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 SL 2 pZq, up to nontrivial error terms; however, their domains (the upper half-plane H, and the rationals 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.

In 2007, G.E. Andrews introduced the (n+1)-variable combinatorial generating function R n (x 1 , x 2 , · · · , x n ; q) for ranks of n-marked Durfee symbols, an (n + 1)-dimensional multisum, as a vast generalization to the ordinary two-variable partition rank generating function. Since then, it has been a problem of interest to understand the automorphic properties of this function; in special cases and under suitable specializations of parameters, R n has been shown to possess modular, quasimodular, and mock modular properties when viewed as a function on the upper half complex plane H, in work of Bringmann, Folsom, Garvan, Kimport, Mahlburg, and Ono. Quantum modular forms, defined by Zagier in 2010, are similar to modular or mock modular forms but are defined on the rationals Q as opposed to H, and exhibit modular transformations there up to suitably analytic error functions in R; in general, they have been related to diverse areas including number theory, topology, and representation theory. Here, we establish quantum modular properties of R n .

No abstract

No abstract