Genuine Lines, Screening, and Charge Lattices

A line is genuine when its definition needs no auxiliary surface after its intrinsic orientation, framing or spin data, and renormalization prescription have been fixed. Screening asks a different question: can dynamical endpoint excitations change the line’s charge? A third test is mutual locality: can the proposed electric, magnetic, or dyonic lines coexist without a nontrivial Dirac-surface phase? These tests determine admissible charge sectors, but they do not determine whether a line has an area law, a perimeter law, or a topological limit. The main setting below is an absolute, oriented four-dimensional theory with compact gauge group and an invertible Abelian reduced charge sector; boundaries, relative theories, and noninvertible fusion require additional data.

Required background. Wilson Lines and Loops supplies honest representation labels, endpoint covariance, and line renormalization. Disorder Operators and Singular Boundary Conditions supplies magnetic cocharacters, flux singularities, and the distinction between a physical attachment and a gauge presentation. Genuine Line Spectra, Discrete Theta Data, and Theory Specification supplies the reduced charge group, maximal-isotropic criterion, and discrete theta choices used here as inputs.

Computational companion. No runnable charge-lattice explorer is currently available. The quotient, pairing, and junction calculations below are therefore shown analytically and are self-contained.

Genuineness, screening, locality, and topology are different tests

Let $C$ be an oriented line support. A surface-attached candidate has the schematic form

$$\widetilde L_\ell(C;D), \qquad \partial D=C,$$

where $D$ is an auxiliary oriented surface. If $D$ and $D'$ have the same boundary, their difference closes to

$$\Sigma=D'\cup_C\overline D.$$

In a group-like sector, changing the attachment can act by a closed surface operator:

$$\widetilde L_\ell(C;D') =U(\Sigma)\,\widetilde L_\ell(C;D)$$

inside correlation functions with the same other insertions. If this action is nontrivial and cannot be absorbed into intrinsic line data, the complete object is the line together with its chosen surface; the line alone is not genuine. This does not make the relative object inconsistent. Open topological surfaces ending on non-genuine lines and the dependence on the global completion are exhibited in Gaiotto, Kapustin, Seiberg, and Willett 2015, § 4.1, arXiv v2, pp. 14–17, especially eqs. (4.1)–(4.2), Open PDF.

The operational distinctions are compactly summarized below.

Independent tests for a proposed line operator

Property	Operational test	What passing the test does not imply
Allowed global label	The representation, cocharacter, or dyonic pair exists for the actual global gauge group.	That the line needs no attached surface.
Genuine	No auxiliary bounding surface is needed to define the line.	That the charge is unscreened or the line is topological.
Endable or screened	A declared dynamical endpoint can change or remove the line charge.	That the corresponding closed line is non-genuine.
Mutually local	The oriented unit-linking phase with every other chosen genuine line is trivial.	That one unique complete spectrum has been selected.
Complete	No additional charge can be added while preserving locality in the declared ambient group.	That boundaries or relative bulk–boundary systems use the same maximal set.
Topological	Allowed support deformations leave correlation functions unchanged.	Genuineness merely from having a well-defined insertion.

Endpoint charges define the screening quotient

First suppose that an Abelian group $\Lambda_{\mathrm{gen}}$ of genuine line charges has already been chosen. Let $S\subseteq\Lambda_{\mathrm{gen}}$ be the subgroup generated by charges carried by declared dynamical endpoints. Fusion with those excitations identifies

$$\ell\sim\ell+s, \qquad s\in S,$$

so the unscreened charge sectors form

$$\mathcal A_{\mathrm{line}} =\frac{\Lambda_{\mathrm{gen}}}{S}.$$

The zero class means that the charge can be completely screened. It does not mean that a closed representative was never a genuine operator, nor that its finite-size correlation function equals that of the identity. Screening is an equivalence of charge sectors, not an equality of fully renormalized line operators.

The same construction can start one stage earlier, before a mutually local genuine set has been selected. Let $D_0$ be a finite Abelian reduced defect group with a perfect alternating pairing

$$b:D_0\times D_0\longrightarrow\mathbb Q/\mathbb Z.$$

If $S\subset D_0$ is an isotropic subgroup generated by dynamical endpoint charges, a probe must be local with every element of $S$. The compatible charges are therefore

$$S^\perp =\bigl\{x\in D_0\mathrel{:}b(x,s)=0 \text{ for every }s\in S\bigr\}.$$

After screening, the reduced group is not generally $D_0/S$ but

$$D_{\mathrm{scr}}=S^\perp/S.$$

The pairing descends to

$$\bar b([x],[y])=b(x,y).$$

This is well defined because adding $s\in S$ to $x$ changes the right-hand side by $b(s,y)=0$ for $y\in S^\perp$. Perfectness of $b$ gives $(S^\perp)^\perp=S$, so $\bar b$ is again perfect. In particular,

$$\lvert D_{\mathrm{scr}}\rvert =\frac{\lvert D_0\rvert}{\lvert S\rvert^2}.$$

This finite-group refinement is useful when screening and the choice of an absolute line spectrum must be made together. The general endpoint quotient and its Pontryagin-dual symmetry interpretation are developed in Bhardwaj et al. 2024, §§ 3.1–3.2, arXiv v2, pp. 26–38, especially eqs. (3.1)–(3.7), (3.16)–(3.17), and (3.44)–(3.47), Open PDF.

Mutual locality selects an absolute line spectrum

Fix the convention that the second oriented line sweeps positively around the first. For electric–magnetic charges $\gamma=(e,m)$ and $\gamma'=(e',m')$, define

$$\Omega(\gamma,\gamma') =\langle e,m'\rangle-\langle e',m\rangle.$$

The sweep phase is

$$B(\gamma,\gamma') =\exp\!\bigl(2\pi i\,\Omega(\gamma,\gamma')\bigr).$$

Two lines can belong to one mutually local genuine spectrum only if this phase is one. Reversing either line or the sweep changes the sign of $\Omega$ and complex-conjugates $B$, so the condition $B=1$ is orientation-independent.

For a cyclic reduced group $D_0=\mathbb Z_k^2$, the pairing is

$$b\bigl((e,m),(e',m')\bigr) =\frac{em'-me'}{k}\pmod 1.$$

These conventions, the reduced electric–magnetic labels, and the mutual locality congruence are given in Aharony, Seiberg, and Tachikawa 2013, § 1.1, arXiv v5, pp. 3–4, eqs. (1.1)–(1.4), Open PDF.

A complete absolute theory in the finite perfect-pairing setting chooses a maximal isotropic subgroup $L\subset D_{\mathrm{scr}}$. Equivalently,

$$L=L^\perp.$$

If $x\notin L$, maximality implies that some $\ell\in L$ has $\bar b(x,\ell)\ne0$. Sweeping an auxiliary surface for the putative line $x$ through the genuine line $\ell$ changes the correlator by $\exp(2\pi i\bar b(x,\ell))$. This detects why $x$ cannot simply be added as another genuine charge. It does not by itself construct the needed relative line–surface object. Completeness as a maximal mutually local choice is stated in Aharony, Seiberg, and Tachikawa 2013, § 1.3, arXiv v5, p. 6, Open PDF; the hard prerequisite develops the reduced spectra and discrete-theta examples in detail.

Maximal isotropy is not an all-purpose classification theorem. The ambient charge group, global form, tangential structure, spin or framing labels, and allowed boundaries must already have been declared. In relative bulk–boundary constructions, which lines are genuine can depend on the boundary condition Gaiotto, Kapustin, Seiberg, and Willett 2015, § 6, arXiv v2, pp. 35–36, especially the discussion around eq. (6.10), Open PDF.

Charge-N matter leaves a finite surface network

Consider a compact-$U(1)$ theory on a closed oriented spin four-manifold, with faithfully normalized connection $a=gA$, curvature $f=da$ locally, and gauge parameter $\lambda\sim\lambda+2\pi$. Work at $\theta=0$ and choose the standard absolute spectrum in which every integer pair $(n,m)$ labels a genuine Wilson–’t Hooft line. Let all dynamical electric charges generate $N\mathbb Z$ with $N\ge 2$, and assume there are no dynamical magnetic monopoles. Wilson and magnetic labels are

$$W_n(C)=\exp\!\left(i n\oint_C a\right), \qquad \frac{1}{2\pi}\int_{S^2}f=m, \qquad (n,m)\in\mathbb Z^2.$$

For two such lines,

$$\Omega\bigl((n,m),(n',m')\bigr) =nm'-mn'\in\mathbb Z,$$

so the integral charge lattice passes the mutual-locality test. The oriented spin-manifold $U(1)$ charge lattice and Dirac pairing are developed in Ang, Roumpedakis, and Seifnashri 2020, § 2.1, arXiv v2, pp. 4–5, eqs. (2.1)–(2.7), Open PDF.

The screening subgroup and quotient are

$$S=N\mathbb Z\times\{0\}, \qquad \mathcal A_{\mathrm{line}} \cong\mathbb Z_N\times\mathbb Z.$$

The first factor is electric charge modulo $N$; the magnetic integer remains because magnetic endpoints were excluded. If $r=[n]_N$, then $W_n$ and $W_{n+N}$ have the same unscreened electric class.

The endpoint test is explicit. Let $P:y\to x$ and let a charge-$N$ field transform as $\psi_N\mapsto e^{iN\lambda}\psi_N$. Then

$$W_N(P) \longmapsto e^{iN[\lambda(x)-\lambda(y)]}W_N(P),$$

and

$$\overline\psi_N(x)\,W_N(P)\,\psi_N(y)$$

is gauge invariant. Thus the closed $W_N$ is genuine but screenable; the bare open transporter is not a standalone gauge-invariant observable. Under the declared spectrum, $W_1$ is genuine and cannot end. For one charge-$N$ field, the quotient $\mathbb Z/N\mathbb Z$ and the residual electric one-form symmetry are derived in Bhardwaj et al. 2024, § 3.2.1, arXiv v2, pp. 29–31, Fig. 9 and eqs. (3.10)–(3.19), Open PDF.

Now assume that the residual electric $\mathbb Z_N$ one-form symmetry has an exact, non-anomalous, invertible group-like surface network. Let $U_\alpha(\Sigma)$, $\alpha\in\mathbb Z_N$, be a closed oriented symmetry surface, and let $C$ and $\Sigma$ be disjoint closed supports in a region where the integer linking number is defined. With all other insertions $\mathcal X$ outside the swept region,

$$\begin{aligned} &\left\langle U_\alpha(\Sigma)W_n(C)\mathcal X \right\rangle \\ &\qquad= \exp\!\left[ \frac{2\pi i}{N}\,\alpha r\, \operatorname{Lk}(\Sigma,C) \right] \left\langle W_n(C)\mathcal X\right\rangle . \end{aligned}$$

The orientation and fusion rules are

$$U_\alpha(\overline\Sigma)=U_{-\alpha}(\Sigma), \qquad W_n(\overline C)=W_{-n}(C),$$

and

$$U_\alpha\otimes U_\beta \simeq U_{\alpha+\beta\;\mathrm{mod}\;N}.$$

If two incoming sheets and one outgoing sheet meet along an oriented junction line $K$, the incidence data are

$$J_{\alpha,\beta}^{\ \gamma}: U_\alpha\otimes U_\beta\longrightarrow U_\gamma, \qquad \alpha+\beta-\gamma=0\pmod N.$$

The congruence is necessary; it does not construct, normalize, or prove the coherence of the junction. The group-like surface action and character pairing follow from Gaiotto, Kapustin, Seiberg, and Willett 2015, § 3, arXiv v2, pp. 11–13, eqs. (3.1)–(3.4), Open PDF and Bhardwaj et al. 2024, § 2.2.2, arXiv v2, pp. 18–19, especially eqs. (2.63) and (2.69)–(2.70), Open PDF. The trivalent incidence condition is the additive $\mathbb Z_N$ specialization of the finite-group network law, with the outgoing sheet orientation reversed Gaiotto, Kapustin, Seiberg, and Willett 2015, § 2, arXiv v2, pp. 6–8, especially eq. (2.2) and the junction paragraph preceding eq. (2.6), Open PDF.

There are three distinct surfaces in this discussion. The closed $U_\alpha(\Sigma)$ measures a line charge. An auxiliary $D$ with $\partial D=C$ is part of the definition of a surface-attached line. The sheets meeting at $K$ are fusion-network strata. Calling all three a “Dirac surface” would erase the operational distinctions.

For a concrete arithmetic check, take $N=6$, $n=8$, and hence $r=2$. A positively linked $U_4$ gives

$$\exp\!\left(\frac{2\pi i}{6}\,4\cdot2\right) =\exp\!\left(\frac{2\pi i}{3}\right).$$

The junction $U_4\otimes U_5\to U_3$ passes the incidence test because $4+5-3=6\equiv0\pmod6$.

Matter changes the non-Abelian reduced lattice

The same quotient logic gives a useful reduced center-charge calculation without repeating the full discrete-theta classification. Begin with the universal $\mathfrak{su}(N)$ reduced group

$$D_0=\mathbb Z_N\times\mathbb Z_N, \qquad b\bigl((e,m),(e',m')\bigr) =\frac{em'-me'}{N}\pmod1.$$

Suppose the dynamical matter representations have $N$-alities $r_1,\ldots,r_s$, and define

$$d=\gcd(N,r_1,\ldots,r_s).$$

The electric screening subgroup is $S=\langle(r_i,0)\rangle$. The condition $b((e,m),S)=0$ forces $m=(N/d)\mu$ with $\mu\in\mathbb Z_d$, while quotienting by $S$ reduces $e$ modulo $d$. Consequently,

$$\begin{aligned} D_{\mathrm{scr}} &\simeq\mathbb Z_d\times\mathbb Z_d, \\ \bar b\bigl((e,\mu),(e',\mu')\bigr) &=\frac{e\mu'-e'\mu}{d}\pmod1. \end{aligned}$$

Fundamental matter gives $d=1$ and removes this reduced center-charge group. For $\mathfrak{su}(4)$ matter of $N$-ality two, $d=2$ and the surviving reduced group is $\mathbb Z_2^2$. The chosen actual global group and discrete-theta data must still select a maximal mutually local genuine subgroup; the reduced quotient does not make that choice automatically. Matter screening and the gcd formula for the residual center symmetry are given in Bhardwaj et al. 2024, § 3.3.4, arXiv v2, pp. 48–49, especially eqs. (3.137)–(3.145), Open PDF.

This calculation tracks center charges only. It is not a classification of full representation fusion, dyonic dressings, monopole bubbling, line-local degrees of freedom, or categorical line types.

What the lattice does not determine

The sequence of inputs matters:

1. Fix the actual global gauge group, allowed bundles, tangential structure, and boundary domain.
2. Determine honest electric representations and magnetic cocharacters.
3. Specify which candidates are genuine, including discrete-theta or other surface-attachment data.
4. Impose mutual locality and completeness in the declared ambient reduced group.
5. Quotient by the charges of the dynamical endpoints actually present.
6. Only then interpret the surviving character group as an exact higher-form symmetry, after checking its topological network and anomaly.

Changing the order can erase essential data. In particular, a low-energy effective theory may omit heavy particles or extended defects that still screen ultraviolet line charges. A 2026 semiclassical study gives an explicit example in which an Abelianized description misses heavy-$W$ and twist-vortex screening and would otherwise assign spurious selection rules Hayashi and Tanizaki 2026, § 2.3.2 and §§ 3.1–3.4, arXiv v1, pp. 14–19, especially eqs. (3.1)–(3.7), Open PDF. This current result is a scoped preprint example, not a general theorem about every effective gauge theory.

The scientific literature used for this status check was reviewed through 9 August 2026. It supports the bounded conclusion above: the charge quotient is exact only for the declared spectrum and background category. It does not determine a screening length, an RG endpoint, or an area/perimeter law.

For the symmetry operators that measure the surviving sectors, continue to Electric and Magnetic One-Form Symmetries. For the geometry of the pairing, continue to Linking, Braiding, and Framing. Phase realization, confinement, and long-distance line diagnostics belong to Line Operators, Screening, and Generalized-Symmetry Diagnostics. For categorical completeness and coherent line/junction classification, see Defects on Stratified Spacetimes and Higher-Categorical Composition.

Common pitfalls

Genuine means unscreened. A closed $W_N$ in the compact-$U(1)$ example is genuine because it needs no surface, but charge-$N$ matter screens it. Genuineness concerns definition; screening concerns dynamical endpoints.

A zero screening class deletes the operator. It deletes a protected charge distinction. Renormalized line observables at finite scales can still contain nontrivial local and dynamical information.

Every candidate charge can be made genuine. Charges with nontrivial Dirac pairing cannot all be inserted as mutually local genuine lines in one absolute theory. Some require surface attachments or belong to a different global-form/discrete-theta choice.

A conserved quotient fixes a phase of matter. The quotient can rule out or identify charge sectors, but area laws, perimeter laws, string breaking, and confinement require dynamical analysis.

Label conservation constructs a junction. The congruence at a junction is only an incidence condition. Existence, normalization, associativity, and possible anomaly phases are additional data.

Check your understanding

1. In the compact-$U(1)$ example with $N=5$, find the unscreened electric class of $W_{12}$ and the phase produced by a positively linked $U_2$.
2. Explain how $W_5$ can be both genuine and screened, while $W_1$ is genuine and unscreened under the declared matter spectrum.
3. Start with electric matter charges $6$ and $10$. Find the residual electric one-form group and identify which Wilson charges are completely screened.
4. For $D_0=\mathbb Z_4^2$, show that the candidate charges $(1,0)$ and $(0,1)$ cannot both lie in one mutually local genuine subgroup.
5. In a $\mathbb Z_5$ surface network, two incoming sheets have labels $2$ and $4$. Find the outgoing label, then explain why this arithmetic does not prove that a coherent junction exists.

Solutions

For $N=5$, $12\equiv2\pmod5$. A unit link with $U_2$ gives

$$\exp\!\left(\frac{2\pi i}{5}\,2\cdot2\right) =\exp\!\left(\frac{8\pi i}{5}\right).$$

The closed $W_5$ is defined by holonomy alone, so it is genuine. A charge-$5$ endpoint makes it trivial in the screening quotient. No declared endpoint has charge $1$, so $W_1$ remains in a nonzero class.

Charges $6$ and $10$ generate $\gcd(6,10)\mathbb Z=2\mathbb Z$. Wilson charges are therefore identified modulo $2$, the unscreened electric group is $\mathbb Z_2$, and precisely the even charges are completely screened.

For $k=4$,

$$b\bigl((1,0),(0,1)\bigr)=\frac14\pmod1,$$

so the unit-linking phase is $e^{\pi i/2}\ne1$. The two candidates cannot both lie in one mutually local genuine subgroup.

The outgoing surface label is $2+4\equiv1\pmod5$. This satisfies the incidence condition, but it does not supply a junction operator or prove its normalization and associativity.

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 2013, no. 8 (2013): 115. DOI. Open PDF, arXiv v5.
• Ang, J. P., Konstantinos Roumpedakis, and Sahand Seifnashri. “Line Operators of Gauge Theories on Non-Spin Manifolds.” Journal of High Energy Physics 2020, no. 4 (2020): 087. DOI. Open PDF, arXiv v2.
• Bhardwaj, Lakshya, Lea E. Bottini, Ludovic Fraser-Taliente, Liam Gladden, Dewi S. W. Gould, Arthur Platschorre, and Hannah Tillim. “Lectures on Generalized Symmetries.” Physics Reports 1051 (2024): 1–87. DOI. Open PDF, arXiv v2.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, no. 2 (2015): 172. DOI. Open PDF, arXiv v2.
• Hayashi, Yui, and Yuya Tanizaki. “Wilson–’t Hooft Classification and the Perimeter Law for Dyonic Loops in 3d Monopole Semiclassics.” arXiv:2601.02058v1 [hep-th], submitted 5 January 2026. YITP-25-197. Stable record. Open PDF, arXiv v1.