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arXiv · 2510.10678

A proof of Witten's asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres

Jørgen Ellegaard Andersen·Li Han·Yong Li·William Elbæk Mistegård·David Sauzin·Shanzhong Sun
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Abstract

Let $X$ be a general Seifert fibered integral homology $3$-sphere with $r\ge3$ exceptional fibers. For every root of unity $\zeta\not=1$, we show that the SU(2) WRT invariant of $X$ evaluated at $\zeta$ is (up to an elementary factor) the non-tangential limit at $\zeta$ of the GPPV invariant of $X$, thereby generalizing a result from [Andersen-Mistegard 2022]. Based on this result, we apply the quantum modularity results developed in [Han-Li-Sauzin-Sun 2023] to the GPPV invariant of $X$ to prove Witten's asymptotic expansion conjecture [Witten 1989] for the WRT invariant of $X$. We also prove that the GPPV invariant of $X$ induces a higher depth strong quantum modular form. Moreover, when suitably normalized, the GPPV invariant provides an ``analytic incarnation'' of the Habiro invariant.

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Jørgen Ellegaard Andersen, Li Han, Yong Li, William Elbæk Mistegård, David Sauzin, Shanzhong Sun. 2025-10-12. A proof of Witten's asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres. https://arxiv.org/abs/2510.10678

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