Quantum Modular Ẑᴳ-Invariants

Abstract

We study the quantum modular properties of Ẑᴳ-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 Ẑᴳ is related to a quantum modular form of depth nr. We prove this for 𝐺 = 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 Ẑᴳ-invariants of the same three-manifold with different gauge group 𝐺. We conjecture a recursive relation among the iterated Eichler integrals relevant for Ẑᴳ with 𝐺 = SU(2) and 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 SU(𝑁). We prove the conjecture when the three-manifold is moreover an integral homological sphere.