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This article is cited in 1 scientific paper (total in 1 paper)
Quantum Modular $\widehat Z^G$-Invariants
Miranda C. N. Chengabc, Ioana Comandb, Davide Passaroa, Gabriele Sgroia a Institute of Physics, University of Amsterdam, Amsterdam, The Netherlands
b Institute for Mathematics, Academica Sinica, Taipei, Taiwan
c Korteweg-de Vries Institute for Mathematics, University of Amsterdam,
Amsterdam, The Netherlands
d Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Japan
Abstract:
We study the quantum modular properties of $\widehat Z^G$-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 $\widehat Z^G$ is related to a quantum modular form of depth $nr$. We prove this for $G={\rm SU}(3)$ 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 $\widehat Z^G$-invariants of the same three-manifold with different gauge group $G$. We conjecture a recursive relation among the iterated Eichler integrals relevant for $\widehat Z^G$ with $G={\rm SU}(2)$ and ${\rm SU}(3)$, 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 ${\rm SU}(N)$. We prove the conjecture when the three-manifold is moreover an integral homological sphere.
Keywords:
3-manifolds, quantum invariants, higher depth quantum modular forms, low-dimensional topology.
Received: May 25, 2023; in final form February 7, 2024; Published online March 9, 2024
Citation:
Miranda C. N. Cheng, Ioana Coman, Davide Passaro, Gabriele Sgroi, “Quantum Modular $\widehat Z^G$-Invariants”, SIGMA, 20 (2024), 018, 52 pp.
Linking options:
https://www.mathnet.ru/eng/sigma2020 https://www.mathnet.ru/eng/sigma/v20/p18
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