Séminaire Analyse Harmonique
A proof of Witten’s asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres
27
mai 2025
mai 2025
| Intervenant : | William Elbæk Mistegård |
| Institution : | University of Southern Denmark |
| Heure : | 14h00 - 15h00 |
| Lieu : | Bâtiment 307, salle 2L8 |
Consider a general Seifert fibered integral homology $3$-sphere $Y$ with $r\geq 3$ exceptional fibers. We show that its $SU(2)$ Witten-Reshetikhin-Turaev invariant (WRT) evaluated at any root of unity $\zeta$ is (up to an elementary factor) the non-tangential limit of its Gukov-Pei-Putrov-Vafa invariant (GPPV) as $q$ tends to $\zeta$. We give a parametrization of the moduli space of flat $SU(2)$-connections on $Y$ as a union of moduli spaces of flat $SU(2)$-connections on the Seifert surface. We use the quantum modularity results developed by Han-Li-Sauzin-Sun for functions like the GPPV invariant to prove Witten’s asymptotic expansion conjecture [CMP 1989] for $Y$: the asymptotic behaviour of the WRT invariant at $\exp(2\pi i/k)$ as $k$ tends to infinity is given by a sum of contributions, one for each $SU(2)$ Chern-Simons critical values.
Furthermore, we show that, when going to the variable $\tau$ defined by $q = \exp(2\pi i\tau)$, the GPPV invariant has for each rational $\alpha$ a non-tangential asymptotic expansion that is a resurgent-summable series of $q-\zeta$ where $\zeta=\exp(2\pi i \alpha)$. The formal series of $q-\zeta$ seems to be related to the expansion at $\zeta$ of the Habiro invariant [Inv. 2008]. In the variable $\tau$ all these formal series make up a higher depth strong quantum modular form in the sense of D. Zagier.
This is a joint work with J. E. Andersen, L. Han, Y. Li, D. Sauzin and S. Sun.