SU(3) higher roots and their lattices

Abstract

After recalling the notion of higher roots (or hyper-roots) associated with "quantum modules" of type $(G, k)$, for $G$ a semi-simple Lie group and $k$ a positive integer, following the definition given by A. Ocneanu in 2000, we study the theta series of their lattices. Here we only consider the higher roots associated with quantum modules (aka module-categories over the fusion category defined by the pair $(G,k)$) that are also "quantum subgroups". For $G=SU{2}$ the notion of higher roots coincides with the usual notion of roots for ADE Dynkin diagrams and the self-fusion restriction (the property of being a quantum subgroup) selects the diagrams of type $A_{r}$, $D_{r}$ with $r$ even, $E_6$ and $E_8$; their theta series are well known. In this paper we take $G=SU{3}$, where the same restriction selects the modules ${\mathcal A}_k$, ${\mathcal D}_k$ with $mod(k,3)=0$, and the three exceptional cases ${\mathcal E}_5$, ${\mathcal E}_9$ and ${\mathcal E}_{21}$. The theta series for their associated lattices are expressed in terms of modular forms twisted by appropriate Dirichlet characters.