Mock modular forms and quantum modular forms

Abstract:

In his last letter to Hardy, Ramanujan introduced mock theta functions. For each of his examples $f(q)$, Ramanujan claimed that there is a collection $\{ G_j\}$ of modular forms such that for each root of unity $\zeta$, there is a $j$ such that \[ \lim _{q \to \zeta }(f(q) - G_j(q)) = O(1).\] Moreover, Ramanujan claimed that this collection must have size larger than $1$. In his 2001 PhD thesis, Zwegers showed that the mock theta functions are the holomorphic parts of harmonic weak Maass forms. In this paper, we prove that there must exist such a collection by establishing a more general result for all holomorphic parts of harmonic Maass forms. This complements the result of Griffin, Ono, and Rolen that shows such a collection cannot have size $1$. These results arise within the context of Zagier’s theory of quantum modular forms. A linear injective map is given from the space of mock modular forms to quantum modular forms. Additionally, we provide expressions for “Ramanujan’s radial limits” as $L$-values.