# Why is the 4D $U(1)$ electric 1-form symmetry a global symmetry?

I am reading the paper Generalised Global Symmetries to understand higher-form symmetries. The first example in Section 4 that the authors talk about is the free Maxwell theory in 4d, i.e., pure $$U(1)$$ gauge theory in 4D. In this example, they talk about two 1-form symmetries: electric $$U(1)_e$$, with 2-form current $$j_e \sim \star F$$, and magnetic $$U(1)_m$$, with 2-form current $$j_m \sim F \equiv dA$$. It is mentioned that the action of the electric 1-form symmetry on the gauge field $$A$$ is a shift by a flat connection $$\lambda$$, i.e., $$d\lambda=0$$. A flat connection is not necessarily a constant, then why is this called a global symmetry?

A related fact discussed in the same section is that the topological symmetry operator corresponding to electric 1-form symmetry is given by $$U_\alpha^e(M_2)=\exp\left(i\frac{2\alpha}{g^2}\int_{M_2}\star F\right),$$ where $$M_2$$ is a 2d submanifold in 4d spacetime, $$g$$ is the gauge coupling constant, and $$e^{i\alpha}\in U(1)_e$$ is the corresponding group element. Here, $$\alpha$$ is indeed a constant. How is $$\alpha$$ related to $$\lambda$$ above?

This is what I think makes the $$U(1)$$ electric 1-form symmetry global. The action of shift of the gauge field by a flat connection on a Wilson line operator is
$$A \rightarrow A + \lambda\implies W_n(C) \rightarrow W_n(C) \exp \left( i n \oint_C \lambda \right).$$
Let $$\alpha(C)=\oint_C \lambda$$, which is the group parameter for the above transformation. Now, consider two Wilson operators defined over two different loops $$C$$ and $$C'$$. As long as the difference $$L$$ of $$C$$ and $$C'$$ lies in the trivial class of $$H_1(X)$$, where $$X$$ is the spacetime, we have
$$\left( \oint_C - \oint_{C'} \right)\lambda = \oint_{L=\partial S} \lambda = \int_S d\lambda = 0,$$
because $$\lambda$$ is a flat connection. Here, existence of the 2d manifold $$S$$ is guaranteed by $$L$$ belonging to trivial class in $$H_1(X)$$. So, $$\alpha(C) = \alpha(C')$$ for all such $$C$$ and $$C'$$ which are smoothly connected to each other.
Global then means that the group parameter $$\alpha$$ is independent (in the above sense) of $$C$$ (1D manifold), which is possible whenever $$\lambda$$ is a flat connection. This is generalisation of the notion of global 0-form symmetry where the group parameter $$\alpha$$ is independent of $$x$$ (0D manifold), the position of the local operator.