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In QCD instanton configurations of the gauge field control the experimentally observed vacuum structure of the theory.
One considers QCD on a Minkowski spacetime Wick rotated to a Euclidean $\mathbb{R}^4$ and assumes the field strength to vanish at infinity. This makes the gauge field configurations be equivalently $SU(3)$-principal connections on the 4-sphere $S^4$. The second Chern class of the underlying $SU(3)$-principal bundle is called the Yang-Mills instanton-number of the gauge field configuration. The theoretical/experimentally observed vacuum of QCD is some superposition of gauge fields in various instanton number sectors.
In particular the instanton liquid model in QCD assumes that the vacuum (ground state of QCD) is populated by instanton field configurations of average radius $1/3 fm$ (femtometres) with a density of 1 such instanton per $fm^4$. Some details of this model remain subtle, see for instance (Schaefer-Shuryak) for a good survey.
An instanton vacuum is argued to account for
formation of the gluon condensate (M. Shifman, A. Vainshtein and V. Zakharov, Nucl. Phys. B 147 (1979) 385).
spontaneous chiral symmetry breaking (Diakonov 96)
baryogenesis in the early universe by inducing baryon numbernon-conservation
In the physics literature a style of discussion of instantons is wide-spread where the motivation of the formulas considered is dubious. In these references (for instance Schaefer-Shuryak 96 and Forkel 00 and many more) it is said that
a YM field of minimal energy, hence of vanishing field strength is one that is pure gauge;
that therefore on a Minkowski spacetime $\mathbb{R}^4$ in temporal gauge and assuming fields are vanishing at infinity, these are given by connection 1-forms on $S^3$ of the form $U^{-1} d U$ for $U \colon S^3 \to G$ a smooth function, and
that therefore equivalence classes of such fields are given by homotopy classes of maps $S^3 \to G$, hence by the third homotopy group $\pi_3(G)$.
These points are not really correct, even though the conclusion about the classification of instanton sectors turns out to be right.
The correct statement is that winding-number sectors/instanton sectors are equivalence classes of $G$-principal bundles on the one-point compactification $S^4$ of Minkowski spacetime $\mathbb{R}^4$, and these happen to indeed be given by homotopy classes of maps $S^3 \to G$, with $S^3$ regarded as the “equator” of $S^4$. This is for instance stated explicitly in (Nakahara) and used in many references, such as (Witten 85).
Before we come to this, first notice the problems with the above items from the standard physics reviews:
Gauge fields with vanishing field strength are flat connections. These are classified by group homomorphisms from the fundamental group of the given space or spacetime to the gauge group. Hence not every such field configuration is “pure gauge”, this is only the case if the fundamental group is trivial. Of course this is the case on $\mathbb{R}^n$ and $S^n$, so the first item above is correct in these cases. But beware that these authors usually justify the third item above by referring to gauge transformations between gauge fields, which is then again wrong, see the next point.
Flat connection on $S^3$ are not classified by homotopy classes of maps $S^3 \to G$. Instead, since $\pi_1(S^3)$ is the trivial group, there is only the trivial such flat connection. This is actually manifest in the formula for such connections which is preferred in these references, which is “$U^{-1} d U$” with $U \colon S^3 \to G$ a gauge transformation. That formula exhibits precisely the trivial gauge connection $d$ as gauge equivalent to $U^{-1} d U$.
What would give the intended classification is if one considered connections of the form $U^{-1} d U$ only modulo those gauge transformations $h \colon S^3 \to G$ which can be smoothly extended to maps $D^4 \to G$. This is what these references seem to be implicitly assuming. But this is not the gauge transformation law for connections… this is instead the gauge transformation law for transition functions of $G$-principal bundles.
And this brings us to the correct mathematical interpretation of instantons, as in (Nakahara) and similar:
We are looking for gauge field configurations on $\mathbb{R}^4$ that have field strength vanishing at infinity. This means we are looking for $G$-principal connections on the one-point compactification $S^4$ of $\mathbb{R}^4$ such that the curvature 2-form vanishes at one “pole”.
Now by the discussion at principal bundle, $G$-principal bundles can be constructed for instance by cocycles in Cech cohomology of $S^4$ with coefficients in $G$. In general this means to specify $G$-valued transitions functions on overlaps of a good open cover of $S^4$. But a standard argument shows that for spheres the following covering is already sufficient: we cover the $n$-sphere $S^n$ by two hemi-$n$-spheres that overlap a bit at the equator, which is hence of the form $S^{n-1} \times [-\epsilon, \epsilon]$. A Cech cocycle on such a cover is then simply given by one single $G$-valed transition function $S^{n-1} \to G$. This is known as the clutching construction.
Moreover, a gauge transformation of such a Cech cocycle is given by multiplying with the restriction of $G$-valued functions on the two hemi-$n$-spheres $\simeq D^n$ to the equator. This just means that those transition functions $S^3 \to G$ are gauge-trivial which may be extended smoothly to funtions $D^4 \to G$.
In conclusion, the desired classification of instanton solutions by homotopy classes of maps $U \colon S^3 \to G$ is precisely the clutching construction of $G$-principal bundles.
Finally, in order to add the principal connections to the picture, we think of one of the two semi-$n$-spheres to be the neighbourhood of infinity and hence, since the field strength is demanded to be vanishing at infinity, take the local gauge field to be the vanishing Lie algebra valued differential 1-form there. But then the rules for Cech cohomology with coefficients in the groupoid of Lie algebra valued 1-forms, and using the above clutching construction, it follows that on the equator overlap the gauge field has to be $U^{-1} d U \in \Omega^1( S^3 \times (-\epsilon, \epsilon) )$. This we may then extend in any desired way to a (non-flat) gauge connection over the remaining semi-$n$-sphere $D^4$, hence over the actual spacetime.
Notice that this is precisely the argument which for $G = U(1)$ the circle group and for $n = 2$ is known as the argument of Dirac charge quantization (which is also often misrepresented in the physics literature…)
Physics-style surveys include the introductory lecture notes
Marcos Marino, Instantons and Large $N$ – An introduction to non-perturbative methods in QFT (pdf)
Hilmar Forkel, A Primer on Instantons in QCD (arXiv:hep-ph/0009136)
and the fairly detailed account (with lot of pointers to the literature)
T. Schaefer, E. Shuryak, Instantons in QCD, Rev. Mod. Phys.70:323-426,1998 (arXiv:hep-ph/9610451)
section III D: relation to confinement
Dmitri Diakonov, Instantons at work (arXiv:hep-ph/0212026)
See also the survey in
A more mathematically precise account which identifies instanton solutions explicitly equivalence classes of $G$-principal bundles is around Example 9.12 (p.320) and Section 10.5.5 (p.360) of
This perspective is for instance also the one used in
Discussion in the context of chiral symmetry breaking includes
Further developments on the role of instantons/monopoles in the QCD vacuum for confinement are discussed in