arXiv subjects

Dominik Gutwein

Publications and source records attributed to Dominik Gutwein.

3 recordsLinked to original sources

arXiv2026

The moduli space of conically singular instantons over an SU(3)-manifold

In this article we study the moduli space of conically singular instantons (or Hermitian Yang--Mills connections) with prescribed tangent connections over a 6-manifold equipped with an $\mathrm{SU}(3)$-structure. That is, we develop a Fredholm deformation theory for such $\mathrm{SU}(3)$-instantons in which we fix the tangent connection but allow the underlying principal bundle (and, in particular, the singular set) to vary. This leads to the existence of a Kuranishi structure for this moduli space. Moreover, we investigate the cokernel of the instanton deformation operator and give under certain assumptions a formula for its dimension. Ultimately, we apply our results to conically singular instantons with structure group $\mathbb{P}\mathrm{U}(n)$ and give a formula for the virtual dimension of their moduli space in terms of sheaf cohomology of certain vector bundles over $\mathbb{P}^2$.

math.DG↗

arXiv2025

A smooth family of $G_2$-instantons over a generalised Kummer construction

We construct a smooth 1-parameter family of $G_2$-instantons over a generalised Kummer construction desingularising a $G_2$-orbifold discovered by Joyce. For this we extend the gluing construction for $G_2$-instantons developed by Walpuski to Kummer constructions resolving $G_2$-orbifolds whose singular strata are of codimension 6 and to connections (and entire families of connections) whose linearised instanton operator has a non-trivial cokernel. In order to overcome the corresponding obstructions, we utilise a $\mathbb{Z}_2$-action on the ambient manifold. More precisely, we perturb the (family of) pre-glued almost-instantons inside the class of $\mathbb{Z}_2$-invariant connections, which has the advantage that only the $\mathbb{Z}_2$-invariant locus of the cokernel needs to vanish. We then prove that the instantons that we construct over the resolution of the orbifold found by Joyce are all infinitesimally rigid and non-flat. Moreover, we show that the resulting curve into the moduli space of $G_2$-instantons modulo gauge is injective, that is, no two distinct instantons within the family are gauge-equivalent. To the author's knowledge this is the first example of a smooth 1-parameter family of instantons over a compact $G_2$-manifold.

math.DG↗

arXiv2023

Coassociative submanifolds in Joyce's generalised Kummer constructions

This article constructs coassociative submanifolds in $G_2$-manifolds arising from Joyce's generalised Kummer construction. The novelty compared to previous constructions is that these submanifolds all lie within the critical region of the $G_2$-manifold in which the metric degenerates. This forces the volume of the coassociatives to shrink to zero when the orbifold-limit is approached.

math.DG↗