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Quantum Modular $\widehat Z{}^G$-Invariants

Miranda C. N. Cheng, Ioana Coman, Davide Passaro, Gabriele Sgroi

Abstract

We study the quantum modular properties of $\widehat Z{}^G$-invariants of closed three-manifolds. Higher depth quantum modular forms are expected to play a central role for general three-manifolds and gauge groups $G$. In particular, we conjecture that for plumbed three-manifolds whose plumbing graphs have $n$ junction nodes with definite signature and for rank $r$ gauge group $G$, that $\widehat Z{}^G$ is related to a quantum modular form of depth $nr$. We prove this for $G={\rm SU}(3)$ and for an infinite class of three-manifolds (weakly negative Seifert with three exceptional fibers). We also investigate the relation between the quantum modularity of $\widehat Z{}^G$-invariants of the same three-manifold with different gauge group $G$. We conjecture a recursive relation among the iterated Eichler integrals relevant for $\widehat Z{}^G$ with $G={\rm SU}(2)$ and ${\rm SU}(3)$, for negative Seifert manifolds with three exceptional fibers. This is reminiscent of the recursive structure among mock modular forms playing the role of Vafa-Witten invariants for ${\rm SU}(N)$. We prove the conjecture when the three-manifold is moreover an integral homological sphere.

Quantum Modular $\widehat Z{}^G$-Invariants

Abstract

We study the quantum modular properties of -invariants of closed three-manifolds. Higher depth quantum modular forms are expected to play a central role for general three-manifolds and gauge groups . In particular, we conjecture that for plumbed three-manifolds whose plumbing graphs have junction nodes with definite signature and for rank gauge group , that is related to a quantum modular form of depth . We prove this for and for an infinite class of three-manifolds (weakly negative Seifert with three exceptional fibers). We also investigate the relation between the quantum modularity of -invariants of the same three-manifold with different gauge group . We conjecture a recursive relation among the iterated Eichler integrals relevant for with and , for negative Seifert manifolds with three exceptional fibers. This is reminiscent of the recursive structure among mock modular forms playing the role of Vafa-Witten invariants for . We prove the conjecture when the three-manifold is moreover an integral homological sphere.
Paper Structure (16 sections, 15 theorems, 131 equations, 1 table)

This paper contains 16 sections, 15 theorems, 131 equations, 1 table.

Table of Contents

  1. Introduction and summary
  2. Introduction
  3. Summary of results
  4. Notation guide
  5. Generalized $\boldsymbol{A_2}$ false theta functions
  6. Identities
  7. Radial limits
  8. Companions
  9. Eichler integrals
  10. Quantum modularity
  11. Properties of $\boldsymbol{\widehat{Z}^{{\rm SU}(3)}}$
  12. Topology
  13. Companions
  14. Examples
  15. Example: $\boldsymbol{ M\bigl(-1;\frac{1}{4},\frac{3}{5},\frac{1}{7}\bigr)}$
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $G={\rm SU}(3)$. For a negative Seifert manifold $M_3$ with three exceptional fibers and for all the allowed ${\vec{\underline{b}}}$, the invariant $\widehat{Z}^{G}_{{\vec{\underline{b}}}}\left(M_3;\tau\right)$ is a sum of depth-one and depth-two quantum modular forms.

Theorems & Definitions (21)

  • Conjecture 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Proposition 3.5
  • Definition 3.6
  • ...and 11 more