  This review paper contains a concise introduction to highest weight
representations of infinite dimensional Lie algebras, vertex operator algebras
and Hilbert schemes of points, together with their physical applications to
elliptic genera of superconformal quantum mechanics and superstring models. The
common link of all these concepts and of the many examples considered in the
paper is to be found in a very important feature of the theory of infinite
dimensional Lie algebras: the modular properties of the characters (generating
functions) of certain representations. The characters of the highest weight
modules represent the holomorphic parts of the partition functions on the torus
for the corresponding conformal field theories. We discuss the role of the
unimodular (and modular) groups and the (Selberg-type) Ruelle spectral
functions of hyperbolic geometry in the calculation of elliptic genera and
associated $q$-series. For mathematicians, elliptic genera are commonly
associated to new mathematical invariants for spaces, while for physicists
elliptic genera are one-loop string partition function (therefore they are
applicable, for instance, to topological Casimir effect calculations). We show
that elliptic genera can be conveniently transformed into product expressions
which can then inherit the homology properties of appropriate polygraded Lie
algebras.
