Quantum Modular ZbG-Invariants

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