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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?

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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.

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