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

Kac and Wakimoto introduced the admissible highest weight representations in order to classify all modular invariant representations of the Kac--Moody algebras. For the Kac--Moody algebra $A_1^{(1)}$ the string functions of admissible representations are allowed to have certain rational levels and were realized by Ahn, Chung, and Tye as the characters of the generalized Fateev--Zamolodchikov parafermionic theories. For the $1/2$-level string functions, we present their mixed mock modular properties as well as elegant mock theta conjecture-like identities involving two mock theta functions from Ramanujan's Lost Notebook. In addition, we demonstrate that the negative level string functions can be evaluated in terms of false theta functions.