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The fundamental difference between spinors and tensors is that spinors are sensitive to the homotopy classes of paths through the rotation group $SO(3)$: \begin{equation} \pi_1(SO(3)) = \mathbb{Z}_2, \end{equation} and so rotations of $2\pi$, which belong to the non-trivial homotopy class, differ from the identity by a minus sign. In two dimensions, the rotation group is topologically a circle, and so there are infinitely many homotopy-inequivalent rotations indexed by winding number.

Does something similar happen for gauge groups?

For example, the gauge group of QED is $U(1)$, which is topologically a circle with \begin{equation} \pi_1(U(1)) = \mathbb{Z}, \end{equation} and so there are infinitely many homotopy-inequivalent paths through the gauge group. I know that gauge invariance is not observable, but are there observable consequences of the topology of the gauge group?

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Absolutely. There are many circumstances where topology has important implications for gauge theories.

Perhaps the most notable example concerns the classification of solitons in a gauge theory which has undergone the Higgs mechanism. If a gauge group $G$ is Higgsed to a subgroup $H$ by a scalar field $\phi$, one can attempt to find stable field configurations which cannot be continuously deformed to the vacuum configuration (without infinite cost in energy). For a field configuration of finite energy in $\mathbb{R}^{d-1}\times \mathbb{R}_\text{time}$, $\phi$ must asymptote to a zero energy configuration at the boundary of space. Then $\phi$ defines a map from an infinitely large sphere in space $S^{d-2}$ to $G/H$. These maps are classified by $\pi_{d-2}(G/H)$.

The simplest example appears in the Abelian Higgs model in $2+1$ dimensions. We have a U(1) gauge theory with the action $$ S = \int \mathrm{d}^3 x\, \left [ - \frac{1}{4} F_{\mu\nu}F^{\mu\nu} + |D_\mu \phi|^2 - V(|\phi|) \right],$$ with $$V(|\phi|) = \frac{\lambda}{4} (|\phi|^2 - v^2)^2.$$ The classical minima occur for $|\phi| = v$, and the gauge symmetry is completely Higgsed. The Hamiltonian is $$H = \int \mathrm{d}^2\mathbf{x} \left[\frac{1}{2} (E^2 + B^2)+ |D_i \phi|^2 + V(|\phi|) \right].$$ We look for a field configuration which cannot be continuously deformed to the vacuum solution. If the configuration is to have finite energy, $V(|\phi|)$ must go to zero at spatial infinity, so $|\phi|$ must approach $v$. The phase of $\phi$ is not fixed, however, $$\phi(r,\theta) \overset{r\to\infty}{\longrightarrow} v e^{i\sigma(\theta)}.$$ Thus, $\phi$ defines a map from the circle at spatial infinity to U(1), and so defines a class $[\phi]\in\pi_1(U(1))=\mathbb{Z}$. This integer is the winding number of the map $\phi$. Since an integer cannot change continuously under continuous deformations of $\phi$, a field configuration with non-zero winding number cannot be continuously deformed to the vacuum configuration (which has winding number zero). One must also of course be careful to see that the remaining terms in the Hamiltonian are finite. Having done so, one can deform this topologically non-trivial field configuration until the energy is minimized and thereby obtain a stable field configuration which is not connected to the vacuum. This soliton is called a vortex. For more details, see e.g. Preskill, Lectures on Vortices and Monopoles.

I'll give one other, somewhat more abstract, example application. A gauge theory with gauge group $G$ over a spacetime manifold $M$ is described mathematically by a principal $G$-bundle $P$ over $M$, $G \to P \to M$. Such a bundle is classified by characteristic classes in the cohomology groups $\left\{H^k(M,\pi_{k-1}(G))\right\}_{k=1}^{\dim M}$. For a four dimensional manifold, these are the groups $$H^1(M,\pi_0(G))\\ H^2(M,\pi_1(G))\\H^3(M,\pi_2(G))\\H^4(M,\pi_3(G)).$$ Let the gauge group be connected, so that $\pi_0(G) = 0$ and $H^1(M,\pi_0(G)) = 0$. For any Lie group $G$, $\pi_2(G) = 0$, so $H^3(M,\pi_2(G))$ is also irrelevant. For all the classical Lie groups $\pi_3(G) = \mathbb{Z}$ (except for $G = \mathrm{SO(4)}$). Then the class in $H^4(M,\mathbb{Z})$ specifies the instanton number of the bundle.

Finally, we have $H^2(M,\pi_1(G))$. If the gauge group is not simply-connected ($\pi_1(G) \neq 0$), a class in this group provides additional topological data necessary to specify the gauge theory beyond the usual instanton number. A characteristic class in this group is called the discrete magnetic flux or 't Hooft flux of the gauge theory. For some more discussion of these kinds of ideas, see e.g. Witten, Supersymmetric Index in Four-Dimensional Gauge Theories.

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