A proof of Witten's asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914089596354560 |
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| author | Andersen, Jørgen Ellegaard Han, Li Li, Yong Mistegård, William Elbæk Sauzin, David Sun, Shanzhong |
| author_facet | Andersen, Jørgen Ellegaard Han, Li Li, Yong Mistegård, William Elbæk Sauzin, David Sun, Shanzhong |
| contents | Let $X$ be a general Seifert fibered integral homology $3$-sphere with $r\ge3$ exceptional fibers. For every root of unity $ζ\not=1$, we show that the SU(2) WRT invariant of $X$ evaluated at $ζ$ is (up to an elementary factor) the non-tangential limit at $ζ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10678 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A proof of Witten's asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres Andersen, Jørgen Ellegaard Han, Li Li, Yong Mistegård, William Elbæk Sauzin, David Sun, Shanzhong Complex Variables Mathematical Physics Classical Analysis and ODEs Differential Geometry Geometric Topology 57R56 Let $X$ be a general Seifert fibered integral homology $3$-sphere with $r\ge3$ exceptional fibers. For every root of unity $ζ\not=1$, we show that the SU(2) WRT invariant of $X$ evaluated at $ζ$ is (up to an elementary factor) the non-tangential limit at $ζ$ 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. |
| title | A proof of Witten's asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres |
| topic | Complex Variables Mathematical Physics Classical Analysis and ODEs Differential Geometry Geometric Topology 57R56 |
| url | https://arxiv.org/abs/2510.10678 |