Vector-valued modular functions for the modular group and the hypergeometric equation
P. Bantay, T. Gannon
TL;DR
The paper develops a complete theory of vector-valued modular functions for finite-dimensional representations $\rho$ of the modular group, including half-integer weights, by embedding the spaces into a free $\mathbb{C}[J]$-module generated by a fundamental matrix $\mathbf{\Xi}(\tau)$. It recasts the compatibility between recursion and differentiation as an abstract hypergeometric equation in a modular-invariant variable $\mathfrak{z}$, whose monodromy equals $\rho$ and yields a solution of the Riemann–Hilbert problem; it further provides an explicit inversion formula to reconstruct any $\mathbb{X}\in\mathcal{M}(\rho)$ from its principal part, and derives signatures, dimension formulas, and rationality/integrality criteria tied to congruence subgroups. The work includes concrete low-dimensional examples and discusses positivity/integrality constraints, with potential extensions to other genus-zero groups, arbitrary weight, and infinite-dimensional representations, as well as a reduction procedure to simplified representations capturing the essential modular data. Applications to conformal field theory and vertex operator algebras are highlighted, illustrating how modular data control character vectors and their $q$-expansions.
Abstract
A general theory of vector-valued modular functions, holomorphic in the upper half-plane, is presented for finite dimensional representations of the modular group. This also provides a description of vector-valued modular forms of arbitrary half-integer weight. It is shown that the space of these modular functions is spanned, as a module over the polynomials in J, by the columns of a matrix that satisfies an abstract hypergeometric equation, providing a simple solution of the Riemann-Hilbert problem for representations of the modular group. Restrictions on the coefficients of this differential equation implied by analyticity are discussed, and an inversion formula is presented that allows the determination of an arbitrary vector-valued modular function from its singular behavior. Questions of rationality and positivity of expansion coefficients are addressed. Closed expressions for the number of vector-valued modular forms of half-integer weight are given, and the general theory is illustrated on simple examples.