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This review paper contains a concise introduction to highest weightrepresentations of infinite dimensional Lie algebras, vertex operator algebrasand Hilbert schemes of points, together with their physical applications toelliptic genera of superconformal quantum mechanics and superstring models. Thecommon link of all these concepts and of the many examples considered in thepaper is to be found in a very important feature of the theory of infinitedimensional Lie algebras: the modular properties of the characters (generatingfunctions) of certain representations. The characters of the highest weightmodules represent the holomorphic parts of the partition functions on the torusfor the corresponding conformal field theories. We discuss the role of theunimodular (and modular) groups and the (Selberg-type) Ruelle spectralfunctions of hyperbolic geometry in the calculation of elliptic genera andassociated $q$-series. For mathematicians, elliptic genera are commonlyassociated to new mathematical invariants for spaces, while for physicistselliptic genera are one-loop string partition function (therefore they areapplicable, for instance, to topological Casimir effect calculations). We showthat elliptic genera can be conveniently transformed into product expressionswhich can then inherit the homology properties of appropriate polygraded Liealgebras.

String Partition Functions, Hilbert Schemes, and Affine Lie Algebra Representations on Homology Groups