On string functions of the generalized parafermionic theories, mock theta functions, and false theta functions, II
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.