Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 018, 52 pages      arXiv:2304.03934

Quantum Modular $\widehat Z{}^G$-Invariants

Miranda C.N. Cheng abc, Ioana Coman bd, Davide Passaro b and Gabriele Sgroi b
a) Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands
b) Institute of Physics, University of Amsterdam, Amsterdam, The Netherlands
c) Institute for Mathematics, Academica Sinica, Taipei, Taiwan
d) Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Japan

Received May 25, 2023, in final form February 07, 2024; Published online March 09, 2024

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.

Key words: 3-manifolds; quantum invariants; higher depth quantum modular forms; low-dimensional topology.

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