Intertwining operators and vector-valued modular forms for minimal models

TL;DR

This paper explores the modular group representations arising from intertwining operators in Virasoro minimal model vertex operator algebras, analyzing their kernels and algebraic structures through vector-valued modular forms.

Contribution

It classifies low-dimensional representations and provides arithmetic criteria for when these representations have noncongruence kernels, advancing understanding of modular forms in VOAs.

Findings

Classified all irreducible module representations of dimension less than four.

Identified conditions for kernels to be congruence or noncongruence subgroups.

Compared spaces of 1-point functions with holomorphic vector-valued modular forms.

Abstract

Using the language of vertex operator algebras (VOAs) and vector-valued modular forms we study the modular group representations and spaces of 1-point functions associated to intertwining operators for Virasoro minimal model VOAs. We examine all representations of dimension less than four associated to irreducible modules for minimal models, and determine when the kernel of these representations is a congruence or noncongruence subgroup of the modular group. Arithmetic criteria are given on the indexing of the irreducible modules for minimal models that imply the associated modular group representation has a noncongruence kernel, independent of the dimension of the representation. The algebraic structure of the spaces of 1-point functions for intertwining operators is also studied, via a comparison with the associated spaces of holomorphic vector-valued modular forms.