  Every four-dimensional ${\cal N}=2$ superconformal field theory comes
equipped with an intricate algebraic invariant, the associated vertex operator
algebra. The relationships between this invariant and more conventional
protected quantities in the same theories have yet to be completely understood.
In this work, we aim to characterize the connection between the Higgs branch of
the moduli space of vacua (as an algebraic geometric entity) and the associated
vertex operator algebra. Ultimately our proposal is simple, but its correctness
requires the existence of a number of nontrivial null vectors in the vacuum
Verma module of the vertex operator algebra. Of particular interest is one such
null vector whose presence suggests that the Schur index of any ${\cal N}=2$
SCFT should obey a finite order modular differential equation. By way of the
"high temperature" limit of the superconformal index, this allows the Weyl
anomaly coefficient $a$ to be reinterpreted in terms of the representation
theory of the associated vertex operator algebra. We illustrate these ideas in
a number of examples including a series of rank-one theories associated with
the "Deligne-Cvitanovi\'c exceptional series" of simple Lie algebras, several
families of Argyres-Douglas theories, an assortment of class ${\cal S}$
theories, and ${\cal N}=4$ super Yang-Mills with $\mathfrak{su}(n)$ gauge group
for small-to-moderate values of $n$.
