The modular properties of $\mathfrak{sl}(2)$ torus $1$-point functions

TL;DR

This paper addresses the modular properties of torus 1-point functions in chiral CFTs for the simple affine $\mathfrak{sl}(2)$ VOA $L(k,0)$ at $k \in \mathbb{Z}_{\ge0}$, extending analysis beyond vacuum torus 1-point functions by allowing insertions from simple modules. It develops a framework that identifies these traces with vector-valued modular forms of weight depending on the insertion, realized via representations $\rho_\lambda$ and multiplier systems $\nu_{h_\lambda}$, and organizes them using modular differential operators in the algebra $\mathcal{R}$. The main results include dimensions and bases for spaces of torus 1-point functions, explicit generators for vector-valued modular forms of dimensions up to three, and a concrete description of the $\mathsf{SL}(2,\mathbb{Z})$ action in terms of categorical data (twists and fusing matrices) from the modular tensor category of $L(k,0)$-modules; this yields both congruence and non-congruence representations and, in some cases, irreducible four-dimensional representations. The work thus ties VOA/intertwining-operator theory, analytic vector-valued modular forms, and modular tensor-category data, providing tools for higher-genus CFTs and potential new invariants beyond the usual $\mathsf{S}$- and $\mathsf{T}$-matrix data.

Abstract

Conformal field theory and its axiomatisation in terms of vertex operator algebras or chiral algebras are most commonly considered on the Riemann sphere. However, an important constraint in physics and an interesting source of mathematics is the fact that conformal field theories are expected to be well defined on any Riemann surface. To this end, a thorough understanding of chiral torus 1-point functions, ideally including explicit formulae, is a prerequisite for a detailed understanding of higher genera. These are distinguished from characters or vacuum torus 1-point functions because the insertion point is explicitly allowed to be labelled by any module over the vertex operator algebra rather than just the vertex operator algebra itself. Compellingly, chiral torus 1-point functions exhibit interesting modular properties, which we explore here in the example of the simple affine $\mathfrak{sl}(2)$ vertex operator algebras at non-negative integral levels. We determine the dimension of the space spanned by such functions, choose a natural basis to construct vector-valued modular forms and describe the congruence properties of these forms. In particular, we find explicit generators for the spaces of all vector-valued modular forms of dimension at most three, when the insertion comes from a simple module other than the vertex operator algebra. Finally, we use the fact that categories of modules over rational vertex operator algebras are modular tensor categories to give explicit formulae for the action of the modular group on chiral torus 1-point functions entirely in terms of categorical data. The usual modular $\mathsf{S}$ and $\mathsf{T}$ matrices of characters are known not to be complete invariants of modular tensor categories, so these generalised modular data are good candidates for more fine-grained invariants.