Abstracts

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.

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.

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.

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.

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.