Andrew Clarke: Instantons on the manifolds of Bryant and Salamon

We give a construction of G2 instantons on one of the manifolds of Bryant and Salamon, using an ansatz of spherical symmetry coming from the space being a rank-4 vector bundle. We show that the connections are asymptotic to a HYM connection on the nearly Kähler S3 x S3.

Derek Harland: Perturbations of nearly Kähler instantons

Nearly Kähler six-manifolds are almost hermitian manifolds whose cone metric has holonomy G2. The best-known example is the unit sphere in the imaginary octonions. Instantons on nearly Kähler six-manifolds are critical points both of the Yang-Mills energy and of a Chern-Simons-type functional; it is known that at least one instanton exists on the tangent bundle of any such manifold.

In this talk I will discuss deformations of instantons on nearly Kähler six-manifolds. A deformation theory will be presented based on elliptic operators, which suggests that instantons typically do not admit deformations. I will illustrate the theory with some simple explicit examples of instantons, including on the six-sphere. This is joint work with B. Charbonneau.

Andriy Haydys: Fueter sections and gauge theory

I will describe a relation between Fueter sections and a generalisation of the Seiberg-Witten equations. This in turn leads to a relation between some gauge theoretic problems including G2 instantons and the Seiberg-Witten equations.

Marcos Jardim: Gauge Theories in Higher Dimensions

Recent results by Tikhomirov, Markushevish and Tikhomirov and by the author and Verbitsky have answered old questions about the geometry of the moduli space I(c) of rank 2 instanton bundles of charge c on P3: we now know that this is an irreducible, rational, non-singular affine variety of dimension 8c-3 carying a geometric structure we called trisymplectic structure. The next step is to study its compactification. Since every rank 2 instanton bundle on IP3 is stable, I(c) can be regarded as an open subset of the Gieseker-Maruyama scheme M(2,0,c,0) of semistable rank 2 torsion free sheaves on I(c) with Chern classes c1=c3=0 and c2=c. One can then consider the closure of I(c) within M(2,0,c,0). In this talk we show that the singular locus of non-locally free rank 2 instanton sheaves on P^3 have pure dimension 1. We then describe certain irreducible components of the boundary of I(c) with dimension 8c-4. Such components consist of stable, non-locally free rank 2 instanton sheaves whose singular loci are rational curves. The results presented are joint work with M. Gargate and with D. Markushevich and A. Tikhomirov.

Thomas Walpuski: A compactness theorem for the Seiberg–Witten equation with multiple spinors in dimension three

In a recent preprint of Andriy Haydis’ and mine we proved that a sequence of solutions of the Seiberg–Witten equation with multiple spinors in dimension three can degenerate only in two ways: either by becoming reducible or by converging to a Fueter section of a bundle of moduli spaces of ASD instantons. I will discuss the relevance of this result for gauge theory on G2–manifolds and if time permits explain some ideas of the proof.