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Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from.
We take an orthogonal frame bundle $P$ of $TX$, a $\textrm{spin}^\mathbb{C}$ structure $\tilde{P}$ with determinant line bundle $\mathcal{L}$, the complex $\pm$ spin bundles $S^\pm(\tilde{P})$ associated to $\tilde{P}$, a unitary connection $A$ on $\mathcal{L}$, and then BAM:
for a spinor $\psi\in C^\infty(S^+(\tilde{P}))$. From here we can consider the space of solutions (monopoles) and do some Floer theory stuff and whatnot.

I only read that these equations come from Witten's famous paper Monopoles and 4-Manifolds (along with two others joint with Seiberg)... however, unless I am mistaken, he simply writes them down and starts arguing for their similarity/duality to Donaldson's theory (with instanton solutions). I then try and go to the standard references of Donaldson, which don't seem to suggest how the SW equations come about (nor do I even really see how the instantons come about). Although I have studied physics for a long time, I seem to just juggle around these papers, without ever finding a natural "blooming" of the SW equations.

Even if it's in the language of String Theory, I would like to know the general story / understanding of the "blooming" of the SW equations, and how exactly they are "dual" to the instanton-scenario of Donaldson, perhaps even for the "blooming" of these instantons. (For instance, I don't see a set of equations for instantons). This post may not be stated in its clearest form, but I will try my best to make appropriate edits.

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A suggestion from one confused soul to another: you might try looking at Naber's books "Topology, Geometry and Gauge Fields" (there are two of them). The goal of those books is to explain the rudiments of Donaldson theory and Seiberg-Witten theory to someone who hasn't necessarily seen the definition of a manifold. The treatment of these topics is obviously not very deep, but it might be a good place to look for intuition and historical background. –  Paul Siegel Feb 14 '12 at 15:49
Deane, when you said Turaev, I think you meant Tyurin (and the someone else was Pidstrigatch). They proposed a certain "master theory" that has instanton and SW moduli spaces as boundary strata. Their proposal was carried out in a series of papers by Feehan and Leness. –  Tim Perutz Feb 15 '12 at 3:30
@Tim Perutz, so it is safe to say (and rigorously shown) that no new information is obtained from SW Theory over Donaldson Theory (and vice versa)? –  Chris Gerig Feb 15 '12 at 5:30
@Chris Gerig To paraphrase Yoggi Berra, in theory there is no difference between SW and YM. In practice there is. There are things that SW theory sees more clearly than YM. (The work of Taubes on symplectic $4$-manifolds, and contact $3$-manifolds is a good example.) As an analogy, try proving Kodaira's vanishing theorem using the sheaf theoretic description of cohomology but not the Hodge theoretic version. In this classical situation you have two theories describing the same object, but in concrete situations one theory may be more useful than the other. –  Liviu Nicolaescu Feb 15 '12 at 10:31
@Chris: picking up Liviu's point, the idea that SW and Donaldson theories are interchangeable is not true, in either direction. What is true is that the primary invariants of these theories are equal. There are further invariants (notably, the homotopical SW invariants) which are only defined in one of the theories; and each sees geometry which is not visible in the other. Kronheimer-Mrowka's proofs of "Property P for knots", or Andrei Teleman's results on curves in Class VII surfaces, say, use instantons to see thing that monopoles don't. –  Tim Perutz Feb 15 '12 at 14:07

2 Answers 2

up vote 5 down vote accepted

After thinking, and reading other references and re-reading the papers I mentioned, I may have found a sufficient explanation (at least to my care): Both instantons/monopoles are solutions to corresponding equations of motions from associated actions, and they "bloom" from an overarching SUSY action.

Witten formulated "twisted N=2 Supersymmetric Yang-Mills", a TQFT with SUSY (supersymmetry), which leads to the Donaldson invariants. This used an $SU(2)$-bundle over $X$ along with a gauge field (connection $\omega$) and matter fields (bosonic $\phi,\lambda$ and fermionic $\eta,\psi,\zeta$), and gave the Donaldson-Witten action functional $S_{DW}=\int_Xtr(\mathcal{L})$,
$\mathcal{L}=\frac{1}{4}F_\omega\wedge(\ast F_\omega+F_\omega)-\frac{1}{2}\zeta\wedge[\zeta,\phi]+id^\omega\psi\wedge\zeta-2i[\psi,\ast\psi]\lambda+i\phi d^\omega{\ast d^\omega}\lambda-\psi\wedge\ast d^\omega\eta$.
This has associated partition function $Z_{DW}=\int e^{-S_{DW}/g^2}D\Phi$ (here $\Phi$ denotes the space of aforementioned fields), where $g$ is a coupling constant that is the key here for answering our question. The "blooming" of this action functional is beyond the scope of my intentions and probably of MathOverflow, so I won't question it.

In weak coupling ($g\rightarrow 0$, known to physicists as the ultraviolet region), the action localizes to the classical Yang-Mills $S_{YM}=\int_Xtr(F_\omega\wedge\ast F_\omega)$ and have the equations of motion $d^\omega\ast F_\omega=0$. The global-minima solutions are $F_\omega=\pm\ast F_\omega$ (as Oliver clarifies in a comment). These solutions are the Donaldson instantons.

Now apparently, when we instead look at strong-coupling ($g\rightarrow\infty$, known to physicists as the infrared-region), the Seiberg-Witten equations should arise (a "duality" in Witten's TQFT). Indeed, Seiberg and Witten showed that this infrared limit of the above theory is equivalent to a weakly-coupled $U(1)$-gauge theory (the $SU(2)$-gauge group is spontaneously broken down to the maximal torus). Perhaps here is where a better understanding would be desirable (buzzwords 'asymptotic freedom' and 'symmetry breaking' appear).
Anyway, some physics-technique stuff happens (the previous paragraph can be described as "condensation of monopoles"), and we must consider a spin-c structure (which all of our oriented 4-manifolds have, whereas a spin structure would not allow us to consider all 4-manifolds); note that $Spin^c(4)=(SU(2)\times SU(2))\times_{\mathbb{Z}_2} U(1)$. This gives the data: $U(1)$-gauge field $A$ and positive spinor field $\psi$ (as written in the original post). The pair $(A,\psi)$ is a monopole when it minimizes an action $S_{SW}$, i.e. are time-independent solutions to equations of motions (the Seiberg-Witten equations). The action here is $S_{SW}=\int_X(|d^A\psi|^2+|F_A^+|^2+\frac{R}{4}|\psi|^2+\frac{1}{8}|\psi|^4)$, with scalar curvature $R$.

I hope this post is not too confusing.

[[Edit/Update]]: I just came across a book chapter by Siye Wu, The Geometry and Physics of the Seiberg-Witten Equations. These lectures tell the physical origin completely! (i.e. completely details my sketch). The SW equations and action functional pops up on pg191. http://www.springerlink.com/content/q37322037j466218/

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@Chris Thanks very nice post ! –  Alexander Chervov Feb 15 '12 at 5:59
This corresponds with my (limited) understanding of the subject. There has been some work in a more mathematically oriented way, by Nekrasov and others (arxiv.org/abs/hep-th/0306238 arxiv.org/abs/math/0306198 arxiv.org/abs/math/0609841 ) in the case when the 4-manifold is sufficiently nice. –  Jan Jitse Venselaar Feb 15 '12 at 9:36
@Jan Jitse Venselaar - I have not seen that SW-monopole equations are discussed in Nekrasov's papers - although, I might be wrong, I'll try to ask his coauthor some day latter to check this. Nekrasov's paper is about giving mathematical meaning and proving of the Seiberg-Witten's calculation of the "prepotential F", which was physically defined as something like "beta-function..." with Feynmann integral, regularization and all that ill-defined stuff. But it seems it was unclear what this thing means mathematically before Nekrasov's work. –  Alexander Chervov Feb 15 '12 at 9:47
@Alexander Chervov - I just checked, you are right, the monopole equation do not appear in these papers, it's been a few years since I studied this. If I understand correctly, these papers do give a better understanding of the equivalence or "duality" between Donaldson and the Seiberg-Witten theories, which was originally motivated by considerations on the prepotential. There does seem to be a gap in the literature on this subject though, since I recall juggling a great deal of partially overlapping papers back when I was writing my master thesis on the subject. –  Jan Jitse Venselaar Feb 15 '12 at 11:38
I realize you are merely sketching things in this very helpful answer but one minor point seems worth adding. The equations of motion of the classical Yang-Mills action you mention are in fact $d_\omega \star F_\omega = 0$. A connection solving the (anti-)self-dual equations $F_\omega = \pm \star F_\omega$ solves $d\star F_\omega = 0$ by the Bianchi identity. However such connections are just the (anti-)self-dual solutions and correspond to global minima of the action. The action can have other critical points which are not global minima. –  Oliver Nash Feb 15 '12 at 13:18

Try this paper

Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory

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Yea, this is one of the papers I was juggling; it seems to only construct an outlined S-theory (like T-duality for the Homological Mirror Symmetry conjecture), in which case the monopoles and instantons are 'supposed' to be dual, but without constructing the corresponding equations. –  Chris Gerig Feb 14 '12 at 19:24

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