  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.
