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

TL;DR

This review explores the interplay between string theory, Lie algebra representations, and geometric invariants, emphasizing modular properties and their applications to elliptic genera and quantum field theories.

Contribution

It provides a comprehensive overview of the connections between infinite dimensional Lie algebras, vertex operator algebras, and Hilbert schemes, highlighting their role in physical and mathematical invariants.

Findings

Elliptic genera relate to string partition functions and geometric invariants.

Modular properties of Lie algebra characters underpin their physical applications.

Product expressions of elliptic genera inherit homology properties of Lie algebras.

Abstract

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.