On string functions of the generalized parafermionic theories, mock theta functions, and false theta functions, II
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917977164611584 |
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| author | Borozenets, Nikolay Mortenson, Eric T. |
| author_facet | Borozenets, Nikolay Mortenson, Eric T. |
| contents | Kac and Wakimoto introduced the admissible highest weight representations as a conjectural classification of all modular-invariant representations of the affine Kac--Moody algebras. For the affine Kac--Moody algebra $A_1^{(1)}$ their conjectural construction has been proved. Using their construction, Ahn, Chung, and Tye introduced the generalized Fateev--Zamolodchikov parafermionic theories. The characters of these parafermionic theories are string functions of admissible representations of $A_1^{(1)}$ up to a simple appropriate factor. Determining modular properties or explicitly calculating string functions and branching coefficients is an important yet wide-open problem. Outside of initial works of Kac, Peterson, and Wakimoto, little is known. Here we take a new approach by first developing a quasi-periodic notion of admissible string functions and then calculating the Zagier--Zwegers' polar-finite decomposition for the admissible characters. As an application of the decomposition, we extend the results of our paper (Borozenets and Mortenson, 2024) for the affine Kac--Moody algebra $A_1^{(1)}$, in that we obtain families of new mock theta conjecture-like identities for $1/3$ and $2/3$-level string functions in terms of Ramanujan's mock theta functions $f_3(q)$ and $ω_3(q)$. We also obtain an analogous family of new identities for the $1/5$-level string functions in terms of Ramanujan's four tenth-order mock theta functions. In addition, we give a heuristic argument for an expansion of the general positive-level admissible string functions in terms of Appell functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03074 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On string functions of the generalized parafermionic theories, mock theta functions, and false theta functions, II Borozenets, Nikolay Mortenson, Eric T. Number Theory Mathematical Physics Representation Theory Kac and Wakimoto introduced the admissible highest weight representations as a conjectural classification of all modular-invariant representations of the affine Kac--Moody algebras. For the affine Kac--Moody algebra $A_1^{(1)}$ their conjectural construction has been proved. Using their construction, Ahn, Chung, and Tye introduced the generalized Fateev--Zamolodchikov parafermionic theories. The characters of these parafermionic theories are string functions of admissible representations of $A_1^{(1)}$ up to a simple appropriate factor. Determining modular properties or explicitly calculating string functions and branching coefficients is an important yet wide-open problem. Outside of initial works of Kac, Peterson, and Wakimoto, little is known. Here we take a new approach by first developing a quasi-periodic notion of admissible string functions and then calculating the Zagier--Zwegers' polar-finite decomposition for the admissible characters. As an application of the decomposition, we extend the results of our paper (Borozenets and Mortenson, 2024) for the affine Kac--Moody algebra $A_1^{(1)}$, in that we obtain families of new mock theta conjecture-like identities for $1/3$ and $2/3$-level string functions in terms of Ramanujan's mock theta functions $f_3(q)$ and $ω_3(q)$. We also obtain an analogous family of new identities for the $1/5$-level string functions in terms of Ramanujan's four tenth-order mock theta functions. In addition, we give a heuristic argument for an expansion of the general positive-level admissible string functions in terms of Appell functions. |
| title | On string functions of the generalized parafermionic theories, mock theta functions, and false theta functions, II |
| topic | Number Theory Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2502.03074 |