Quantum Modular Ẑ^G-Invariants

We study the quantum modular properties of Ẑ^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 Ẑ^G 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 Ẑ^G-invariants of the same three-manifold with different gauge group G. We conjecture a recursive relation among the iterated Eichler integrals relevant for Ẑ^G 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.