Gauge theories are considered to live on $G$-principal bundles $P$ over the spacetime $\Sigma$. For convenience, the usual text often either compactify $\Sigma$ or assume it is already compact.

An instanton is now a "false vacuum" of the theory - a local minimum of the action functional. In four dimensions, instantons are the (anti-)self-dual configurations with $\star F = \pm F$, and the Yang-Mills action then is just the integral over $\mathrm{Tr}(F \wedge F)$, which is the characteristic class of the bundle, also called its Chern class, which are connected through Chern-Weil theory. This class is a topological invariant of the principal bundle associated to a field configuration.

As this math.SE post indicates, the isomorphism classes of bundles over a manifold are in bijection with its first Cech cohomology, which, for smooth manifolds, agrees with the usual other cohomology theories if $G$ is abelian. The question is now twofold:

  1. If $G = \mathrm{U}(1)$, does the existence of instanton solutions imply non-trivial first ordinary (singular, DeRham, whatever) cohomology of spacetime? Or is it rather the case that the instanton/non-trivial bundle configuration only indicates that the gauge theory only holds on the spacetime with points (or possibly more) removed, indicating the presence of magnetic monopoles at these points rather than anything about spacetime?

  2. If $G$ is non-abelian, does the existence of instanton solutions, and hence the non-vanishing of the "non-abelian Cech cohomology" imply anything about the topological structure of spacetime? Perhaps something about the non-abelian homotopy groups rather than the abelian homology? Or, again, does this indicate a non-abelian analogon of monopoles?


First, the reasoning in the question about isomorphism classes of bundles is wrong, because the $\check{H}^1(M,G)$ from the linked math.SE post is not the cohomology of $M$ with coefficients in $G$, but actually the Čech cohomology of $M$ for the sheaf $\mathscr{G} : U\mapsto C^\infty(U,G)$.

However, this indeed has a relation to the cohomology of $M$ itself for $G = \mathrm{U}(1)$, via $$ 0 \to \mathbb{Z} \to \mathbb{R} \to \mathrm{U}(1) \to 0$$ which turns into $$ 0 \to C^\infty(U,\mathbb{Z})\to C^\infty(U,\mathbb{R}) \to C^\infty(U,\mathrm{U}(1))\to 0$$ since $C^\infty(M,-)$ is left exact and one may convince oneself that this particular sequence is still exact since the map $C^\infty(M,\mathbb{R})\to C^\infty(M,\mathrm{U}(1))$ works by just dividing $\mathbb{Z}$ out of $\mathbb{R}$. Considering this as a sheaf sequence $0\to \mathscr{Z}\to \mathscr{R} \to \mathscr{G} \to 0$, $\mathscr{Z} = \underline{\mathbb{Z}}$ for $\underline{\mathbb{Z}}$ the locally constant sheaf since $\mathbb{Z}$ is discrete, and the sheaf of smooth real-valued functions on a manifold is acyclic due to existence of partitions of unity, so taking the sheaf cohomology one gets $$ \dots \to 0 \to H^1(M,\mathscr{G}) \to H^2(M,\underline{\mathbb{Z}})\to 0 \to \dots$$ and thus $H^1(M,\mathscr{G}) = H^2(M,\underline{\mathbb{Z}}) = H^2(M,\mathbb{Z})$ where the last object is just the usual integral cohomology of $M$. Hence, $\mathrm{U}(1)$ bundles are indeed classified fully by their first Chern class which is physically the (magnetic!) flux through closed 2-cycles, and the existence of non-trivial $\mathrm{U}(1)$-bundles would imply non-trivial second cohomology of spacetime (or rather of one-point compactified spacetime $S^4$ since one should be able to talk about the field configuration "at infinty" and the bundle being framed at infinity). Indeed, since $H^2(S^4) = 0$, the existence of $\mathrm{U}(1)$-instantons would contradict the idea that spacetime is $\mathbb{R}^4$.

For general compact, connected $G$, it turns out the possible instantons are pretty much independent of the topology of $M$ because a generic instanton is localized around a point, as the BPST instanton construction shows - the instanton has a center, and one may indeed imagine the Chern-Simons form to be a "current" that flows out of that point, giving rise to a nontrivial $\int F\wedge F$.

Topologically, one may understand this by imagining $S^4$, and giving a bundle by giving the gauge fields on the two hemisphere, gluing by specifying a gauge transformation on the overlap of the two, which can be shrunk to $S^3$, i.e. the bundle is given by a map $S^3\to G$, and the homotopy classes of such maps are the third homotopy group $\pi_3(G)$, which is $\mathbb{Z}$ for semi-simple compact $G$. Since the "equator" can be freely moved around the $S^4$, or even shrunk arbitrarily close to a point, this construction does not in fact depend of the global properties of $S^4$, it can be done "around a point".

Thus, instantons in general do not tell us anything about the topology of spacetime.

This answer has been guided by the PhysicsOverflow answer to the same question.


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