the usual roots for ADE Dynkin diagrams. We consider the associated lattices when G = SU(3) and determine their theta functions in a number of cases; these functions can be expressed as modular forms twisted by appropriate Dirichlet characters. arXiv:1708.00560v1 [math.QA] 2 Aug 2017The ADE correspondence between indecomposable module-categories of type SU(2) and simplylaced Dynkin diagrams was first obtained by theoretical physicists in the framework of conformal field theories (classification of modular invariant partition functions for the WZW models of type SU(2), [1], [8]). Its relation with subfactors was studied in [19] and it was set in a categorical framework by [16,24]. In plain terms, the diagrams encoding the action of the fundamental representation of SU(2) at level k (which is classically 2-dimensional) on the simple objects of the various module-categories existing at that level, are the Dynkin diagrams describing the simplylaced simple Lie groups with Coxeter number k + 2.At a deeper level, there is a correspondence between fusion coefficients of the SU(2) modulecategory described by a Dynkin diagram E and the inner products between all the roots of the simply-laced Lie group associated with the same Dynkin diagram. In the non-ADE cases one can also define the action of an appropriate ring on modules associated with the chosen Dynkin diagrams and still obtain a correspondence between structure coefficients describing this action and inner products between roots -one has only to introduce scaling coefficients in appropriate places.The correspondence relating fusion coefficients for module-categories of type SU(2) and inner products between weights and/or roots of root systems was clearly stated (but not much discussed) in [20]; in a different context, some of its aspects were already present in the article [10]. The correspondence was used and described in some detail in one section of [3]. As observed in [20], one can start from the SU(2) fusion categories and their modules 1 to recover or define the usual root systems, and associate with each of them a periodic quiver describing, in particular, the inner products between all the roots. It is also observed in the same reference that the construction can be generalized: replacing SU(2) by an arbitrary simple Lie group G leads, for every choice of a module-category E of type 2 G, to a system of "higher roots" that we call "hyper-roots of type G". Usual root systems are therefore hyper-root systems of type SU(2). A usual root system gives rise, in particular, to an Euclidean lattice. The same is true for hyper-root systems. Given a lattice, one may consider its theta series whose n th coefficient gives the number of vectors of g...