Two $2/5$-level mock theta conjecture-like identities

TL;DR

This paper discovers two new families of identities related to $2/5$-level string functions in Kac--Moody algebras, expanding the understanding of mock theta functions and their modular properties.

Contribution

It introduces two novel families of mock theta conjecture-like identities specifically for $2/5$-level string functions, building on prior work on fractional-level string functions.

Findings

Identified two new families of mock theta conjecture-like identities.

Each family involves Ramanujan's tenth-order mock theta functions.

The identities include simple theta function quotients.

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.