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

ABSTRACT

We recall the definition of the hyper-roots that can be associated with 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.

Keywords:

• lattices
• roots
• modular forms
• quantum groups
• affine Lie algebras
• fusion categories
• conformal field theories

2010 AMS Subject Classification:

• 17B37
• 17B67
• 18D10
• 22E46
• 52C07
• 81R10
• 81T40

Notes

¹ They are called “quantum subgroups” of SU(2) in the latter reference.

² Here and below, this means that the underlying monoidal category is ${\mathcal{A}}_k\left(G\right)$, whose definition is briefly recalled at the beginning of Section 2.1

3 Section 4.2, about D⁺₆, discusses some properties of the latter; it may have an independent interest.

4 or semi-simple, but k is then a multiplet of positive integers.

5 The proof of equivalence given in the first two references assumed a negative level. The fact that it holds in all cases has been part of the folklore for a long time because it could be verified on a case by case basis. Its general validity is now considered as a consequence of the Huang's proof of the Verlinde conjecture [Huang 05].

6 This amounts to say [Ostrik 03] that we are given a monoidal functor from ${\mathcal{A}}_k\left(G\right)$ to the category of endofunctors of an abelian category ${\mathcal{E}}_k\left(G\right)$.

7 The F_n are sometimes called “annular matrices” when ${\mathcal{A}}_k\left(G\right)$ and ${\mathcal{E}}_k\left(G\right)$ are distinct (if they are the same, then F_n = N_n), and the τ_a are sometimes called “essential matrices.”

8 This is the shifted Weyl action: w · n = w(n + ρ) − ρ where ρ is the Weyl vector.

9 meaning that we consider the fusion category ${\mathcal{A}}_k\left(G\right)$ or one of its module-categories

10 The terminology “ribbon” comes from A. Ocneanu.

11 for reasons explained in Section 2.2.4

12 actually we follow the reversed red arrows in Figure 2, which means that we use the opposite Λ, but this choice is purely conventional and plays no role in the sequel.

13 This harmonicity property is illustrated for ${\mathcal{A}}_2\left(SU3\right)$ in Figure 4.

14 Using equation (2-5(2-5) $$<\alpha ,\beta >=\sum _{w\in \mathcal{W}}ϵ\left(w\right)F_{m-n+w\rho -\rho ,a,b}.$$(2-5) ), one could define a periodic inner product on $\Lambda {\times }_{\mathcal{Z}}\mathcal{E}$ that would not be positive definite because of the periodicity, but we consider directly its non-degenerate quotient, naturally defined on the ribbon $D{\times }_{\mathcal{Z}}\mathcal{E}$.

15 The group G may be simply laced or not, but for the modules considered in this paper (choices of $\mathcal{E}$), all hyper-roots have only one possible length.

16 This general result was claimed in the last two slides of [Ocneanu 00b] and it can be explicitly checked in all the cases that we consider below.

17 If G is not simply laced, one should be careful not to use here the basis of simple co-roots.

18 It may be useful to enlarge these pictures, using an online version of the present paper.

19 Some properties of this matrix and of its inverse are investigated in one section of [Coquereaux and Zuber 16], see also [Coquereaux and Schieber 07].

20 This parameter q is not related to the root of unity, called $\mathfrak{q}$, that appears in Section 2.1

21 As Γ₁(ℓ)⊂Γ₀(ℓ), one can sometimes use modular forms (and bases of spaces of modular forms) twisted by Dirichlet characters on the congruence subgroup Γ₁(ℓ).

22 Warning: A simple counting argument shows that the lattice of hyper-roots obtained by taking k = 0 for G = SU(n) and n > 3 cannot be identified with the usual root lattice of SU(()n).