The traditional story in the physics textbooks (copied endlessly from one textbook to the next, over generations) of SU(2)-instantons (BPST instantons) tends to fail to highlight some key global points, without which the whole construction really collapses. The following text means to explain the correct description (using the mathematics of Cech cohomology cocycles via the clutching construction on the one-point compactification of Minkowski spacetime) but presented and phrased such that the folklore physics story becomes fully visible – including its crucial fix.
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Instantons
For $G$ any gauge group (a Lie group), a $G$-Yang-Mills instanton on some 4-dimensional spacetime (a pseudo-Riemannian manifold) $X$ is a $G$-principal bundle on $X$ (whose class is going to be the “instanton sector”), equipped with a $G$-principal connection $\nabla$ (this is the actual gauge field) such that this is self-dual, in that its curvature form $F_\nabla$ satisfies $F_\nabla = \star F_{\nabla}$, where $\star$ denotes the Hodge star operator for the given metric (which represents the field of gravity).
The ADHM construction gives an essentially unique self-dual principal connection $\nabla$ on every isomorphism class of $SU(2)$-principal bundles (ie.: “in every instanton sector”). Therefore classifying and counting instantons amounts to classifying and counting $G$-principal bundles.
This is what makes instantons a “topological” structure in the parlance of physics, meaning that they do not depend on Riemannian metric information, after all.
We now take the spacetime to be Minkowski spacetime $\mathbb{R}^{3,1}$ and the gauge group to be the special unitary group $SU(2)$. This is the case of “BPST-instantons”. Of course other choices are possible and may lead to richer situations, but this simple case is what the physics textbooks tend to focus on (not the least, of course, because these choices are relevant for phenomenology of the weak nuclear force as seen in accelerator experiments) and it already serves to highlight key points. Since, as we just said, we may disregard all metric properties, this means now that we regard spacetime to be just the Cartesian space $\mathbb{R}^4$.
Or almost. At this point one needs to be careful with the boundary conditions in order to get the topology right.
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Vanishing at infinity
The physical energy condition on an instanton is that the field strength $F_\nabla$ vanishes “at infinity”. Mathematically, a continuous function on a locally compact topological space $X$ vanishes at infinity precisely if it extends to the “one-point compactification” $X^+ \coloneqq X \cup \{\infty\}$ of $X$ – where we literally adjoin the “point at infinity” “$\infty$” and glue it to $X$ by defining a suitable topology on $X \cup \{\infty\}$.
Hence we may formalize the boundary condition by saying that our $SU(2)$-principal connection actually exists on the one-point compactification of $\mathbb{R}^4$. This is the 4-sphere
It is this passage from $\mathbb{R}^4$ to $S^4$ which takes care of the subtleties that often tend to be glossed over.
With the boundary conditions “at infinity” taken care of this way, an $SU(2)$-instanton now is the isomorphism class of an $SU(2)$-principal bundle on the 4-sphere $S^4$.
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Instanton number
There is a classifying space for $SU(2)$-principal bundles, denoted $B SU(2)$. This being a classifying space means that for $X$ any paracompact topological space, we have that the isomorphism classes of $SU(2)$-principal bundles on $X$ are in bijection with homotopy classes of continuous functions from $X$ to $B SU(2)$.
Now for $X = S^4$, it follows that
is the 4th homotopy group of the classifying space $B SU(2)$.
This is an integer, and so this integer labels isomorphism classes of instantons on $S^4$ (hence on $\mathbb{R}^4$). We see below that Chern-Weil theory identifies this number with the instanton number.
But this counting of instantons works more generally, if we use a suitable counting function. First of all, there is a topological space whose only non-trivial homotopy group is $\pi_4$, and such that this is the group of integers. This is the Eilenberg-MacLane space $K(\mathbb{Z},4)$:
This space has the following remarkable property: homotopy classes of continuous functions into it compute ordinary cohomology with integer coefficients:
Now there is a continuous function
called the universal second Chern class. This hence sends $SU(2)$-principal bundles, classified by some map
to classes in degree-4 cohomology
This cohomology class
is hence called the second Chern class of the $SU(2)$-principal bundle.
This is one in a whole sequence of characteristic classes which are carried by $SU(2)$-principal bundles, the Chern classes.
But in the special case that the base space $X$ is 4-dimensional, we have that only a single one of these classes may be non-trivial, namely the second Chern class $c_2$. Therefore this class completely characterizes $SU(2)$-principal bundles in 4d.
In conclusion: Where a BPST-instanton is manifested by an SU(2)-principal bundle on a 4-dimensional manifold, the instanton number is the second Chern class of this bundle.
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Constructing instantons from gauge transformations
We may construct the bundles, that are classified this way, explicitly by using Cech cohomology. This says that we get such a bundle by
choosing an open cover $\{U_i \to X\}$ of $X$ (by “charts”)
choosing transition functions $g_{i j} \colon U_i \cap U_j \to SU(2)$ on each double overlap of two charts
such that on triple overlaps the cocycle condition $g_{i j} \cdot g_{j k} = g_{i k}$ holds.
Now comes the major fact which makes this general theory look like the structures that appear in the physics books: the clutching construction.
Namely, for general $X$ one needs the charts $\{U_i \to X\}$ to form a good open cover in order to guarantee that all isomorphism classes of gauge bundles are captured by the construction via transition functions.
But the clutching construction says that whenever $X$ happens to be a sphere, then it is sufficient to cover it by two hemispheres that overlap a little:
But since everything is topological now, it doesn’t matter that these charts are literally hemi-spheres in the metric sense. In order to get the standard picture we instead make $U_+$ maximally large and take it to cover all of $S^4$ except the “north pole” (which is really the “point at infinity”, due to the one-point compactification above), while we take $U_-$ to be a tiny open neighbourhood of that point, sitting there as a tiny ice cap around the north pole. So
Topologically this is homeomorphic to the situation before, and hence just as good.
So now back to the general prescription of building principal bundles via Cech cohomology, we are to choose transition functions on all overlaps of charts. But thanks to the clutching construction, there is now just a single such overlap, namely
Moreover, there is no non-trivial triple overlap, hence no cocycle condition that our transition function is to satisfy.
In conclusion, Cech cohomology and the clutching construction jointly now say that $SU(2)$-instantons are classified by maps
But notice here that the intersection of two hemi-4-spheres that overlap slightly is just a 3-sphere times a slight thickening:
Moreover, the thickening direction here is trivial, and so one finds that instantons are classified by homotopy classes of maps
This is where this “gauge transformation at infinity” in the physics textbooks really comes from. The key point here is that it is indeed gauge transforming, namely the restriction of our bundle to $U_+$ is gauge transformed to its restriction to the “neighbourhood of infinity” $U_-$.
But, as topological spaces, $SU(2) \simeq S^3$ and so
This way we get the same classification from Cech cohomology that we got from classifying space theory, as it must be. But now we know how that second Chern class may be concretely embodied in the “gauge transformation at infinity”.
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Gauge fields vanishing at infinity
Now we bring in connections. As discussed before, we may just as well consider any principal connection. In the Cech cohomology picture and still using the clutching construction this now means to choose
$\mathfrak{su}(2)$-valued 1-forms $A_{\pm}$ on $U_\pm$
such that on the tiny intersection of the two charts at infinity we have
Now observe that in the given situation of using the clutching construction on a sphere, we may always choose
Because then the above just says that $A_+$ on $U_+$ becomes gauge trivial “at infinity”, by the given gauge transformation $g \colon S^3 \to SU(2)$ (the one whose winding number counts our instantons).
(In the present case this is really simple; but also generally there is a partition of unity-argument to construct connections on any bundle in generalization of this simple situation, see here.)
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Counting instantons by integrating $tr(F_\nabla \wedge F_\nabla)$
So far we have derived the physics picture of an instanton: An $SU(2)$-gauge field which becomes gauge trivial “at infinity”, witnessed by a gauge transformation on the “annulus at infinity” $S^3 \to SU(2)$, whose winding number is the instanton number. But the key point is that we see that the little neighbourhood of infinity $U_-$ is part of the picture, and that is necessary now to understand the Chern 4-form.
Namely to every $\mathfrak{su}(2)$-valued 1-form $A$ we may assign the ordinary (abelian)differential 4-form
where $F_A$ is the curvature form of $A$.
Now a general fact of Chern-Weil theory is that the 4-form built this way from a single $A$ is always exact, a potential is given by the Chern-Simons form $CS(A)$:
But beware that this is only true on a single chart. And just because our chart $U_+$ covers “everything except infinity”, we must not forget that there is a second chart $U_-$, the “neighbourhood of infinity”.
Namely the 4-form $\langle F_\nabla \wedge F_\nabla\rangle$ that is defined on the whole of the 4-sphere $S^4$, this 4-form is only locally exact (as every closed 4-form is, by the Poincare lemma). Generally, we define it chart-wise by
and
This does indeed give a globally defined 4-form, no matter what the local connection forms $A_{\pm}$ are, as long as they satisfy the required condition that they are related by a gauge transformation on the overlap $U_+ \cap U_-$. Because the 4-form is gauge invariant.
So the 4-form thus defined
has
a potential 3-form $CS(A_+)$ when restricted to $U_+$;
a potential 3-form $CS(A_-)$ when restricted to $U_-$;
but it does not have a potential 3-form on all of $S^4$, unless the instanton number vanishes.
Put this way this should be very obvious now. But it is easy to get confused about this, due to the sheer convenience of the clutching construction used above: we actually were allowed to choose $A_- = 0$!
This might make it superficially look like there is only a single local gauge potential $A_+$ around, and that the 4-form is locally exact. But this is not the case: there are two local gauge potentials on two different charts, and just because one of them happens to be equal to zero still does not mean that the extension of the 4-form to all of the 4-sphere has a global potential. It has a potential $CS(A_+)$ only after restricting it to $U_+$, even if this means “only removing the point at infinity”.
With it thus being understood that $\langle F_\nabla \wedge F_\nabla\rangle$ is not globally exact, it becomes believable that the integral
is generally non-vanishing, and is in fact yet another incarnation of the same integer that we had before, the instanton number. That this is so is given to us by Chern-Weil theory.
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Outlook: Chern-Simons 2-Gerbes
In fact the full story is nicer still. Namely the local Chern-Simons 3-forms $CS(A_\pm)$ together with the gauge transformation at infinity form a Cech cohomology cocycle for a circle 3-bundle with connection (a bundle 2-gerbe). This is the Chern-Simons 2-gerbe of the gauge field. And the fourth incarnation of the instanton number is: the Dixmier-Douady class of this 2-gerbe
Last revised on March 5, 2024 at 01:14:06. See the history of this page for a list of all contributions to it.