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:


$F\_A^+=\psi\otimes\psi^\*-\frac{1}{2}|\psi|^2$  

$D\_A\psi=0$


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.



Meaning/origin of Seiberg-Witten equations/invariants
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.
[Link](https://doi.org/10.1007/978-1-4612-0067-3_7)


