New polar-finite forms of generalized Euler identities for $A_{1}^{(1)}$-string functions and mock theta conjecture-like identities

TL;DR

This paper develops new finite sum identities for affine Kac--Moody algebra $A_{1}^{(1)}$ string functions, revealing modular and mock modular structures, and introduces identities akin to mock theta conjectures for specific levels.

Contribution

It introduces polar-finite forms of generalized Euler identities for $A_{1}^{(1)}$ string functions, connecting them to mock modular and theta functions, and extends these identities to admissible levels with new mock theta-like identities.

Findings

Finite sum representations for string functions at integral levels.

New mock theta conjecture-like identities for levels 1/2, 1/3, 2/3.

Expressions involving Ramanujan's mock theta functions.

Abstract

Determining the explicit forms and modularity for string functions and branching coefficients for Kac--Moody algebras after Kac, Peterson, and Wakimoto is an important problem. For positive admissible-level string functions for the affine Kac--Moody algebra $A_{1}^{(1)}$, very little is known. Here we apply the notion of quasi-periodicity to a generalized Euler identity of Schilling and Warnaar for the affine Kac--Moody algebra $A_{1}^{(1)}$. For integral-level string functions the classical periodicity reduces the infinite sum of string functions in the generalized Euler identity to a finite sum of string functions with theta function coefficients. For admissible-level, we similarly reduce to an analogous finite sum of string functions, but we also gain an additional finite sum of the form \begin{equation*} \sum_{i}\Phi_{i}(q)\Psi_{i}(q), \end{equation*} where the $\Phi_i(q)$'s are modular and depend only on the spin and the $\Psi_{i}(q)$'s are (mixed) mock modular Hecke-type double-sums and depend only on the quantum number. For levels $1/2$, $1/3$, and $2/3$, we shall also see that the $\Psi_{i}(q)$'s give us families of mock theta conjecture-like identities for symmetric Hecke-type double-sums. Our work here focuses on evaluating the $\Psi_{i}(q)$'s, and our expressions utilize Ramanujan's second-order mock theta function $\mu_2(q)$ and third-order mock theta functions $f_{3}(q)$, $\omega_3(q)$, $\psi_{3}(q)$, and $\chi_3(q)$.