Geodesic
Quantum Modular $\widehat Z^G$-Invariants
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {Miranda C. N. Cheng and Ioana Coman and Davide Passaro and Gabriele Sgroi},
     title = {Quantum {Modular} $\widehat Z^G${-Invariants}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a17/}
}
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Miranda C. N. Cheng; Ioana Coman; Davide Passaro; Gabriele Sgroi. Quantum Modular $\widehat Z^G$-Invariants. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a17/

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