SU(3) higher roots and their lattices

TL;DR

This paper explores the theta series of higher root lattices associated with quantum modules of SU(3), expressing them through modular forms and extending known results from SU(2) to more complex structures.

Contribution

It introduces the study of theta series for higher roots related to SU(3) quantum modules, expanding the understanding of their lattice structures and modular form representations.

Findings

Theta series expressed via modular forms with Dirichlet characters

Identification of specific modules for SU(3) with well-defined theta series

Extension of SU(2) root lattice results to SU(3) case

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.