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The Line operators in the G2HDM model

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper argues that the discrete center quotient $\Gamma$ of the G2HDM gauge group — invisible to local experiments — determines the spectra of Wilson, 't Hooft, and dyonic line operators, the periodicity of the five CP-violating…

desk verdict Solid extension of Tong's line-operator program to G2HDM, but the mixed U(1)_V/U(1)_A section drops the charge-lattice parity constraint and overcounts spectra. read the letter →

arxiv 2412.14949 v2 pith:OE64C34G submitted 2024-12-19 hep-ph hep-th

Rui Yang , Jiaming Guo , Linghai Li , Xun Xue , Tzu-Chiang Yuan This is my paper · ORCID
classification hep-phhep-th PACS 11.15.-q12.60.-i
keywords G2HDMlineoperatorsWilsonlines'tHooftdyonicthetaanglesgeneralizedDiracquantizationgaugegroupglobalstructure
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks a question that local experiments cannot answer: the Gauged Two-Higgs-Doublet Model has a gauge symmetry whose universal cover is $\tilde{G} = U(1)_Y \times SU(2)_L \times SU(3)_C \times U(1)_X \times SU(2)_H$, but the real gauge group could be any quotient $G = \tilde{G}/\Gamma$ by a discrete subgroup of the center. It argues that this choice is physically meaningful: $\Gamma$ controls which Wilson, 't Hooft, and dyonic line operators exist, how far the five CP-violating $\theta$-angles can run, and what the smallest electric and magnetic charges are after the two Higgs condensations. If the argument is right, global structure becomes a probe of the dark sector rather than a convention, and observations of monopoles or neutral colored Wilson lines could in principle identify which $\Gamma$ Nature picked. The paper's contribution is a systematic catalogue of these spectra and charge tables for the main quotient patterns.

What carries the argument

The engine is the generalized Dirac quantization condition, Eq. (18) (with the $U(1)_V\times U(1)_A$ version in Eq. (26)), read as a mod-6 constraint on pairs of electric and magnetic labels: $(z_e^2,z_e^3,q; z_m^2,z_m^3,g)$ for the visible factor and $(x_e^2,h; x_m^2,k)$ for the dark factor. Quotienting by $\Gamma$ is implemented by demanding invariance of Wilson lines under the quotient generators, which restricts the electric weights; the magnetic spectrum is then solved from the quantization condition. For the $\theta$-angles the machinery is the combined $U(1)\times SU(N)\times SU(M)$ $\theta$-term of Eq. (31), from which the periodicities and CP-invariant values are read off; for symmetry breaking it is the Gell-Mann-Nishijima-style formulas $Q_{\rm em}=q/6+\lambda_e^2/2$ and $Q_D=h/2+\rho_e^2/2$ together with the deconfinement conditions $6g\equiv z_m^2 \pmod 2$ and $2k\equiv x_m^2 \pmod 2$.

What would settle it

Compute the full set of allowed line operators for one quotient, say $G = (U(1)_Y\times SU(2)_L\times SU(3)_C)/Z_{6L} \times (U(1)_X\times SU(2)_H)/1$, directly from the weight and co-weight lattices of the covering group without importing Eq. (18) as an axiom; if the resulting lattice differs from the paper's Fig. 8, the spectra change. Experimentally, discover a magnetic monopole and measure whether it satisfies the electromagnetic Dirac condition for leptons but not for quarks: the paper predicts the quotient must then contain $Z_{6L}$ or $Z_{6H}$, so a monopole with exactly the opposite pattern would falsify the central claim.

Watch

Extended reading notes

Core claim

The central claim is that the G2HDM's global gauge structure is not invisible: the discrete quotient $\Gamma$ changes the lattice of allowed non-local operators and the topological angles attached to them. Concretely, for each quotient pattern the allowed Wilson and 't Hooft lines are the subsets of the covering-group lattices left invariant by $\Gamma$, with coexisting dyonic lines constrained by the generalized Dirac quantization condition $-6gq + 3z_e^2 z_m^2 + 2z_e^3 z_m^3 + 3x_e^2 x_m^2 - 6kh \equiv 0 \pmod 6$ (and its $U(1)_V \times U(1)_A$ variant). The same data determine the $\theta$-angle periodicities, which range from $2\pi$ up to $72\pi$ for $\tilde\theta_Y$ and $8\pi$ for $\tilde\theta_X$ depending on $\Gamma$. After the two-step breaking $SU(2)_H \times U(1)_X \to U(1)_D$ and $SU(2)_L \times U(1)_Y \to U(1)_{\rm em}$, the minimal charges shift: for instance $Q_{\rm em}$ can be $1/6$, $1/3$, or $1/2$, with magnetic counterpart $G_{\rm em}$ equal to $1$, $2$, $3$, or $6$. When $\Gamma$ contains the $SU(3)_C$ center, the minimal electromagnetic monopole violates the pure Dirac condition and must carry color magnetic charge, so the low-energy group becomes $U(3)_C$; when the mixed $U(1)_V \times U(1)_A$ basis is quotiented by $Z_{2L}\times Z_{2H}$ or $Z_{6L}\times Z_{2H}$, even the dark Dirac condition can fail. The paper also identifies the residual dark QED angle $\theta_{\rm dem} = (\tilde\theta_X + 2\theta_{2H})/4$ as the physical, chiral-rotation-invariant counterpart of $\theta_{\rm em} = (\tilde\theta_Y + 18\theta_{2L})/36$.

Load-bearing premise

The analysis treats the generalized Dirac quantization condition of Eq. (18) (and its Eq. (26) variant) with the chosen integer charge normalizations $q=6Y$, $h=2X$ as the complete constraint on allowed line operators, and the spectra and charge tables would change if additional quantization constraints apply to products with two U(1) factors.

Editorial extensions

If this is right

  • Two G2HDM copies with identical local physics but different $\Gamma$ are distinguished by their line-operator spectra: Wilson, 't Hooft, and dyonic lattices are different for different center quotients.
  • If $\Gamma$ contains a $Z_3$ or $Z_{6L}$, the minimal Dirac monopole compatible with leptons is incompatible with quark fractional charges; the consistent monopole must carry color magnetic charge and the low-energy gauge group is $U(3)_C$, not $U(1)_{\rm em}\times SU(3)_C$.
  • The dark QED theta-angle $\theta_{\rm dem}=(\tilde\theta_X+2\theta_{2H})/4$ is physical and cannot be rotated away, just like the SM residual $\theta_{\rm em}$; its allowed range depends on $\Gamma$ through the minimal dark charge $Q_D$.
  • The minimal electric and magnetic charges after the two Higgs breakings are $\Gamma$-dependent: $Q_{\rm em}\in\{1/6,1/3,1/2\}$, $G_{\rm em}\in\{1,2,3,6\}$, $Q_D\in\{1/2,1\}$, $G_D\in\{1,2\}$ across the tables, so measurements of monopole or dyon charges could pick out the quotient.
  • Neutral colored Wilson lines ('neutral quarks') exist only for specific $\Gamma$; their observation would pin the global structure, while their absence is consistent with the larger quotients.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the same center-quotient technique transfers directly to the left-right symmetric model and the Pati-Salam gauge group, which the paper cites as future directions; the charge tables for those models could be produced by the same mod-arithmetic recipe.
  • Beyond the paper: a measurement that finds a monopole obeying the lepton Dirac condition but not the quark condition would, by the paper's logic, force $\Gamma$ to contain $Z_{6L}$ or $Z_{6H}$; this makes the global structure a concrete, falsifiable input to dark-sector model building.
  • Beyond the paper: if the assumption that Eq. (18) is complete for products with two U(1) factors fails, the qualitative claim survives but the Section VI minimal-charge tables would need revision; checking this by a direct weight-lattice construction is a well-defined follow-up calculation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper extends David Tong's analysis of line operators in the Standard Model to the Gauged Two-Higgs-Doublet Model (G2HDM), whose covering group is U(1)_Y x SU(2)_L x SU(3)_C x U(1)_X x SU(2)_H. The authors classify quotients of this covering group by discrete subgroups of the center, then compute spectra of Wilson, 't Hooft, and dyonic lines for several quotient patterns, the periodicity of the five theta-angles and the CP-invariant values of these angles, and the minimal electric and magnetic charges after the two-stage Higgs symmetry breaking. The central message is that different choices of Gamma produce physically different global data: line-operator spectra, theta-angle periodicities, and post-breaking charge quanta all depend on the quotient.

Significance. The topic is well motivated: global-structure ambiguities of gauge theories are physically meaningful, and extending the SM analysis of Tong to a concrete dark-sector model is a useful contribution to the phenomenology-oriented generalized-symmetry literature. The paper contains no parameter fitting and the working examples are algebraic consequences of the stated model and the imported generalized Dirac quantization condition. The theta-angle periodicity derivation in Section V is clear and internally consistent, and the first quotient example in Section IV is solved explicitly. If the issues identified below are corrected, the paper could provide a reliable reference for global-structure effects in G2HDM. The present version, however, overstates its systematic character: most of the 20 quotient cases are not actually derived, and the mixed U(1)_V x U(1)_A analysis is affected by a lattice mismatch.

major comments (2)
  1. [Section IV, Eqs. (22)-(26), Figs. 19-26, Table VII] The mixed-basis analysis treats q+ = q + h and q- = q - h as independent integers, but the transformation (q,h) -> (q+,q-) has determinant -2 and maps the physical charge lattice Z^2_{q,h} to the sublattice {q+ ≡ q- (mod 2)}. The paper does not impose this sublattice condition in the generators (24), the GDQC (25)-(26), or the spectra in Figs. 19-26 and Table VII. For example, the Abelian generator (q+,q-) = (1,0) shown in Fig. 19 corresponds to (q,h) = (1/2,1/2), which is not a representation of U(1)_Y x U(1)_X with the Table I normalization. This is not a typographical issue: the Wilson and 't Hooft spectra for all U(1)_V x U(1)_A cases are overcounted, and the minimal-charge entries derived from them in Table VII are not charges of the G2HDM covering group. The derivation must impose q+ ≡ q- (mod 2) before solving the center-invariance and Dirac-quantization constraints.
  2. [Section IV, p. 24; Section VII, Tables VIII-X] The paper explicitly states that 17 of the 20 quotient patterns are not discussed in detail, yet the abstract and summary claim a systematic characterization, and Tables VIII-X list the allowed Gamma for neutral quarks for all three cases (A), (B), and (C). No derivation is provided for most entries of these tables; the text only says the method is similar. This is a load-bearing gap because the systematic claim rests on these tables. The authors should either supply the missing derivations (an appendix or supplementary material) or explicitly restrict the claims to the worked examples.
minor comments (4)
  1. [Eq. (17)] Equation (17) is difficult to read: the factors '6g' and '2k' appear as stray multiplicative terms rather than as part of the exponents, which makes the equivalence to Eq. (18) hard to verify. The typesetting should be corrected.
  2. [Figures 19-26] The figure captions for the mixed U(1) cases do not specify the complete set of allowed charges; the reader must infer the spectra from the green circles. Since the spectra are the main quantitative output, a short explicit list or a table of allowed charge pairs would improve reproducibility.
  3. [Title and text] The manuscript contains several OCR-style artifacts, e.g. 'T wo-Higgs-Doublet Model' in the title, 'suffer' in Section II, and 'g auge' in some headings. These should be cleaned before publication.
  4. [Section VI, Eq. (39)] The notation lambda_m^2 is used before its relation to the earlier z_m^2 is fully explained; the reader must interpolate between the line-operator lattice variables and the Gell-Mann-Nishijima-type charges. A sentence connecting lambda_{e,m}^2 to z_{e,m}^2 from Section IV would remove ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the line-operator spectra, θ-periodicities, and minimal charges are algebraic consequences of the stated model, the center-invariance condition, and the imported generalized Dirac quantization condition; there is no fitted parameter renamed as a prediction.

full rationale

The paper's central derivation chain is self-contained given its stated inputs. The universal covering group and matter content are defined in Eqs. (1)-(3) and Table I, with charge normalizations q = 6Y and h = 2X chosen so all matter charges are integral. The line-operator spectra in Section IV are then obtained by imposing center invariance on Wilson lines and solving the generalized Dirac quantization condition, Eq. (18) (and Eq. (26) in the mixed U(1) basis), for each quotient Γ. These are algebraic constraints; no parameter is fitted to data and no output is used to define an input. The θ-angle periodicities in Section V and Tables III-IV are imported from Tong's SM analysis [11] via Eq. (31); importing an external, independently derived result is a dependency, not circularity, and the paper explicitly extends rather than redefines those results. The authors' own earlier G2HDM papers [21-25] supply the model's matter content, which is the object being studied, not evidence for the line-operator conclusions. The skeptical concern about the mixed U(1)_V × U(1)_A basis dropping the sublattice constraint q+ ≡ q- (mod 2) is a potential correctness or completeness issue in the derivation, not a circularity: even if some displayed lines are spurious, the claimed Γ-dependence of the spectra does not reduce to an input by construction. No self-citation chain is load-bearing for the main claims, and no known result is merely renamed. Hence the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on established line operator formalism, the fixed G2HDM matter content, and discrete choices of Gamma. There are no fitted parameters and no new entities. The main load-bearing assumptions are the charge normalizations and the generalized Dirac quantization condition, both standard in the field.

assumptions (4)
  • domain assumption The G2HDM matter content in Table I is anomaly-free and defines the model.
    Used to specify the matter representations for the line operator analysis; anomaly cancellation is shown in Section II.
  • standard math Wilson and 't Hooft lines are classified by center charges and the generalized Dirac quantization condition of Corrigan-Olive.
    This is the established framework from [11,35] used throughout Sections III, IV, and VI.
  • standard math For a quotient G = \tilde{G}/Gamma, the periodicity of theta-angles is extended by factors related to N and M as in Table II (from Tong [11]).
    Used in Section V to derive Tables III and IV; imported from prior literature.
  • ad hoc to paper The symmetry breaking pattern is SU(2)_L x U(1)_Y to U(1)_em and SU(2)_H x U(1)_X to U(1)_D with small mixings neglected.
    This simplification is stated in footnote 5 of Section VI and affects the minimal charge analysis for the mixed U(1)_V/U(1)_A quotients.

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Pith. "Pith review of The Line operators in the G2HDM model." pith.science (2026). https://pith.science/paper/OE64C34G

@misc{pith2026241214949,
  author       = {Pith},
  title        = {Pith review of: The Line operators in the G2HDM model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OE64C34G}},
  note         = {Machine review of arXiv:2412.14949}
}
abstract

We investigate the global structure of the Gauged Two-Higgs-Doublet Model (G2HDM), a framework that extends the Standard Model by introducing a dark sector governed by the gauge symmetry $U(1)_X \times SU(2)_H$. The full gauge symmetry of the theory, including the visible sector, is given by the universal covering group $\tilde{G} = U(1)_Y \times SU(2)_L \times SU(3)_C \times U(1)_X \times SU(2)_H$. However, the true gauge group may instead be a quotient $G = \tilde{G} / \Gamma$, where $\Gamma$ is the center of $\tilde{G}$ or a subgroup thereof, leading to different global structures that cannot be distinguished by local experiments.We explore the physical implications of these global structures, analyzing their effects on Wilson, 't Hooft, and dyonic line operators, as well as the periodicity of the CP-violating $\theta$-angles associated with each group factor. Furthermore, we determine the minimal electric and magnetic charges that arise after electroweak symmetry breaking, highlighting their dependence on the choice of $\Gamma$. These findings provide a systematic characterization of the G2HDM's global properties and their potential phenomenological consequences.

Figures

Figures reproduced from arXiv: 2412.14949 by the authors.

Figure 1
Figure 1. The spectra of line operators for SU(2) in (a) and SU(2)/Z2 at θ = 0 (b), 2π (c), respectively. Last case of (c) represent the Witten effect. See [11] for more detailed discussions. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. The spectra of line operators for SU(3) in (a) and SU(3)/Z3 at θ = 0 (b), 2π (c), and 4π (d), respectively. Last two cases of (c) and (d) represent the Witten effect. See [11] for more detailed discussions. In the case of U(N) ∼= (U(1) × SU(N))/ZN theory, the U(1) factor θ-angle, denoted as ˜θ, extends to [0, 2πN2 ), instead of the usual [0, 2π). Meanwhile, the periodicity of the SU(N) θ-angle, denoted as θN , remai… view at source ↗
Figure 3
Figure 3. Γ = 1 × 1 for G = U(1)Y ×SU(2)L×SU(3)C Γp × U(1)X×SU(2)H Γm . Abelian lines generated by (q, g) = {(0, 1),(1, 0)} and (h, k) = {(0, 1),(1, 0)} can be included. Γ = Z2L × 1: Wilson lines χ are unrestricted, thus as in previous case, the electric charges h ∈ Z with x e 2 = 0, 1 and magnetic charges k ∈ Z with x m 2 = 0 mod 2 are all allowed. On the other hand, the Wilson lines ξ, invariant under ξ 3 , are restricted b… view at source ↗
Figures from the paper (28 more)
Figure 4
Figure 4. Figure 4: Γ = Z2L × 1 for G = U(1)Y ×SU(2)L×SU(3)C Γp × U(1)X×SU(2)H Γm . Abelian lines generated by (q, g) = {(0, 1),(2, 0)} and (h, k) = {(0, 1),(1, 0)} can be included. the other hand, Wilson lines χ, invariant under χ, are restricted by h = x e 2 mod 2. The GDQC demands magn…
Figure 5
Figure 5. Figure 5: Γ = 1 × Z2H for G = U(1)Y ×SU(2)L×SU(3)C Γp × U(1)X×SU(2)H Γm . Abelian lines generated by (q, g) = {(0, 1),(1, 0)} and (h, k) = {(0, 1),(2, 0)} can be included. z e 3 z m 3 z e 2 z e 2 z e 2 z e 2 z e 2 z e 2 z e 2 z e 2 z e 2 z m 2 z m 2 z m 2 z m 2 z m 2 z m 2 z m 2…
Figure 6
Figure 6. Figure 6: Γ = Z3 × 1 for G = U(1)Y ×SU(2)L×SU(3)C Γp × U(1)X×SU(2)H Γm . Abelian lines generated by (q, g) = {(0, 1),(3, 0)} and (h, k) = {(0, 1),(1, 0)} can be included. invariant under generator ξ. This implies that the Abelian electric charges satisfy q = 3z e 2 − 2z e 3 mod …
Figure 7
Figure 7. Figure 7: Γ = Z2L × Z2H for G = U(1)Y ×SU(2)L×SU(3)C Γp × U(1)X×SU(2)H Γm . Abelian lines generated by (q, g) = {(0, 1),(2, 0)} and (h, k) = {(0, 1),(2, 0)} can be included. comes more abundant, particularly the ’t Hooft lines, which can take SU(2)L × SU(3)C magnetic charges 6g …
Figure 8
Figure 8. Figure 8: Γ = Z6L × 1 for G = U(1)Y ×SU(2)L×SU(3)C Γp × U(1)X×SU(2)H Γm . Abelian lines generated by (q, g) = {(0, 1),(6, 0)} and (h, k) = {(0, 1),(1, 0)} can be included. z e 3 z m 3 z e 2 z e 2 z e 2 z e 2 z e 2 z e 2 z e 2 z e 2 z e 2 z m 2 z m 2 z m 2 z m 2 z m 2 z m 2 z m 2…
Figure 9
Figure 9. Figure 9: Γ = Z3 × Z2H for G = U(1)Y ×SU(2)L×SU(3)C Γp × U(1)X×SU(2)H Γm . Abelian lines generated by (q, g) = {(0, 1),(3, 0)} and (h, k) = {(0, 1),(2, 0)} can be included. Let’s consider yet another one of the 8 possibilities of case (A): G = U(1)X × SU(2)H × SU(3)C Γp × U(1)Y …
Figure 10
Figure 10. Figure 10: Γ = Z6L × Z2H for G = U(1)Y ×SU(2)L×SU(3)C Γp × U(1)X×SU(2)H Γm . Abelian lines generated by (q, g) = {(0, 1),(6, 0)} and (h, k) = {(0, 1),(2, 0)} can be included. This pattern is similar to Eq. (14) with the SM U(1)Y × SU(2)L and its dark replica U(1)X × SU(2)H in th…
Figure 11
Figure 11. Figure 11: Γ = 1 × 1 for G = U(1)X×SU(2)H×SU(3)C Γp × U(1)Y ×SU(2)L Γm . Abelian lines generated by (q, g) = {(0, 1),(1, 0)} and (h, k) = {(0, 1),(1, 0)} can be included. z e 3 z m 3 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x m 2 x m 2 x m 2 x m 2 x m 2 x m 2 x m 2 …
Figure 12
Figure 12. Figure 12: Γ = 1 × Z2L for G = U(1)X×SU(2)H×SU(3)C Γp × U(1)Y ×SU(2)L Γm . Abelian lines generated by (q, g) = {(0, 1),(2, 0)} and (h, k) = {(0, 1),(1, 0)} can be included. now mix together. We assume the “vector” hypercharge U(1)V to be q+ = q + h and the other “axial” hypercha…
Figure 13
Figure 13. Figure 13: Γ = Z2H × 1 for G = U(1)X×SU(2)H×SU(3)C Γp × U(1)Y ×SU(2)L Γm . Abelian lines generated by (q, g) = {(0, 1),(1, 0)} and (h, k) = {(0, 1),(2, 0)} can be included. z e 3 z m 3 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x m 2 x m 2 x m 2 x m 2 x m 2 x m 2 x m …
Figure 14
Figure 14. Figure 14: Γ = Z3 × 1 for G = U(1)X×SU(2)H×SU(3)C Γp × U(1)Y ×SU(2)L Γm . Abelian lines generated by (q, g) = {(0, 1),(1, 0)} and (h, k) = {(0, 1),(3, 0)} can be included. generators. The possible Γ for quotienting in Eq. (22) is Γ ={1 × 1 × 1, 1 × 1 × Z2L, 1 × Z2H × 1, Z3 × 1 ×…
Figure 15
Figure 15. Figure 15: Γ = Z2H × Z2L for G = U(1)X×SU(2)H×SU(3)C Γp × U(1)Y ×SU(2)L Γm . Abelian lines generated by (q, g) = {(0, 1),(2, 0)} and (h, k) = {(0, 1),(2, 0)} can be included. z e 3 z m 3 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x m 2 x m 2 x m 2 x m 2 x m 2 x m 2 x …
Figure 16
Figure 16. Figure 16: Γ = Z6H × 1 for G = U(1)X×SU(2)H×SU(3)C Γp × U(1)Y ×SU(2)L Γm . Abelian lines generated by (q, g) = {(0, 1),(1, 0)} and (h, k) = {(0, 1),(6, 0)} can be included. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]
Figure 17
Figure 17. Figure 17: Γ = Z3 × Z2L for G = U(1)X×SU(2)H×SU(3)C Γp × U(1)Y ×SU(2)L Γm . Abelian lines generated by (q, g) = {(0, 1),(2, 0)} and (h, k) = {(0, 1),(3, 0)} can be included. z e 3 z m 3 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x e 2 x m 2 x m 2 x m 2 x m 2 x m 2 x m 2 x m…
Figure 18
Figure 18. Figure 18: Γ = Z6H × Z2L for G = U(1)X×SU(2)H×SU(3)C Γp × U(1)Y ×SU(2)L Γm . Abelian lines generated by (q, g) = {(0, 1),(2, 0)} and (h, k) = {(0, 1),(6, 0)} can be included. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]
Figure 19
Figure 19. Figure 19: Γ = 1 × 1 × 1 for G = U(1)V ×SU(3)C Γp × U(1)A×SU(2)H Γm × SU(2)L Γn . Abelian lines generated by (q+, g+) = {(0, 1),(1, 0)} and (q−, g−) = {(0, 1),(1, 0)} can be included. There are still 17 types of G with different central quotient divisions that have not been disc…
Figure 20
Figure 20. Figure 20: Γ = 1 × 1 × Z2L for G = U(1)V ×SU(3)C Γp × U(1)A×SU(2)H Γm × SU(2)L Γn . Abelian lines generated by (q+, g+) = {(0, 1),(1, 0)} and (q−, g−) = {(0, 1),(1, 0)} can be included. z e 3 z m 3 x e 2 x m 2 z m 2 z m 2 q− = 0 q− = 1 g− = 0 g− = 1/2 q+ = 0 q+ = 0 q+ = 0 g+ = 0…
Figure 21
Figure 21. Figure 21: Γ = 1 × Z2H × 1 for G = U(1)V ×SU(3)C Γp × U(1)A×SU(2)H Γm × SU(2)L Γn . Abelian lines generated by (q+, g+) = {(0, 1),(1, 0)} and (q−, g−) = {(0, 1),(2, 0)} can be included. z e 3 z m 3 x e 2 x m 2 z m 2 z m 2 q− = 0 q− = 0 g− = 0 q+ = 0 q+ = 1q+ = 2 g+ = 0 g+ = 2/3 …
Figure 22
Figure 22. Figure 22: Γ = Z3 × 1 × 1 for G = U(1)V ×SU(3)C Γp × U(1)A×SU(2)H Γm × SU(2)L Γn . Abelian lines generated by (q+, g+) = {(0, 1),(3, 0)} and (q−, g−) = {(0, 1),(1, 0)} can be included. lines are (2g = x m 2 mod 2; x m 2 = 0, 1 mod 2), (2k = z m 2 mod 2; z m 2 = 0, 1 mod 2) and (…
Figure 23
Figure 23. Figure 23: Γ = 1 × Z2H × Z2L for G = U(1)V ×SU(3)C Γp × U(1)A×SU(2)H Γm × SU(2)L Γn . Abelian lines generated by (q+, g+) = {(0, 1),(1, 0)} and (q−, g−) = {(0, 1),(2, 0)} can be included. z e 3 z m 3 x e 2 x m 2 z m 2 z m 2 q− = 0 q− = 0 g− = 0 q+ = 0 q+ = 1q+ = 2 g+ = 0 g+ = 2/…
Figure 24
Figure 24. Figure 24: Γ = Z3 × 1 × Z2L for G = U(1)V ×SU(3)C Γp × U(1)A×SU(2)H Γm × SU(2)L Γn . Abelian lines generated by (q+, g+) = {(0, 1),(3, 0)} and (q−, g−) = {(0, 1),(1, 0)} can be included. z e 3 z m 3 x e 2 x m 2 z m 2 z m 2 q− = 0 q− = 1 g− = 0 g− = 1/2 q+ = 0 q+ = 1q+ = 2 g+ = 0…
Figure 25
Figure 25. Figure 25: Γ = Z3 × Z2H × 1 for G = U(1)V ×SU(3)C Γp × U(1)A×SU(2)H Γm × SU(2)L Γn . Abelian lines generated by (q+, g+) = {(0, 1),(3, 0)} and (q−, g−) = {(0, 1),(2, 0)} can be included. z m 3 mod 3; z m 2 = 0 mod 2; z m 3 = 0, 1, 2 mod 3) and (g+ = x m 2 mod 2; x m 2 = 0, 1 mod…
Figure 26
Figure 26. Figure 26: Γ = Z3 × Z2H × Z2L for G = U(1)V ×SU(3)C Γp × U(1)A×SU(2)H Γm × SU(2)L Γn . Abelian lines generated by (q+, g+) = {(0, 1),(3, 0)} and (q−, g−) = {(0, 1),(2, 0)} can be included. 2z m 3 ) mod 6; z m 2 = 0, 1 mod 2; z m 3 = 0, 1, 2 mod 3) and (g+ = x m 2 mod 2; x m 2 = …
Figure 27
Figure 27. Figure 27: Witten effect for the Abelian lines in U(1)×SU(2) Z2 where ˜θ = [0, 8π) – (1/3). 50 [PITH_FULL_IMAGE:figures/full_fig_p050_27.png]
Figure 28
Figure 28. Figure 28: Witten effect for the Abelian lines in U(1)×SU(2) Z2 where ˜θ = [0, 8π) – (2/3). 51 [PITH_FULL_IMAGE:figures/full_fig_p051_28.png]
Figure 29
Figure 29. Figure 29: Witten effect for the Abelian lines in U(1)×SU(2) Z2 where ˜θ = [0, 8π) – (3/3). 52 [PITH_FULL_IMAGE:figures/full_fig_p052_29.png]
Figure 30
Figure 30. Figure 30: Witten effect for the Abelian lines in U(1)×SU(3) Z3 where ˜θ = [0, 18π). 53 [PITH_FULL_IMAGE:figures/full_fig_p053_30.png]
Figure 31
Figure 31. Figure 31: Witten effect for the Abelian lines in U(1)×SU(2)×SU(3) Z6 where ˜θ = [0, 72π). 55 [PITH_FULL_IMAGE:figures/full_fig_p055_31.png]

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