Do higher homotopy groups play any role in gauge theory?

Question

As is more-or-less well-known, the magnetic monopoles of a gauge theory are classified by the first homotopy group of the gauge group, $\pi_1(G)$ (cf. Lubkin (1963)). The second homotopy group is trivial for any Lie group. For compact $G$, the third group is $\pi_3(G)=\mathbb Z$, which IIRC is related to the instanton number. Do higher-order groups $\pi_{i>3}(G)$ play any role in gauge theories? Is there any real-life effect that is characterised/controlled by these groups?

Answer 1

Higher homotopy groups do play roles in gauge theory.

In many cases, we consider a space or a compactified space space-time $M$ with spherical topology $S^n$. In the case that our gauge theory is defined on a trivial bundle $P = S^n\times G$, the group of gauge transformations (automorphism group of the principal bundle) is the group $\mathcal{G}=\mathrm{Map}(S^n, G)$ of smooth maps from $S^n$ to $G$. Homotopy groups can be used to topologically classify the points of this space which are configurations of gauge transformation. Here we have the identity: $$\pi_m(\mathrm{Map}(S^n, G)) = \pi_{m+n}(G)$$ Examples:

1. Let me first consider the example given in the question $$\pi_0(\mathrm{Map}(S^3, G)) = \pi_3(G)$$ Thus the instanton number determines the connected component (i.e., the element of $\pi_0$)of the gauge configuration from the equator $S^3$ of space time (assumed to be $S^4$).

2. Low energy effective theory of QCD in $4$ dimensions (please see Witten's global aspects). In this case, the configurations of the group of gauge transformation of the (flavor) $SU(3)$ become the Nambu-Goldstone Bosons at low energy, the action can be extended to a five dimensional manifold whose boundary is space-time because $\pi_4(SU(3))=0$. The non-equivalent quantizations are then classified by $\pi_5(SU(3))$, which turns out to be the number of colors of the color group (which is otherwise absent from the low energy description as all exciattions belong to the trivial color representation).

3. Witten's $SU(2)$ anomaly, which was the first global anomaly discovered. Here an $SU(2)$ gauge theory on a compactified space time $S^4$ with an odd number of fermion species is anomalous because $\pi_4(SU(2)) = \pi_0(\mathrm{Map}(S^4, SU(2)= \mathbb{Z}_2$. Please see the exposition by Catenacci and Lena.

Fukui Fujiwara Hatsugai found that the same homotopy group is responsible for the $\mathbb{Z}_2$ invariant in time reversal invariant topological insulators. Here the group $SU(2) \cong SP(1)$ is the gauge group of the induced Berry connection of a Kramer's doublet.

4. The above examples treat matter fields in the background of Yang-Mills fields. Higher homotopy groups appear also in the problem of quantization of pure yang-Mills. Here, since the space of all Yang-Mills configurations $ \mathcal{A}$ (including gauge copies) is contractible, the short homotopy exact sequence implies that the homotopy groups of the space of gauge inequivalent connections are given by:

$$\pi_k(\mathcal{A}/\mathcal{G}) = \pi_{k-1}(\mathcal{G})$$ The non-equivalent quantizations (supeselection sectors) of any configuration space $\mathcal{M}$ corresponds to a direct sum of $H^2(\mathcal{M})$ which counts Dirac magnetic charges and $\mathrm{Map} (\pi_1(\mathcal{M}), U(1))$ which corresponds to Aharonov-Bohm fluxes. Thus in the case of (Hamiltonian) Yang-Mills defined on the space manifold $S^3$, we have: $$\pi_2(\mathcal{A}/\mathcal{G}) = \pi_{1}(\mathcal{G}) = \pi_{1}(\mathrm{Map}(S^3, SU(N))) = \pi_4(SU(N)) $$ which is trivial, however: $$\mathrm{Map} (\pi_1(\mathcal{A}/\mathcal{G} ), U(1))= \mathrm{Map} (\pi_3(SU(N)) , U(1)) = \mathrm{Map} (\mathbb{Z} , U(1)) = \mathbb{R}/2 \pi\mathbb{Z}$$ (sorry that this example includes only a third homotopy group), the representative of these inequivalent representations are the $\theta-$ vacua. The generalization of the Aharonov-Bohm flux to this infinite dimensional quantization problem was coined by Wu and Zee as an Abelian gauge structure. It is a functional Abelian gauge field living inside the configuration space of Non-Abelian Yang-Mills.