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

  • Amanda Folsom
  • Min-Joo Jang
  • Sam Kimport
  • Holly SwisherEmail author
Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 19)

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.

In this paper we study the (n + 1)-variable combinatorial rank generating function Rn(x1, x2, …, xn;q) for n-marked Durfee symbols. These are n + 1 dimensional multi-sums for n > 1, and specialize to the ordinary two variable partition rank generating function when n = 1. The mock modular properties of Rn when viewed as a function of \(\tau \in \mathbb {H}\), with q = e2πiτ, for various n and fixed parameters x1, x2, ⋯ , xn, have been studied in a series of papers. Namely, by Bringmann and Ono when n = 1 and x1 a root of unity; by Bringmann when n = 2 and x1 = x2 = 1; by Bringmann, Garvan, and Mahlburg for n ≥ 2 and x1 = x2 = ⋯ = xn = 1; and by the first and third authors for n ≥ 2 and the xj suitable roots of unity (1 ≤ j ≤ n).

The quantum modular properties of R1 readily follow from existing results. Here, we focus our attention on the case n ≥ 2, and prove for any n ≥ 2 that the combinatorial generating function Rn is a quantum modular form when viewed as a function of \(x \in \mathbb {Q}\), where q = e2πix, and the xj are suitable distinct roots of unity.

Keywords

Quantum modular forms Mock modular forms Modular forms Durfee symbols Combinatorial rank functions Partitions 

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Copyright information

© The Author(s) and The Association for Women in Mathematics 2019

Authors and Affiliations

  • Amanda Folsom
    • 1
  • Min-Joo Jang
    • 2
  • Sam Kimport
    • 3
  • Holly Swisher
    • 4
    Email author
  1. 1.Department of Mathematics and Statistics, Amherst CollegeAmherstUSA
  2. 2.Department of MathematicsThe University of Hong KongPokfulamHong Kong
  3. 3.Department of MathematicsStanford UniversityStanfordUSA
  4. 4.Department of MathematicsOregon State UniversityCorvallisUSA

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