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.

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.

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 spin-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.

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.