Magnetic one-form charge of an external U(1) line

Use natural units $\mathrm{\hslash }=c=1$ and $M={\mathbb{R}}^4$ with its standard orientation and Euclidean metric. Let $C$ be an admitted closed smooth oriented curve and set $X=M\setminus C$. The selected field model is pure compact Maxwell theory with actual gauge group $U(1)=\{e^{i\alpha }:\alpha \in \mathbb{R}\mathrm{/}2\pi \mathbb{Z}\}$, no dynamical electric or magnetic matter, and an ordinary smooth right principal $U(1)$-bundle $P\to X$ with smooth connection, using the defining character and the standard compact-circle normalization. The model admits nonzero genuine external magnetic line probes $T_n(C^{\mathrm{\prime }})$ for every $n\in \mathbb{Z}$ on its admitted closed smooth oriented curves $C^{\mathrm{\prime }}$, with flux $2\pi n$ on sufficiently small linking spheres; in each correlator sector considered below, one such line is fixed and is the only magnetic source. In particular, the complete admitted insertion data include a unit-charge probe $T_1$, which is used to make the $U(1)$ symmetry action effective. A principal connection is a smooth $\mathfrak{u}(1)$-valued one-form reproducing each fundamental vertical vector and equivariant under the right group action. Identify $\mathfrak{u}(1)=i\mathbb{R}$ with $\mathbb{R}$ by $ia\mapsto a$, so its local potentials $A_i$ have holonomy $\exp (i\int A_i)$ and the real curvature $F$ obeys $F{\mathrm{\mid }}_{U_i}=d{A}_i$, since the Abelian Lie bracket vanishes. The model fixes the action convention $S[A]=(2e^2{)}^{-1}\int F\wedge \ast F$ with dimensionless $e>0$, integrating outside any prescribed line-core tube with its declared boundary condition. It also fixes the allowed bundle sectors, boundary behavior at infinity, and any regulator or measure; no continuum measure is constructed here. With this normalization $[A_i]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-1}$, $[F]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-2}$, and ${\int }_{\mathrm{\Sigma }}F$ is dimensionless.

Let $C$ be the closed image of an orientation-preserving smooth embedding $c:\mathcal{C}\hookrightarrow M$ from an oriented smooth one-manifold $\mathcal{C}$ without boundary, and in the sector being calculated let it carry the admitted genuine external magnetic line $T_m(C)$, where genuine means that no auxiliary surface is attached to define the insertion. At each $x\in C$, orient the normal quotient $N_{x}C:=T_{x}M\mathrm{/}{T}_{x}C$ by requiring that a positive tangent $u\in T_{x}C$, followed by lifts of a positive basis of $N_{x}C$, gives the positive orientation of $T_{x}M$. Orient every small linking sphere $S_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}^2$ as the boundary of a small ball in $N_{x}C$, with the outward normal first. The line's integer charge $m\in \mathbb{Z}$ is specified by ${\int }_{S_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}^2}F=2\pi m$ on every sufficiently small such sphere. In this sector the model has no other magnetic source. Its complete input domain for this fixed sector has the external line $T_m(C)$, any finite list of compact oriented surface insertions $U_{{\alpha }_j}({\mathrm{\Sigma }}_j)$ disjoint from $C$, and any finite list of admitted gauge-invariant ordinary field functionals $f_k$ with specified supports. The data $b$ record the boundary behavior at infinity, any excised core-tube boundary condition, bundle sectors and regulator choices. Let ${\mathcal{F}}_{C,m,b}$ be the complex vector space of actual functions on the admitted bundle-and-connection configurations generated by finite products of these $f_k$ and the surface functions defined below, with equality understood pointwise. The selected model supplies a complex-linear functional ${\mathcal{L}}_{C,m,b}$ on that space, with its prescribed normalization. Its complete correlator rule is

$$C_b(T_m(C);U_{{\alpha }_1}({\mathrm{\Sigma }}_1),\dots ,U_{{\alpha }_r}({\mathrm{\Sigma }}_r);f_1,\dots ,f_n)={\mathcal{L}}_{C,m,b}\text{\hspace{0.17em}⁣}(\prod _{j=1}^r{U}_{{\alpha }_j}({\mathrm{\Sigma }}_j)\prod _{k=1}^n{f}_k)\mathrm{.}$$

There is no assumption that ${\mathcal{L}}_{C,m,b}(1)$ is nonzero. The functional, admissibility and normalization are conditional model inputs; no continuum measure is constructed. Gauge changes transport configurations and $b$, with the model's functional required to satisfy the corresponding pullback covariance. Allowed support isotopies transport all orientations and line boundary data, keep $b$ fixed, and avoid the other insertions. Pointwise equalities of the displayed functions therefore imply equalities for every such complete correlator by linearity, and a constant pointwise phase factors out of it.

For a compact smooth oriented two-manifold $\mathrm{\Sigma }$ without boundary and a smooth embedding $\iota :\mathrm{\Sigma }\hookrightarrow X$, and $\alpha \in \mathbb{R}\mathrm{/}2\pi \mathbb{Z}$, define the magnetic surface operator using any representative $\tilde{\alpha }\in \mathbb{R}$:

$$U_{\alpha }(\mathrm{\Sigma })=\exp \text{\hspace{0.17em}⁣}\left(\frac{i\tilde{\alpha }}{2\pi }{\int }_{\mathrm{\Sigma }}F\right)\mathrm{.}$$

For this example, a smooth singular three-chain is a finite integer linear combination of smooth maps from the standard 3-simplex, with boundary the alternating sum of its face maps. For a source-free move, the model supplies a compact smooth singular three-chain $V$ in $M\setminus C$ with $\mathrm{\partial }V={\mathrm{\Sigma }}_1-{\mathrm{\Sigma }}_0$. For a move across the line, the model supplies a compact oriented smooth three-dimensional track $W$ in $M$, represented as a finite smooth singular chain, with endpoint boundary $\mathrm{\partial }W={\mathrm{\Sigma }}_1-{\mathrm{\Sigma }}_0$; its support is a smooth embedded three-manifold near each of its finitely many intersections with $C$, and the intersections are transverse interior points. Transverse means that the tangent space of this three-manifold and the tangent line of $C$ span $T_{x}M$. At an intersection $x_j$, let $u$ be a positive tangent vector to $C$ and $(w_1,w_2,w_3)$ a positive basis of $T_{x_j}W$. Set ${ϵ}_j=+1$ when $(u,w_1,w_2,w_3)$ is positive in $T_{x_j}M$ and ${ϵ}_j=-1$ otherwise; this ordered convention defines $I(W,C):={\sum }_j{ϵ}_j$. The quotient map $T_{x_j}W\to N_{x_j}C$ then has orientation sign ${ϵ}_j$. Orient each small linking sphere $S_j^2$ as the boundary of a small ball in the already oriented normal space, with the outward normal first. Since the inner boundary of the punctured track has the opposite boundary orientation, excising a small open three-ball at each intersection gives a punctured chain $V$ in $M\setminus C$ with $\mathrm{\partial }V={\mathrm{\Sigma }}_1-{\mathrm{\Sigma }}_0-{\sum }_j{ϵ}_j{S}_j^2$. The claimed action is that a source-free move leaves $U_{\alpha }$ unchanged, while a move across the line multiplies it by $e^{i\tilde{\alpha }mI(W,C)}$. In particular, the line carries the character ${\chi }_m(e^{i\alpha })=e^{i\alpha m}$ of the magnetic one-form $U(1)$ symmetry.

Gaiotto, Kapustin, Seiberg, and Willett, §4.1, PDF pp. 14–16

Bhardwaj et al., §2.2.2 Example 2.3 and §4.1 Example 4.1