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 the modular group, up to nontrivial error terms; however, their domains (the complex upper half-plane, and the rationals, respectively) are notably different.
In this talk, we consider the (n+1)-variable combinatorial rank generating function R_n for n-marked Durfee symbols. These are n+1 dimensional multisums for n>1, and specialize to the ordinary two-variable partition rank generating function when n=1. The mock modular properties of R_n for various n and fixed parameters x_i have been previously studied by Bringmann and Ono; Bringmann; Bringmann, Garvan, and Mahlburg; and Folsom and Kimport. The quantum modular properties of R_1 follow from existing results. In our work, we prove that the combinatorial generating function R_n is a quantum modular form when viewed as a function of rationals.
This work is joint with Amanda Folsom, Min-Joo Jang, and Sam Kimport.