Elliptic Genera and Characteristic $q$-Series of Superconformal Field Theory

TL;DR

This paper explores the analytic forms of various $q$-series related to superconformal field theory, revealing deep connections with elliptic modular forms, spectral functions, and algebraic identities.

Contribution

It establishes a novel correspondence between characteristic series of the Witten genus, $KO$-theory, and algebraic structures like superconformal algebras and affine Lie superalgebras.

Findings

$q$-series relate to spectral Ruelle functions and elliptic modular forms.

Deep correspondence between characteristic series and algebraic identities.

Representation of series as double series via Hecke-Rogers identity.

Abstract

We analyze the characteristic series, the $KO$ series and the series associated with the Witten genus, and their analytic forms as the $q$-analogs of classical special functions (in particular $q$-analog of the beta integral and the gamma function). $q$-series admit an analytic interpretation in terms of the spectral Ruelle functions, and their relations to appropriate elliptic modular forms can be described. We show that there is a deep correspondence between the characteristic series of the Witten genus and $KO$ characteristic series, on one side, and the denominator identities and characters of $N=2$ superconformal algebras, and the affine Lie (super)algebras on the other. We represent the characteristic series in the form of double series using the Hecke-Rogers modular identity.