On odd-spin $A_1^{(1)}$-string functions, cross-spin identities, and mock theta conjecture-like identities

Determining the explicit forms and modularity for string functions and branching coefficients for Kac–Moody algebras after Kac, Peterson, and Wakimoto is a long-standing, yet wide-open, problem and recently a connection has been made between positive admissible-level $A_1^{(1)}$-string functions and Ramanujan’s mock theta functions. In this paper we obtain the polar-finite decomposition for the admissible-level $A_1^{(1)}$ character of odd spin, and we also find new mock theta conjecture-like identities for the odd-spin, $2\mathrm{/}3$-level and $2\mathrm{/}5$-level $A_1^{(1)}$-string functions.