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I'm talking specifically about instantons on four-manifolds, but my confusion here is probably of a more general nature. So I'd also appreciate less specific answers!

Okay, so I know that in physics, if you have an action $S[A]$, a configuration is called a "classical solution" if it extremizes the action, or equivalently if it satisfies the equations of motion. In gauge theory on four-manifolds, you construct an honest moduli space of instantons $\mathcal{M}_{\text{inst}}$. There are anti-self dual connections $A$ which can be shown to minimize the Yang-Mills functional $S_{\text{YM}}[A]$. Therefore, I'd naively say that instantons are classical since they minimize an action. (BTW do they even solve the vacuum Yang-Mills equations?)

On the other hand, many beautiful results in both math and physics come by integrating over $\mathcal{M}_{\text{inst}}$ or similarly, computing SUSY invariants of $\mathcal{M}_{\text{inst}}$ like Euler characteristic, elliptic genus etc. Such a global topological invariant of a moduli space sounds pretty quantum to me: it sounds like a path integral with a particular choice of a measure. In addition, I hear people saying things like "instantons are suppressed" which makes them sounds like quantum corrections or something.

So what's the right way to think of all this? Should I think of $\mathcal{M}_{\text{inst}}$ as a "moduli space of classical vacua"? Then what does it mean to integrate over a moduli space of classical configurations vs. a moduli space of all configurations? For example in gauge theory, we write $\mathcal{A}/\mathcal{G}$ as the space of all connections modulo gauge transformations. What is the relation between

$$\int_{\mathcal{M}_{\text{inst}}} \cdots \,\,\,\,\,\,\, \text{and} \,\,\,\,\,\,\, \int_{\mathcal{A}/\mathcal{G}} \cdots \,\,\,\,\,\,?$$

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    $\begingroup$ Yes, instantons solve the YM equations of motion, so they are classical solutions, as you said, and, as such, critical configurations in the (quantum) path integral, and so influence quantum processes such as tunneling. I think you might well review the role of classical solutions in the path integral and leave instantons aside, for the time being. $\endgroup$ – Cosmas Zachos Jun 24 '17 at 22:37
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    $\begingroup$ Related 159098. $\endgroup$ – Cosmas Zachos Jun 24 '17 at 22:39
  • $\begingroup$ Instantons are solutios of the Euclidean version of the theory (QFT or QM) and describe certain types of tunneling processes. For thus reason they are proper of quantum theories where both the Euclidean version and quantum tunnelling make sense. Their use in quantum theory is based on the so-called saddle point method which permits to compute an estimation of Euclidean path integral using Euclidean classical solutions of Euler-Lagrange solutions. However these solurions have no proper classical meaning as they describe evolutions which are not classically permitted. $\endgroup$ – Valter Moretti Jun 25 '17 at 11:26

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