arXiv subjects

William Elbæk Mistegård

Publications and source records attributed to William Elbæk Mistegård.

4 recordsLinked to original sources

arXiv2025

The Automorphism Equivariant Hitchin Index

Let T be the one-dimensional complex torus. We consider the action of an automorphism of a Riemann surface X on the cohomology of the T-equivariant determinant line bundle over the moduli space of rank two Higgs bundles on X with fixed determinant of odd degree. We define and study the automorphism equivariant Hitchin index. We prove a formula for it in terms of cohomological pairings of canonical T-equivariant classes of certain moduli spaces of parabolic Higgs bundles over the quotient Riemann surface.

math.DG↗

arXiv2025

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

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.

math.CV↗

arXiv2024

A proof of The Radial Limit Conjecture for Costantino--Geer--Patureau-Mirand Quantum invariants

For a negative definite plumbed three-manifold, we give an integral representation of the appropriate average of the GPPV invariants of Gukov--Pei--Putrov--Vafa, which implies that this average admits a resurgent asymptotic expansion, the leading term of which is the Costantino--Geer--Patureau-Mirand invariant of the three-manifold. This proves a conjecture of Costantino--Gukov--Putrov.

math.GT↗

arXiv2020

Resurgence Analysis of Quantum Invariants of Seifert Fibered Homology Spheres

For a Seifert fibered homology sphere we show that the q-series Z-hat invariant introduced by Gukov, Pei, Putrov and Vafa is a resummation of the Ohtsuki serie. We show that for every even level k there exists a full asymptotic expansion of Z-hat for q tending to a certain k'th root of unity and in particular that the limit exists and is equal to the WRT quantum invariant. We show that the poles of the Borel transform of the Ohtsuki series coincide with the classical complex Chern-Simons values, which we further show classifies the corresponding components of the moduli space of flat SL(2, C)-connections.

math.DG↗