Linking of a sphere with a Wilson line In the papers such as Ref.[I] and Ref.[II], they have introduced the operator,
$$
U_\alpha (M_{d-2}) = e^{\frac{i\alpha}{g^2}\int_{M_{d-2}}*F} .
$$
They said that the Wilson loop:
$$W_n(\gamma)=e^{in\oint_\gamma A} \ ,$$
is charged under the above operator. For example Ref.[II] claims that:
"
The Ward identity for this
1-form global symmetry implies that such a charge operator supported on an $S^{d−2}$
linking a
Wilson line is equal to the Wilson line up to a phase,
$U_{g=e^{iα}} (S^{d-2}
)W_n(γ) = e^{iαn}W_n(γ)$"
My question is, what is the meaning of linking between $S^{d-2}$ and a Wilson loop. I can imagine an $S^1$ linked with another $S^1$. But how can, for example in 4d, an $S^2$ is linked with an $S^1$?
References:
[I] "Generalized Global Symmetries" by Gaitto et al (https://arxiv.org/abs/1412.5148  [page 14]).
[II] "Non-Invertible Global Symmetries and Completeness of the Spectrum" by Ben Heidenreich et al. (https://arxiv.org/abs/2104.07036   [page 16])
 A: Fact of life: In $d$ a dimensions, a $p$-dimensional closed manifold can link with a $q$ dimensional closed manifold if $p+q+1=d$. (cf. wiki: linking number) $\newcommand{\S}{\mathbb{S}}\newcommand{\R}{\mathbb{R}}$
This fact of life implies that $\S^1$ can link with a closed $(d-2)$-dimensional closed manifold, such as $\S^{d-2}$.
To visualise this in 4 dimensions we can think of the sphere as fibred by circles. Explicitly, let's think of $\mathbb{R}^4$ as $\mathbb{R}^3\times\mathbb{R}_t$, where $\mathbb{R}_t$ is time (or Euclidean time, doesn't matter it is for visualisation purposes only). Then an $\S^2$ in $\R^3\times\R_t$ is an $\S^1$ with time-dependent radius. At $t\to-\infty$ it starts its life as a point, $\S^1_{r=\varepsilon}\subset\R^3\times\{t\to-\infty\}$, then it slowly grows until at $t=0$ is is $\S^1_{r=1}\subset\R^3\times\{t=0\}$ and then it shrinks to zero again at $t\to\infty$, $\S^1_{r=\varepsilon}\subset\R^3\times\{t\to\infty\}$. At any such point, this $\S^1_{r(t)}\subset\R^3\times\{t\}$, can link with another $\color{maroon}{\S^1}$. So, if at $t\to-\infty$ your sphere  starts its life linked with an $\color{maroon}{\S^1}$, it will stay linked until it dies at $t\to\infty$ (cf. figure below)

