On string functions and double-sum formulas

TL;DR

This paper explores new relations between string functions of affine Kac--Moody algebras by expressing them as double-sums and connecting these to Appell--Lerch and theta functions, revealing links to mock theta functions.

Contribution

It introduces novel double-sum formulas for string functions and relates them to special functions like Appell--Lerch and theta functions, expanding understanding of their structure.

Findings

Derived new double-sum formulas for string functions

Expressed Hecke-type double-sums in terms of Appell--Lerch and theta functions

Connected string functions to Ramanujan's mock theta functions

Abstract

String functions are important building blocks of characters of integrable highest modules over affine Kac--Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type $A_{1}^{(1)}$ in terms of Dedekind eta functions. We produce new relations between string functions by writing them as double-sums and then using certain symmetry relations. We evaluate the series using special double-sum formulas that express Hecke-type double-sums in terms of Appell--Lerch functions and theta functions, where we point out that Appell--Lerch functions are the building blocks of Ramanujan's classical mock theta functions.