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

TL;DR

This paper explores the connection between odd-spin $A_{1}^{(1)}$-string functions and mock theta functions, providing explicit decompositions and new identities that extend understanding of modularity in Kac--Moody algebra representations.

Contribution

It derives the polar-finite decomposition for odd-spin admissible-level $A_{1}^{(1)}$ characters and introduces new mock theta conjecture-like identities for specific levels.

Findings

Polar-finite decomposition for odd-spin $A_{1}^{(1)}$-characters.

New mock theta conjecture-like identities at 2/3 and 2/5 levels.

Enhanced understanding of modularity in string functions.

Abstract

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, $2/3$-level and $2/5$-level $A_{1}^{(1)}$-string functions.