Quantum modular forms and plumbing graphs of 3-manifolds

TL;DR

This paper explores quantum modular forms related to quantum invariants of plumbed 3-manifolds, confirming a conjecture for certain graphs and expressing invariants for general cases as linear combinations of quantum modular forms.

Contribution

It explicitly computes quantum invariants for specific 3-manifolds, confirms a quantum modularity conjecture, and generalizes the expression of invariants for broader classes of graphs.

Findings

Confirmed quantum modularity conjecture for 3-leg star graphs.

Computed invariants explicitly for unimodular, positive definite plumbing matrices.

Expressed invariants of n-leg star graphs as linear combinations of quantum modular forms.

Abstract

In this paper, we study quantum modular forms in connection to quantum invariants of plumbed 3-manifolds introduced recently by Gukov, Pei, Putrov, and Vafa. We explicitly compute these invariants for any $3$-leg star plumbing graphs whose associated matrix is unimodular and positive definite. For these graphs we confirm a quantum modularity conjecture of Gukov. We also analyze the invariants for general $n$-leg star graphs with unimodular plumbing matrices, and prove that they can be expressed as linear combinations of quantum modular forms.