2026/02/02 by Stepan Konenkov, Eric T. Mortenson
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models
paper · pdf · doi:10.1007/s40687-026-00645-8
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 A1(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 A1(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 ∑iΦi(q)Ψi(q), where the Φi(q)'s are modular and depend only on the spin and the Ψi(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 Ψi(q)'s give us families of mock theta conjecture-like identities for symmetric Hecke-type double-sums. Our work here focuses on evaluating the Ψi(q)'s, and our expressions utilize Ramanujan's second-order mock theta function μ2(q) and third-order mock theta functions f3(q), ω3(q), ψ3(q), and χ3(q).