Theta functions for lattices of SU(3) hyper-roots

TL;DR

This paper explores the structure of hyper-root lattices associated with SU(3) modules, analyzing their theta functions and expressing them as modular forms with Dirichlet characters, extending previous work on SU(2).

Contribution

It introduces explicit theta function calculations for SU(3) hyper-root lattices, expanding the understanding of their modular properties and generalizing earlier SU(2) results.

Findings

Theta functions expressed as modular forms with Dirichlet characters

Explicit determination of SU(3) hyper-root lattice theta functions

Extension of hyper-root concepts from SU(2) to SU(3)

Abstract

We recall the definition of the hyper-roots that can be associated to modules-categories over the 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.