On Partition Functions of Hyperbolic Three-Geometry and Associated Hilbert Schemes

TL;DR

This paper explores the deep connections between hyperbolic geometry, Lie algebra representations, and partition functions, revealing new insights into their algebraic and geometric structures with applications in quantum field theory and string theory.

Contribution

It establishes a unified framework linking spectral functions, modular forms, and homological identities to analyze partition functions and Hilbert schemes in hyperbolic geometry and physics.

Findings

Partition functions can be expressed as product formulas using spectral functions.

Homological and K-theoretic methods elucidate the structure of partition functions and Hilbert schemes.

Connections between hyperbolic geometry, Lie algebra representations, and modular forms are demonstrated.

Abstract

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 are discussed. Examples of these connection in quantum field theory are considered (in particular, within the AdS/CFT correspondence), as the AdS$_{3}$ 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-Poincar\'e 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 $SL(2, {\mathbb Z})$) to partition functions, Hilbert schemes of points, and symmetric products are investigated by means of homological and K-theory methods.