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

TL;DR

The article develops polar-finite decompositions for $A_{1}^{(1)}$ string functions at positive admissible levels by applying quasi-periodicity to Schilling–Warnaar’s generalized Euler identities. It achieves finite sums plus an auxiliary mock-modular term with coefficients factorizing into modular spins and mock-theta components, with explicit constructions for levels $1/2$, $1/3$, and $2/3$ and for both even- and odd-spin cases. The authors express the double-sum coefficients in terms of Ramanujan’s mock theta functions $ ext{μ}_2(q)$, $f_3(q)$, $\\omega_3(q)$, $\\psi_3(q)$, and $\\chi_3(q)$, yielding families of mock-theta conjecture-like identities for symmetric Hecke-type sums. These results connect affine Kac–Moody string function modularity with Appell-function and mock-theory, providing new polar-finite forms and a broad set of mock-theta identities with potential implications for representation theory and number theory.

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}Φ_{i}(q)Ψ_{i}(q), \end{equation*} where the $Φ_i(q)$'s are modular and depend only on the spin and the $Ψ_{i}(q)$'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 $Ψ_{i}(q)$'s give us families of mock theta conjecture-like identities for symmetric Hecke-type double-sums. Our work here focuses on evaluating the $Ψ_{i}(q)$'s, and our expressions utilize Ramanujan's second-order mock theta function $μ_2(q)$ and third-order mock theta functions $f_{3}(q)$, $ω_3(q)$, $ψ_{3}(q)$, and $χ_3(q)$.

Theorems & Definitions (135)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 1.6
• Remark 1.7
• Theorem 2.1
• Corollary 2.2
• Corollary 2.3