William Elbæk Mistegård

• phd
• PostDoc Position at University of Southern Denmark

In this paper we follow the constructions of Turaev's book [Tu] closely, but with small modifications, to construct of a modular functor, in the sense of Kevin Walker, from any modular tensor category. We further show that this modular functor has duality and if the modular tensor category category is Hermitian or unitary, then the resulting modula...

In this paper we engage in a general study of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants of mapping tori of surface mapping class group elements. We use the geometric construction of the Witten-Reshetikhin-Turaev TQFT via the geometric quantization of moduli spaces of flat connections on surfaces. We identify assumptions o...

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 t...

We motivate and define the zed-hat invariant of a plumbed 3-manifold constructed by Gukov, Pei, Putrov and Vafa. This invariant is a power series with integer coefficients, which is convergent inside the unit disc and have quantum modularity properties. We compute the zed-hat invariant for all Seifert fibered homology spheres via resurgence of the...

This thesis is about the topological quantum field theory (TQFT) invented by Reshetikhin- Turaev and motivated by Atiyah’s axioms, and Witten’s work on quantum Chern-Simons theory and the Jones polynomial. The main theme is the connection between this TQFT and Chern-Simons theory. This thesis contains the results which were obtained during my PhD s...

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 c...

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 L over the moduli space M 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...

This is a presentation about quantization and quantum topology given at the GEOQANT 2021 conference.

For a Seifert fibered homology sphere X$X$, we show that the q$q$‐series invariant Ẑ0(X;q)$\hat{\operatorname{Z}}_0(X;q)$, introduced by Gukov–Pei–Putrov–Vafa, is a resummation of the Ohtsuki series Z0(X)$\operatorname{Z}_0(X)$. We show that for every even k∈N$k \in \mathbb {N}$ there exists a full asymptotic expansion of Ẑ0(X;q)$ \hat{\operatornam...