Genuine Line Spectra, Discrete Theta Data, and Theory Specification

In four dimensions, fixing the Lie algebra and connected global gauge group need not finish the definition of a gauge theory. One must also specify which electric, magnetic, and dyonic line charges occur as a complete mutually local spectrum of genuine operators. For quotient groups, discrete theta data can correlate the magnetic and electric labels. The pair $\operatorname{SO}(3)_+$ and $\operatorname{SO}(3)_-$ is the basic counterexample to group data being sufficient: the two theories have the same algebra and global group but different genuine dyonic spectra. This page works at the reduced topological charge level for compact connected four-dimensional gauge theories; it does not construct renormalized line operators or determine their phase behavior.

Required background. Global Form, Matter Representations, and the Faithful Gauge Group supplies the central-quotient, representation-descent, and bundle data assumed below.

Helpful background. Parallel Transport and Holonomy supplies the holonomy description of Wilson probes. Compact Lie Groups, Roots, Weights, and Weyl Structure supplies the weight and center-character language behind the reduced electric labels.

Genuine lines are additional theory data

A line defect supported on a curve $C$ is genuine when its insertion is defined without choosing an auxiliary topological surface ending on $C$, after any required orientation, framing, and spin data have been fixed. A surface-attached line can still be a meaningful defect, but its correlation functions also depend on a surface $D$ with $\partial D=C$. It is not a member of the theory’s genuine line spectrum by itself.

This distinction is independent of long-distance dynamics. A genuine line need not be topological or unscreened, and the words “genuine” and “confining” are not opposites. Area laws, perimeter laws, string breaking, and screening lengths are dynamical questions. Here the only question is which reduced charge classes can be inserted without an attached surface and can coexist as mutually local operators.

The spectrum is part of the theory specification because two choices can have the same local operators and the same local correlation functions on $\mathbb R^4$ without line insertions, yet differ once line or surface operators are admitted. The need for a complete mutually local line choice is developed in Aharony, Seiberg, and Tachikawa 2013, arXiv v5, §§ 1–1.3, preprint pp. 1–6, Open PDF. The genuine and non-genuine dyonic distinction is reviewed in Bhardwaj et al. 2024, arXiv v2, § 3.4, preprint pp. 53–57, Open PDF.

Mutual locality selects a maximal charge subgroup

Take as input a cyclic reduced charge group $\mathbb Z_k$. Root- and coroot-lattice identifications and any declared dynamical screening relations have already been imposed; the labels below do not classify the full renormalized operators or their fusion category. After choosing generators, a line charge is

$$(e,m)\in\mathbb Z_k\times\mathbb Z_k,$$

where $m$ is the magnetic class and $e$ labels an electric character under the identification $\widehat{\mathbb Z_k}\simeq\mathbb Z_k$.

Fix the convention that the swept surface of the second line links the first once with positive orientation. Define the antisymmetric integer

$$\Delta\bigl((e,m),(e',m')\bigr) =em'-me'.$$

Transporting a line $(e',m')$ around $(e,m)$ produces the phase

$$B =\exp\!\left[ \frac{2\pi i}{k} \Delta \right].$$

If both lines are genuine in one theory, their joint correlation function must return to itself. Hence

$$em'-me'\equiv0\pmod{k}.$$

Reversing the linking convention reverses the sign of the antisymmetric pairing but does not change its vanishing. The linking argument and cyclic congruence appear in Aharony, Seiberg, and Tachikawa 2013, arXiv v5, § 1.1, preprint pp. 3–4, Open PDF.

Let $L\subset\mathbb Z_k^2$ be the allowed reduced charges. Fusion and orientation reversal make $L$ a subgroup. Mutual locality says that the pairing vanishes for every pair in $L$; such a subgroup is called isotropic. Completeness requires $L$ to be maximal under inclusion: no additional charge can be adjoined while preserving mutual locality. For this nondegenerate pairing, let $L^\perp$ denote all charges that pair trivially with every $\ell\in L$. Completeness can then be written

$$L=L^\perp.$$

Indeed, if $x\in L^\perp\setminus L$, then $x$ pairs trivially with $L$ and, by alternation, with itself. The subgroup generated by $L$ and $x$ would be a larger isotropic subgroup, contradicting maximality.

The explicit maximal-completeness requirement is stated in Aharony, Seiberg, and Tachikawa 2013, arXiv v5, § 1.3, preprint p. 6, Open PDF. “Maximal” does not mean that every formal electric and magnetic label is genuine. It means that the mutually local set cannot be enlarged.

For a noncyclic finite Abelian magnetic group $A$, the electric label is a character $\chi\in\widehat A$. Writing characters additively as maps $A\to\mathbb Q/\mathbb Z$, the finite pairing becomes

$$\begin{aligned} \left\langle(\chi,m),(\chi',m')\right\rangle &=\chi(m')-\chi'(m) \\ &\in\mathbb Q/\mathbb Z. \end{aligned}$$

The allowed subgroup must be maximal isotropic for this pairing. A complete cocharacter-lattice classification for arbitrary groups is outside the present scope.

Three Z₂ spectra separate group data from theory data

For $\mathfrak{su}(2)$, use reduced center charges $(e,m)\in\mathbb Z_2^2$. Every two distinct nonzero charge vectors have nontrivial pairing, so each maximal isotropic subgroup contains the identity and exactly one nonzero class. The three choices are

$$\begin{aligned} L_{\operatorname{SU}(2)} &=\langle(1,0)\rangle, \\ L_{\operatorname{SO}(3)_+} &=\langle(0,1)\rangle, \\ L_{\operatorname{SO}(3)_-} &=\langle(1,1)\rangle. \end{aligned}$$

The labels have concrete meanings.

Electric choice. For $\operatorname{SU}(2)$, the half-integer Wilson class $(1,0)$ is genuine. The odd magnetic class is not genuine in that theory.

Pure-magnetic choice. For $\operatorname{SO}(3)_+$, the minimal pure magnetic class $(0,1)$ is genuine. Half-integer Wilson representations do not descend to $\operatorname{SO}(3)$.

Dyonic choice. For $\operatorname{SO}(3)_-$, the minimal genuine line with odd magnetic charge is $(1,1)$. Its odd electric label is not an honest standalone $\operatorname{SO}(3)$ Wilson representation. Neither the formal $(1,0)$ line nor the formal $(0,1)$ line is genuine separately in this theory, but their dyonic combination is genuine.

Thus $\operatorname{SO}(3)_+$ and $\operatorname{SO}(3)_-$ share the same Lie algebra, connected global group, and honest pure Wilson representations. They are nevertheless different QFTs because their complete genuine line spectra differ. These three spectra and their mutual-locality interpretation are given in Aharony, Seiberg, and Tachikawa 2013, arXiv v5, §§ 1.2–1.3, preprint pp. 4–6, Open PDF.

Discrete theta data correlate electric and magnetic charge

For a quotient $G=\widetilde G/\mathcal Z$, a discrete theta choice can associate a magnetic class $m\in\mathcal Z$ with an electric character $\gamma(m)\in\widehat{\mathcal Z}$. This is the general reason that a minimal magnetic defect may have to carry an electric dressing. The map $\gamma$ is induced by admissible discrete-theta data; its consistency conditions depend on the group and spacetime structure, so it is not an arbitrary homomorphism in every setting Bhardwaj et al. 2024, arXiv v2, § 3.4, preprint p. 57, Open PDF.

The $\operatorname{SO}(3)$ example is especially explicit. Let $X$ be a closed oriented spin four-manifold and $P\to X$ an $\operatorname{SO}(3)$ bundle. Its lift obstruction is $w_2(P)\in H^2(X,\mathbb Z_2)$, and its Pontryagin square is

$$\mathcal P\!\left(w_2(P)\right) \in H^4(X,\mathbb Z_4).$$

On a spin four-manifold, the integral of this class is even modulo $4$. Define its reduced integral by

$$I(P) \equiv \int_X\mathcal P\!\left(w_2(P)\right) \pmod 4.$$

The two discrete choices can then be represented by the path-integral factor

$$W_p(P) =\exp\!\left[ \frac{i\pi p}{2}I(P) \right], \qquad p\in\mathbb Z_2.$$

The choice $p=0$ gives the $+$ spectrum and $p=1$ gives the $-$ spectrum in this normalization. The spin assumption and this Pontryagin-square term are explained in Bhardwaj et al. 2024, arXiv v2, § 3.4, preprint pp. 53–57, Open PDF.

The ordinary theta angle acts on the reduced line labels through the Witten effect. With the present charge convention,

$$\begin{gathered} \theta\longmapsto\theta+2\pi, \\ (e,m)\longmapsto(e+m,m). \end{gathered}$$

It sends $\langle(0,1)\rangle$ to $\langle(1,1)\rangle$ and sends the latter back after a second shift. Equivalently, one may parameterize the spin $\operatorname{SO}(3)$ family by

$$(\theta,p)\sim(\theta+2\pi,p+1).$$

A $2\pi$ shift therefore permutes $\operatorname{SO}(3)_+$ and $\operatorname{SO}(3)_-$; it is not a period of either fixed line spectrum. The fixed theory returns after $4\pi$ on spin manifolds in this normalization Aharony, Seiberg, and Tachikawa 2013, arXiv v5, § 1.2, preprint p. 5, Open PDF.

On a non-spin manifold, $\mathcal P(w_2)$ need not be even. The simplified $p\in\mathbb Z_2$ factor above then does not apply unchanged; line statistics also require further choices. When arbitrary oriented non-spin backgrounds and quarter-instanton sectors are admitted, the cited $\operatorname{SO}(3)$ normalization gives an $8\pi$ fixed-theory return. No universal periodicity statement should be exported beyond the declared spin setting Aharony, Seiberg, and Tachikawa 2013, arXiv v5, § 1.2, preprint p. 5, Open PDF.

A bounded SO(3) system in three descriptions

Consider adjoint-matter $\mathfrak{su}(2)$ Yang–Mills theory on

$$M=\mathbb R_t\times\Sigma, \qquad \Sigma=[r_-,r_+]\times S^2.$$

Interpret the inner boundary $\mathbb R_t\times S^2$ as a small tube around an excised timelike line. Choose boundary data that preserve the defect sector, permit the relevant $\operatorname{SO}(3)$ bundle, and do not supply extra boundary degrees of freedom or an attached surface that changes which bulk line is genuine.

Orbit description. For a bundle $P_m\to\Sigma$, define the magnetic class on the linking sphere by

$$m =\left\langle w_2(P_m),[S^2]\right\rangle \in\mathbb Z_2.$$

The reduced connection-orbit space in that sector is

$$\mathcal C_m =\frac{\mathcal A(P_m)}{\mathcal G_0(P_m)},$$

where $\mathcal G_0(P_m)$ restricts to the identity at the boundary. $\operatorname{SU}(2)$ has only $m=0$ in this lift-obstruction label, while $\operatorname{SO}(3)$ can admit $m=1$. A boundary trivialization would remove the $m=1$ sector, so it is not imposed here. Crucially, $\operatorname{SO}(3)_+$ and $\operatorname{SO}(3)_-$ have the same classical connection-orbit space for a fixed $m$. The orbit quotient alone does not choose the line spectrum.

Charge description. The defect boundary condition carries a reduced pair $(e,m)$. Assume that the boundary phase space preserves this label and does not absorb the line. When $m=1$, the $+$ theory admits the pure magnetic pair $(0,1)$ as genuine, while the $-$ theory admits the dyonic pair $(1,1)$. The $e=1$ entry in the latter is not a separate half-integer $\operatorname{SO}(3)$ Wilson endpoint; only the combined dyonic defect is genuine. A boundary condition that can absorb the auxiliary surface or add charged boundary degrees of freedom can alter this conclusion, which is why the boundary assumptions are explicit.

Gauge-fixed description. In a common local patch, the gauge potential, adjoint ghosts, Faddeev–Popov operator, and perturbative vertices depend on the same algebra $\mathfrak{su}(2)$. The $m=1$ sector still requires patched or singular representatives, and the $p$ choice changes the permitted defect and topological phase rather than the local gauge-fixing formula. Matching local propagators in a shared perturbative patch therefore cannot distinguish the two spectra.

The closed-manifold Pontryagin-square formula must not be inserted naively on $M$ with boundary. For two histories $X_1$ and $X_2$ on a finite time slab with identical fixed boundary data—including matching spin structures, restricted bundles, and a chosen boundary bundle isomorphism—glue one to the orientation reverse of the other:

$$Y=X_1\cup_{\partial}\overline{X_2}.$$

Let $P_Y\to Y$ be the $\operatorname{SO}(3)$ bundle obtained by this gluing. The closed spin manifold $Y$ defines the relative discrete-theta phase

$$\Phi_p(X_1,X_2) =\exp\!\left[ \frac{i\pi p}{2} \int_Y\mathcal P\!\left(w_2(P_Y)\right) \right].$$

An absolute amplitude on the bounded spacetime additionally requires a boundary trivialization, reference state, or boundary counterterm. This gluing construction keeps the bulk formula within its closed-manifold hypotheses.

Scope limits

• The scalar congruence $em'-me'\equiv0\pmod{k}$ is only the cyclic reduced charge case. Product centers require the corresponding finite Abelian pairing.
• A charge class does not construct a line operator, establish its renormalization, or determine its fusion category.
• The genuine spectrum does not determine confinement, screening dynamics, area or perimeter laws, or vacuum energy.
• Disconnected gauge groups, higher groups, and an all-group classification of discrete theta terms are outside this page.
• On a manifold with boundary, bulk genuineness and the discrete phase depend on a specified boundary completion.

Common pitfalls

Equating genuine with unscreened or topological. Genuineness asks whether an auxiliary surface is required. Screening and long-distance behavior are separate dynamical properties.

Adjoining every formal charge. A pure electric class and a pure magnetic class with nonzero Dirac pairing cannot both be genuine in one mutually local theory.

Treating the odd electric label in $\operatorname{SO}(3)_-$ as a standalone Wilson representation. The separate half-integer Wilson line does not descend to $\operatorname{SO}(3)$. Only its dyonic combination with the odd magnetic defect is genuine.

Calling a $2\pi$ theta shift a fixed-theory period. On a spin $\operatorname{SO}(3)$ theory it exchanges the $+$ and $-$ line spectra. A second shift is required to return to the same spectrum.

Using a closed-manifold topological action without boundary data. On a bounded region, only a relative gluing phase is fixed until a boundary completion is supplied.

Check your understanding

1. List all maximal mutually local subgroups of $\mathbb Z_2^2$ under the pairing $em'-me'$. Identify the $\operatorname{SU}(2)$, $\operatorname{SO}(3)_+$, and $\operatorname{SO}(3)_-$ choices. Apply one $2\pi$ theta shift to each nonzero generator.
2. For the shell system with $m=1$, state the genuine reduced charge in each $\operatorname{SO}(3)$ theory. Why do the connection-orbit space and a local gauge-fixed propagator fail to distinguish the two choices? How is a discrete-theta phase compared without applying the closed-manifold formula directly to the shell?

Solution

The three nonzero vectors in $\mathbb Z_2^2$ are $(1,0)$, $(0,1)$, and $(1,1)$. Any two distinct ones have pairing $1$ modulo $2$, so no mutually local subgroup contains two of them. Each nonzero vector therefore generates one maximal isotropic subgroup:

$$\begin{gathered} \langle(1,0)\rangle=L_{\operatorname{SU}(2)}, \\ \langle(0,1)\rangle=L_{\operatorname{SO}(3)_+}, \\ \langle(1,1)\rangle=L_{\operatorname{SO}(3)_-}. \end{gathered}$$

Under one theta shift, $(e,m)\mapsto(e+m,m)$. Thus $(1,0)$ is fixed, $(0,1)$ maps to $(1,1)$, and $(1,1)$ maps to $(0,1)$. The $\operatorname{SU}(2)$ spectrum is unchanged, while the two $\operatorname{SO}(3)$ spectra are exchanged.

For $m=1$, $\operatorname{SO}(3)_+$ admits the genuine pair $(0,1)$ and $\operatorname{SO}(3)_-$ admits $(1,1)$. Both theories use the same $\operatorname{SO}(3)$ bundles and hence the same connection-orbit space in a fixed magnetic sector. A local gauge-fixing operator sees $\mathfrak{su}(2)$, not which maximal subgroup or discrete phase was chosen. To compare two bounded histories with the same boundary data, glue one to the orientation reverse of the other and evaluate the Pontryagin-square phase on the resulting closed spin four-manifold. An absolute phase needs additional boundary data.

Where the remaining data are developed

Large Gauge Transformations and Topological Sectors is the next step: it asks how disconnected gauge transformations and action phases act within the chosen theory.

Genuine Lines, Screening, and Charge Lattices constructs line operators and develops screening beyond the reduced specification used here. Line Operators, Screening, and Generalized-Symmetry Diagnostics develops phase diagnostics and long-distance dynamics.

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 08 (2013): 115. DOI. Open PDF, arXiv:1305.0318v5
• 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:2307.07547v2