Communications in Number Theory and Physics

Volume 12 (2018)

Number 4

Intertwining operators and vector-valued modular forms for minimal models

Pages: 657 – 686



Matthew Krauel (Department of Mathematics and Statistics, California State University, Sacramento, Calif., U.S.A.)

Christopher Marks (Department of Mathematics and Statistics, California State University, Chico, Calif., U.S.A.)


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.

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Portions of Matthew Krauel’s research and preliminary work for this paper were supported by the European Research Council (ERC) Grant agreement n. 335220– AQSER, as well as by the Japan Society of the Promotion of Science (JSPS), No. P13013.

Received 4 April 2017

Accepted 15 June 2018

Published 14 January 2019