String partition functions, Hilbert schemes and affine Lie algebra representations on homology groups
Abstract
This review paper contains a concise introduction to highest weight representations of infinitedimensional 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 this paper is to be found in a very important feature of the theory of infinitedimensional 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 (Selbergtype) Ruelle spectral functions of hyperbolic geometry in the calculation of elliptic genera and associated qseries. For mathematicians, elliptic genera are commonly associated with new mathematical invariants for spaces, while for physicists elliptic genera are oneloop 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.
This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker’s 75th birthday devoted to ‘Applications of zeta functions and other spectral functions in mathematics and physics’.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 September 2012
 DOI:
 10.1088/17518113/45/37/374002
 arXiv:
 arXiv:1206.0664
 Bibcode:
 2012JPhA...45K4002B
 Keywords:

 High Energy Physics  Theory
 EPrint:
 56 pages, review paper, in honour of J.S.Dowker. arXiv admin note: text overlap with arXiv:0905.1285, arXiv:math/0006201, arXiv:math/0412089, arXiv:math/0403547 by other authors