# Are Instantons Classical or Quantum?

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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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This is a nice question I never had though about in the way you phrase it, I have to admit. Although instanton configurations arise from classical considerations (e.g. Donaldson's work does not require any quantization proceedure), they can also receive quantum corrections. This is what happens in Seiberg-Witten theory, that is $\mathcal{N}=2$ SYM with gauge group $G$ where one usually starts with a UV theory but then, when you allow RG flow, the field configurations get quantum corrections. The same holds for the moduli space of instantons. There is a classical moduli space and a quantum moduli space which is controlled by the energy - the $u$ coordinate if you wish - and it looks different in the UV and different in the IR.  Physically, I am sure you know, the reason is that as the theory flows towards the IR it becomes strongly coupled. Now, the instanton moduli space classically is indeed the moduli space of anti-self dual connections modulo gauge transformations, i.e. $End(E)\equiv \mathcal{G}$ where $E$ is the associated to $G$ vector bundle. What changes in the quantum theory? I am not sure exactly on how $\mathcal{A}/\mathcal{G}$ changes but, if you use Seiberg-Witten theory (which in a sence takes a problem from the moduli space of instantons to the moduli space of - Coulomb - vacua $\mathcal{M}_{\text{vac.}}$ of the theory) and you write down the corresponding algebraic curve you will see that it develops some singularities that are not really present in the UV theory, in specific, two of the three singularities of the moduli space are due to the fact that the theory is not broken by a Higgs mechanism. You can then see that the moduli space has three singularities, and once mapped to the upper half plane, you get two cusps on the real line not present in the classical theory.

To conclude, $\mathcal{M}_{\text{vac.}}$ looks different in the UV and different in the IR. Ok, this is not quite precise because what we really describe in SW theory is the moduli space of gauge inequivalent vacua, but the theory really localizes in the instanton moduli space really. Taking the partition function will give you Donaldson and/or Seiberg-Witten invariants. What I find quite interesting is that for gauge groups whose Cartan subalgebra is of rank higher than 1, these singularites correspond to divisors of the 4-manifold and there are special "fixed" points where the theory becomes a SCFT, the Argyres-Douglas theory, and usually it admits no Lagrangian description. I would like to understand better though the story from the $\mathcal{A}/\mathcal{G}$ point of view.

answered Jun 24, 2017 by (3,625 points)
edited Jun 25, 2017

It seems to me that you are mixing two things, the moduli spaces of instantons and the moduli space of vacua of the theory: these two things are different. Moduli spaces of instantons are geometric spaces which exist by themselves, independent of any classical/quantum considerations. The moduli space of vacua has a classical and a quantum version and the corrections from classcial to quantum precisely come from instantons contributions.

Indeed, my goal is not to mix those two spaces, rather, show that at least in Donaldson theory, these two space are in some sense related: in the sense that they both compute Donaldson invariants. By no means the two spaces are the same. One can see this also from the point of view of Nekrasov partition function. It is formally defined as an integral of the equivariant cohomology of $\mathcal{M}_{\text{inst}}$, i.e. $\oint_{\mathcal{M}_{\text{inst}}} [1]$ but we can reduce this integral to an integral over the coordinates of $\mathfrak{h}$. Correct me if wrong please.

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Instantons are solutions of the classical equation of motions in Euclidean, i.e. imaginary, time, not in Lorentzian, i.e. real, time.

If you are a classical physicist, you look at solutions of the classical equations of motion in real time and so you don't see instantons.

If you are a quantum physicist, you compute partition functions or more generally correlation functions by path integral over all the possible classical configurations (not necessarily solutions of the classical equation of motions) weighted by $e^{iS/g}$ where $S$ is the classical action in real time and where $g$ is some coupling constant which is assumed to be small. In the limit $g \rightarrow 0$, stationary phase approximation tells you that the path integral is dominated by a classical equation of motion, and you can compute corrections in powers of $g$ to the classical contribution: it is usual perturbative physics. But sometimes, perturbative physics do not capture the full physics. For example, some transitions are classically impossible and so there will be no corresponding classical solution, and so the above approximation will give zero as transition amplitude, whereas the true amplitude is maybe not zero. Typical examples are tunneling processes between configurations separated by a potential which is too high to be classically passed. To study this kind of processes, it is useful to do a Wick rotation, i.e. to go from real time to imaginary time, and then to recover results in real time by analytical continuation. The integrand in the path integral in imaginary time is now $e^{-S^{eucl}/g}$ where $S^{eucl}$ is the Euclidean action. Saddle point approximation tells you that this path integral is dominated by stationnary points of $S^{eucl}$, i.e. by solutions of the classical equations of motion in imaginary time, which are classically called instantons when non-trivial. The real time interpretation of instantons is as computing leading order terms for tunnelling effects. The contribution of an instanton is of the form $e^{-S^{eucl}_I /g}$ where $S^{eucl}_I$ is the Euclidean action of the instanton, and so cannot be seen in a power series expansion is $g \rightarrow 0$: ther are non-perturbative effects in $g$.

In a field theoretic context, what I have written above as to treated carefully because of possible quantum renormalizations of $g$.  In particular, the instanton approximation can become useless when the effective coupling becomes strong (as it happens in quantum Yang-Mills theory).

In general, instantons give leading terms in a saddle path approximation to a Euclidean path integral. In general, they are not the exact answer: they are quantum perturbative effects around instantons, effective interactions between instantons... But the miracle of supersymmetric theories is that in such case  the path integrals localize over supersymmetric configurations. If there is enough supersymmetry, path integrals localize to finite dimensional integrals over finite dimensional moduli spaces of supersymmetric instantons.

In non-supersymmetric Yang-Mills theory, the path integral is over the infinite dimensional space $\mathcal{A}/\mathcal{G}$ over gauge equivalence classes of connections. But in 4d Yang-Mills theory with $\mathcal{N}=2$ supersymmetry, this path integral localizes over the finite dimensional moduli spaces $\mathcal{M}_{inst}$ of instantons (it is the original physics interpretation of Donaldson invariants given by Witten).

If you are a mathematician, I don't think that the distinction classical/quantum really makes sense.

answered Jun 25, 2017 by (5,120 points)

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