© 2014 Springer Science+Business Media New York. All rights reserved. Highest-weight representations of infinite-dimensional Lie algebras and Hilbert schemes of points are considered, together with the applications of these concepts to partition functions, which are most useful in physics. Partition functions (elliptic genera) are conveniently transformed into product expressions, which may inherit the homology properties of appropriate (poly)graded Lie algebras. Specifically, the role of (Selberg-type) Ruelle spectral functions of hyperbolic geometry in the calculation of partition functions and associated q-series is discussed. Examples of these connections in quantum field theory are considered (in particular, within the AdS/CFT correspondence), as the AdS3 case where one has Ruelle/Selberg spectral functions, whereas on the CFT side, partition functions and modular forms arise. These objects are here shown to have a common background, expressible in terms of Euler-Poincaré and Macdonald identities, which, in turn, describe homological aspects of (finite or infinite) Lie algebra representations. Finally, some other applications of modular forms and spectral functions (mainly related with the congruence subgroup of) to partition functions, Hilbert schemes of points, and symmetric products are investigated by means of homological and K-theory methods.
|Title of host publication||Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava|
|Number of pages||17|
|Publication status||Published - 1 Nov 2014|