New polar-finite forms of generalized Euler identities for $A_{1}^{(1)}$-string functions and mock theta conjecture-like identities
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| Format: | Preprint |
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2026
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| _version_ | 1866915768872992768 |
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| author | Konenkov, Stepan Mortenson, Eric T. |
| author_facet | Konenkov, Stepan Mortenson, Eric T. |
| contents | 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 $χ_3(q)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_02242 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | New polar-finite forms of generalized Euler identities for $A_{1}^{(1)}$-string functions and mock theta conjecture-like identities Konenkov, Stepan Mortenson, Eric T. Number Theory Representation Theory 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 $χ_3(q)$. |
| title | New polar-finite forms of generalized Euler identities for $A_{1}^{(1)}$-string functions and mock theta conjecture-like identities |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2602.02242 |