Quantum modular forms and singular combinatorial series with repeated roots of unity

Abstract

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 .

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

“…Here, the subgroup Γ n Ď SL 2 pZq under which p Apζ n ; qq transforms is defined by 11) and the Nebentypus character χ γ is given in Lemma 2.1.…”

“…In this setting, as shown in [12], the nonholomorphic completion of q´1 24 R n pζ n ; qq is not modular, but is instead a sum of two (nonholomorphic) modular forms of different weights. We will address this more general case in a forthcoming paper [11].…”

“…Remark 1.10. In a forthcoming joint work [11], we extend Theorem 1.7 to hold for the more general vectors of roots of unity considered in [12], i.e., those with repeated entries. Allowing repeated roots of unity introduces additional singularities, and the modular completion of R n is significantly more complicated.…”

An analogue of $k$-marked Durfee symbols for strongly unimodal sequences

“…, ω k ; qq with k ě 2 is a type of mixed mock modular form. Then in 2018, Folsom, Jang, Kimport, and the fourth author [15,16] proved that R k pω 1 , . .…”

Quantum modularity of partial theta series with periodic coefficients

“…Suitable modifications can be made to restrict the domain of r γ to appropriate subsets of Q and allow both multiplier systems and transformations on subgroups of SL 2 (Z). Since their inception, there has been substantial interest in studying these modular objects which emerge in diverse contexts: Maass forms [9], supersymmetric quantum field theory [12], topological invariants for plumbed 3-manifolds [5], [10], [11], combinatorics [13], [20], unified Witten-Reshetikhin-Turaev invariants [19] and L-functions [24], [26]. For more examples, see Chapter 21 in [4].…”