Two $2/5$-level mock theta conjecture-like identities
Stepan Konenkov, Eric T. Mortenson
TL;DR
The paper extends polar–finite decomposition techniques for fractional-level string functions to the $2/5$-level of the affine algebra $A_1^{(1)}$, producing two new families of mock theta conjecture-like identities. Each family expresses the fractional-level string functions as combinations of theta quotients and Ramanujan’s four tenth-order mock theta functions, with explicit forms given for $(p,p')=(5,12)$ in the cases $(m,ell)=(0,2r)$ and $(m,ell)=(2,2r)$. The approach combines Appell-function representations, theta identities via Frye–Garvan methods, and careful specializations (notably at $z=i$ and $z=iq^5$) to obtain to-the-point identities that parallel earlier $1/3$- and $2/3$-level results. These results deepen the connection between fractional-level string functions and mock modular objects, enriching the modularity landscape of Kac–Moody characters and their branching data. The work also provides explicit machinery—polar–finite decompositions and master theta identities—that can be reused to explore further fractional levels.
Abstract
Determining the explicit forms and modularity for string functions and branching coefficients for Kac--Moody algebras after Kac, Peterson, and Wakimoto is an important problem. In a pair of papers, Borozenets and Mortenson determined the explicit forms for fractional-level string functions for the Kac--Moody algebra $A_{1}^{(1)}$. For positive fractional-level string functions they obtained mock theta conjecture-like identities, and for negative fractional-level string functions, they obtained mixed false theta function expressions. Here we find two new families of mock theta conjecture-like identities but for the $2/5$-level string functions. Each of these two families of identities is composed of the four tenth-order mock theta functions from Ramanujan's Lost Notebook as well as a simple quotient of theta functions.