Quantum Modular Ẑᴳ-Invariants

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2024
• Main Authors: Cheng, Miranda C.N., Coman, Ioana, Passaro, Davide, Sgroi, Gabriele
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2024
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212105
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Quantum Modular Ẑᴳ-Invariants. Miranda C.N. Cheng, Ioana Coman, Davide Passaro and Gabriele Sgroi. SIGMA 20 (2024), 018, 52 pages

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 . 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 , that Ẑᴳ is related to a quantum modular form of depth nr. We prove this for = 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 . We conjecture a recursive relation among the iterated Eichler integrals relevant for Ẑᴳ with = 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(). We prove the conjecture when the three-manifold is moreover an integral homological sphere.