I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction of instantons. At the heart of this is the complex
Vσz→V⊕V⊕Wτz→V.
Where σz=(B1−z1B2−z2j) and τz=(−(B2−z2),B1−z1,I). I am just getting these from Nakajima's book, "Lectures on Hilbert Schemes of Points on Surfaces". And I'll omit explaining the left over notation. Now two key conditions that I find are that τzσz=0 and τzτ∗z=σ∗zσz. The first condition says this is exact, which allows you to look at the quotient Ez:=ker(τz)/im(σz). We also require σz to be injective and τz to be surjective (which is a given assuming ADHM conditions) to assume that Ez never changes rank and in turn will give a vector bundle. So what does τzτ∗z=σ∗zσz tell you? I know that it gives you part of the ADHM condition, but as far as the vector bundle, what does it tell you? I'm pretty sure it gives the Anti-Self-Duality of the induced connection but I'm not sure exactly how to see this. Thanks for the help!!
This post imported from StackExchange MathOverflow at 2015-02-17 11:30 (UTC), posted by SE-user user46348