Quantum Modular Forms and Singular Combinatorial Series with Distinct Roots of Unity

Abstract

Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation properties under suitable subgroups of \(\mathrm {SL}_2(\mathbb {Z})\), up to nontrivial error terms; however, their domains (the upper half-plane \(\mathbb {H}\) and the rationals \(\mathbb {Q}\), respectively) are notably different. Quantum modular forms, originally defined by Zagier in 2010, have also been shown to be related to the diverse areas of colored Jones polynomials, meromorphic Jacobi forms, partial theta functions, vertex algebras, and more.