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The question is as in the title. I have browsed online (Wikipedia, etc) and while they do give me the definition, it gets a bit too much physics-y for me. Therefore I would appreciate it if someone would spoonfeed me the math a little.

I am especially interested in the case of moduli of instantons on a compact complex surface. How does one construct such moduli spaces? How can these objects be used to differentiate between surfaces?

I believe I have a decent enough background in algebraic/complex/differential geometry so a high-tech answer would be appreciated. Thank you!

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    $\begingroup$ These are self-dual connections on principal $SU(2)$-bundles over the given oriented 4-dimensional Riemannian manifold. Do these words mean anything to you? You can read, say, the book by Freed and Uhlenbeck or by Donaldson and Kronheimer to find more about instantons, their moduli spaces and applications to 4d-topology. $\endgroup$ May 20 at 8:54
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    $\begingroup$ @WarlockofFiretopMountain maybe there's been a misunderstanding with the title, sorry about that. I would like to understand what 'moduli space of instantons' means. I study moduli spaces in algebraic geometry all the time so I think this is a natural question to ask. I understand the definition of an instanton but I am nowhere close to working with them, therefore I phrased myself in the title as a beginner. $\endgroup$ May 20 at 10:37
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    $\begingroup$ An instanton is a particular case of connection. Hence the set of all instantons is a subset $S$ of the space of connections $\mathcal C(E)$ ($E\to X^4$ is our bundle). $\mathcal C(E)$ is a topological space, (hence also the set of instantons has a topology), and is acted upon by the gauge group $\mathcal G$. The moduli space of instantons is just the image of $S$ in the quotient $\mathcal C/ \mathcal G$. Under some assumptions it is a smooth manifold, non-compact that can be compactified though. The diffeomorphism class of it is an invariant of the pair $(E,X)$. $\endgroup$ May 20 at 10:46
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    $\begingroup$ Donaldson's invariants are constructed, essentially, by looking at some characteristic classes of bundles constructed over this moduli space. Hence they regard the homotopy type of the moduli space. And give you an invariant of $(E,X)$. They are powerful (and difficult to compute) since they depend on the smooth structure of $X$. $\endgroup$ May 20 at 10:50
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    $\begingroup$ Thank you for the overview! So it's a moduli space just in the sense of points parameterizing the connections (moduli automorphisms). There doesn't appear to be a moduli functor here. $\endgroup$ May 20 at 10:56

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An instanton is a particular case of connection. Hence the set of all instantons is a subset 𝑆 of the space of connections $\mathcal{C}(E) $ ($E\to X^4$ is our bundle). $\mathcal{C}(E)$ is a topological space, (hence also the set of instantons has a topology), and is acted upon by the gauge group $\mathcal G(E)$ (the group of automorphisms of the bundle). The moduli space $\mathcal M$ of instantons is just the image of 𝑆 in the quotient $\mathcal C(E)/\mathcal G(E)$. Under some assumptions $\mathcal M$ has a natural structure of a smooth manifold, non-compact but with a "nice" compactification. The diffeomorphism class of it is an invariant of the pair $(E,X)$.

Invariants like Donaldson's evaluate some characteristic classees of bundles over $\mathcal M$, hence in particular are homotopy invariant of $\mathcal M$. These are powerful (alas hard to compute) invariants of $(E,X)$ because they depend on the smooth structure of $X$.


For $X$ Kähler we have also the following.

Corollary 6.1.6 of Donaldson-Kronheimer's "The Geometry of 4-manifolds". If $E$ is an $SU(2)$ bundle over a compact Kähler surface $X$, the moduli space of irreducible* instantons is naturally identified, as a set, with the set of equivalence classes of stable holomorphic $SL(2,\mathbb C)$ bundles which are topologically equivalent to $E$.

Maybe you can get some functorial properties out of this perspective, idk.

Also, if you are specifically interested in complex surfaces a good reference is Friedman-Morgan's "Smooth 4-manifolds and complex surfaces".

*meaning connections with trivial stabilizer wrt the gauge group action.

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