# Reference request for instantons

I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. But this only works for $\mathbb{R}^4$ and $\mathbb{R}^4_{NC}$. How do we extend these to other manifolds? For example, how can we get instantons on the torus $\mathbb{T}^4 = \mathbb{R}^4 / \Gamma$? Or a complex torus $\mathbb{T}_{\mathbb{C}}^2$? One would think to allow some sort of periodicity which agrees with the lattice (forgive the non-technical terminology), but how is this done and/or what are the obstructions to this?

The ADHM construction is such a beautiful thing and I hope there are analogues for other manifolds.

In summary I am looking for papers/notes/books to read that give an expository and/or technical explanation about finding instantons on other manifolds, particularly on the (complex, ideally) torus, which hopefully have a hand in the ADHM pot as well. And I'm not a physicist nor an analyst so the more algebro-/complex-geometric the better. Thanks for the help.

• If you don't mind a bit of noncommutative differential geometry, then you can take a look at this expository article by Landi and Van Suijlekom for an account of instantons on noncommutative (real) $4$-tori: arxiv.org/abs/hep-th/0603053 The heart of the matter is that if you apply Rieffel's strict deformation quantisation to a compact $\mathbb{T}^N$-manifold $X$, then any $\mathbb{T}^N$-equivariant object over $X$ (e.g., an equivariant vector bundle) can be deformed to an analogous noncommutative-geometric object over the deformation of $X$. – Branimir Ćaćić Jul 29 '14 at 16:49
• It sounds like you're probably past this, but you might take a peek at section 13.2 of amazon.com/Classical-Theory-Fields-Valery-Rubakov/dp/0691059276 – Steve Huntsman Jul 29 '14 at 20:40
• @Branimir - Thanks for the article, and I don't mind looking at noncommutative things. I actually am sort of searching in that direction. Though I find it is hard to grasp sometimes without a proper and thorough introduction. – user46348 Jul 30 '14 at 14:46
• @Steve - Thanks for the recommendation as well. I am always willing to read something even if just to solidify an idea especially if it is well written. – user46348 Jul 30 '14 at 14:47
• You might also look at Chapter 8 of Ward and Wells' book on twistors. – Steve Huntsman Jul 31 '14 at 3:09

I think a good starting place for your question regarding the moduli space for a flat 4-torus is the Fourier-Mukai' correspondence which came out of work of Nahm and which relates the moduli space over one such torus to that of its dual torus. See section 3.2 of the bookThe Geometry of Four-Manifolds' by Donaldson and Kronheimer, and more recently (and chosen somewhat at random from a google search): http://www.ime.unicamp.br/~jardim/publicados/crelle03.pdf