Pith. sign in

REVIEW 3 major objections 5 minor 7 cited by

On discrete gauging and non-invertible selection rules

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For finite discrete groups, gauging an automorphism turns coupling selection rules into non-invertible class-product algebras whose residual symmetries come from the group's inner and outer automorphisms.

desk verdict A solid algebraic extension of non-invertible selection rules to Z3/S3 gaugings; the fusion rules are new and useful, but the loop-protection claim is asserted rather than proven. read the letter →

arxiv 2507.02375 v1 pith:FQWZC5RM submitted 2025-07-03 hep-th hep-ph

Jun Dong , Tim Jeric , Tatsuo Kobayashi , Ryusei Nishida , Hajime Otsuka This is my paper · ORCID
classification hep-thhep-ph
keywords non-invertiblesymmetriesconjugacyclassesdiscretegaugingselectionrulesfinitegroupsouterautomorphismsflavorfusion
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 what selection rules survive when a finite discrete symmetry such as $\mathbb{Z}_N$ is gauged by one of its automorphisms (a $\mathbb{Z}_2$, $\mathbb{Z}_3$, or $S_3$ action), and answers that the gauge-invariant objects are conjugacy classes whose multiplication rules form non-invertible fusion algebras. It then establishes that these class-product algebras carry residual finite Abelian or non-Abelian discrete symmetries inherited from the inner and outer automorphisms of the underlying group, collecting examples such as a $\mathbb{Z}_2$ symmetry for $\mathbb{Z}_2$ gauging of $\mathbb{Z}_N$ with even $N$, a $\mathbb{Z}_3$ symmetry for $\mathbb{Z}_3$ gauging of $\mathbb{Z}_N \times \mathbb{Z}'_N$ when $N/3$ is an integer, and an $S_4$ symmetry for the full conjugacy-class algebra of $\Delta(54)$. Because the non-invertible selection rules are violable by radiative or stringy corrections while the residual automorphism symmetries are not, the residual symmetries mark exactly which coupling restrictions are exact. A sympathetic reader cares because fields labeled by conjugacy classes appear in string orbifold constructions, and these rules yield new fermion mass textures and Higgs-sector structures that ordinary group-theoretic flavor symmetries cannot produce.

What carries the argument

The load-bearing object is the conjugacy class together with its multiplication (fusion) rule. After gauging a finite group by a $\mathbb{Z}_2$, $\mathbb{Z}_3$, or $S_3$ automorphism, the gauge-invariant combinations of group elements are classes $C = \{b^n g b^{-n}\}$, and products of classes expand as sums of classes with integer coefficients, as in Eq. (4.4); this is what turns coupling selection into a non-invertible algebra. The selection rule is then the dictionary that an $n$-point vertex $\prod_i \phi_{k_i}$ is allowed if and only if the class product contains the identity class, and the symmetries of these product tables are identified with the inner and outer automorphism groups of the underlying discrete group, which act on classes by permutation or by diagonal phases (for example $\mathbb{Z}_3$ charges $\omega^p$ and $b \to \omega b$ on the $\Delta(27)$ classes). The same automorphism action, applied to twist fields associated with the conjugacy classes containing $b$, is what produces the large residual symmetries such as $S_4$ for $\Delta(54)$.

What would settle it

Compute the one-loop correction to a coupling that is forbidden by the $\mathbb{Z}_3$ charge in $\mathbb{Z}_3$ gauging of $\mathbb{Z}_3\times\mathbb{Z}'_3$ (e.g., a vertex with total class charge not equal to $\alpha+\beta \bmod 3$): if such an amplitude is non-zero, the claimed loop-protection of the residual symmetry fails; alternatively, compute a tree-level three-point function $\langle \phi_{k,\ell}^3\rangle$ in a $\mathbb{T}^2/\mathbb{Z}_3$ model, which the paper predicts is always allowed, and check that its class product indeed contains the identity — a vanishing result would falsify the selection-rule dictionary.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a systematic derivation of conjugacy-class selection rules for several families of finite discrete groups, in both the gauged and ungauged settings, together with the residual discrete symmetries of those rules. For the gauged classes of $\mathbb{Z}_N \times \mathbb{Z}'_N$ under a $\mathbb{Z}_3$ outer automorphism, the classes $C(k,\ell)$ obey $C(k,\ell)\cdot C(m,n) = C(k+m,\ell+n) + C(k-m+n,\ell-m) + C(k-n,\ell+m-n)$, and when $N/3$ is integral the algebra splits under a $\mathbb{Z}_3$ charge $C_\alpha \cdot C_\beta = C_{\alpha+\beta \bmod 3}$. Analogous explicit tables are given for $\mathbb{Z}_3$ gauging of $\mathbb{Z}_N$, $\mathbb{Z}_N\times\mathbb{Z}'_N\times\mathbb{Z}''_N$, and $S_3$ gauging of $\mathbb{Z}_N\times\mathbb{Z}'_N$, and for the full conjugacy classes of the groups $T_7$, $\Delta(3N^2)$, $\Sigma(3N^3)$, and $\Delta(6N^2)$; the $\Delta(27)$ table is invariant under $((\mathbb{Z}_3\times\mathbb{Z}_3)\rtimes Q_8)\rtimes S_3$ and the $\Delta(54)$ table under $S_4$. The paper reads a physical selection rule off these algebras: an $n$-point coupling of fields labeled by classes $C_i$ is allowed exactly when the class product contains the identity element, and it emphasizes that in $\mathbb{Z}_3$ gauging the two-point self-coupling $\phi_k^2$ is generically forbidden while $\phi_k^3$ is always allowed, in contrast to $\mathbb{Z}_2$ gauging.

Load-bearing premise

The load-bearing premise is that a physical field can be labeled by a conjugacy class of the gauged group and that an $n$-point coupling is allowed exactly when the product of its classes contains the identity, a dictionary the paper takes from earlier non-invertible-symmetry constructions rather than proving from first principles.

Editorial extensions

If this is right

  • For $\mathbb{Z}_3$ gauging of $\mathbb{Z}_N \times \mathbb{Z}'_N$ with $N/3$ integral, the class-product algebra is $\mathbb{Z}_3$ graded, so couplings that conserve this $\mathbb{Z}_3$ charge cannot be generated by loop corrections; the symmetries in Table 14 mark which parts of the selection rules are exact.
  • In $\mathbb{Z}_3$ gauging, a field in a generic class cannot pair with itself at two points, so diagonal entries of neutrino mass matrices from Weinberg operators can vanish, while the cubic self-coupling $(\phi_k)^3$ is always allowed; this yields mass textures, including the $A_1$ texture, that differ from those forced by $\mathbb{Z}_2$ gauging.
  • For $\Delta(54)$, the entire conjugacy-class multiplication table is invariant under $S_4$, meaning all couplings of fields in $\Delta(54)$ classes are organized by this outer-automorphism symmetry.
  • In supersymmetric models, the $\mu$-term can be forbidden or allowed by choosing the classes of $H_u$ and $H_d$; $\mathbb{Z}_3$ gauging permits the superpotential $W = \lambda S H_u H_d + \kappa S^3$ with $S^3$ always allowed, giving a $\mathbb{Z}_3$-invariant NMSSM-type Higgs sector.
  • $S_3$ gauging of $\mathbb{Z}_N\times\mathbb{Z}'_N$ produces six-element classes whose multiplication rules include the $S_3$ structure, and restricting the $\mathbb{Z}_2$ even part of a $\mathbb{Z}_3$-gauged $\mathbb{Z}_3\times\mathbb{Z}'_3$ algebra reproduces the $S_3$-gauged algebra, while a different projection yields a $\mathbb{Z}_3$ Tambara-Yamagami-type fusion rule.

Reading between the lines

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

  • The class-to-coupling dictionary could be tested directly by computing tree-level correlators in an explicit $\mathbb{T}^2/\mathbb{Z}_3$ magnetized orbifold: the paper predicts $\langle \phi_{k,\ell}^3\rangle \neq 0$ and $\langle \phi_{k,\ell}^2\rangle = 0$ for generic classes, so a string computation checking those zeros would settle the dictionary.
  • Because the residual symmetries only exist when the particle content includes the partners that are permuted, model builders can evade loop protection by omitting partner fields; this suggests a sharp selection rule for which residual symmetries are real in a given spectrum, a point the paper notes but does not develop into a criterion.
  • The sequential gauging procedure in Appendix C, gauging a symmetry of an already-gauged class algebra, hints that these algebras form a hierarchy related by condensation, and one could try to characterize which finite automorphism groups can be gauged in this way for arbitrary finite groups without computing tables case by case.
  • It would be natural to check whether the residual symmetries are anomaly-free in the sense of discrete gauge anomalies; an anomalous residual symmetry would not actually protect couplings non-perturbatively.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper studies selection rules of conjugacy classes of finite discrete groups, focusing on discrete gauging. It reviews Z2 gauging of Z_N and D_N, then extends the analysis to Z3 gauging of Z_N, Z_N x Z'_N, and Z_N x Z'_N x Z''_N, and to S3 gauging of Z_N x Z'_N, together with the associated groups Delta(3N^2), Sigma(3N^3), and Delta(6N^2). The main technical content is a set of class multiplication/fusion rules, given in Eqs. (3.16), (4.4), (4.9), (5.12), (6.3), and in a series of tables, together with the identification of symmetries of these multiplication tables (Z2, S2, Z3, S3, GL(2,3), S4, and combinations), which the paper attributes to inner and outer automorphisms of the underlying groups. The paper also sketches phenomenological applications to Weinberg-operator textures and supersymmetric mu-terms, emphasizing differences between Z2 and Z3 gauging.

Significance. If the physical identification of fields with conjugacy classes is granted, the paper provides a useful systematic catalogue of non-invertible selection rules for a wider class of discrete gaugings than the previously studied Z2 case. The algebraic derivations are explicit and transparent; the class multiplication tables can be verified directly by group multiplication and automorphism actions, and the worked examples for N=2 and N=3 appear internally consistent. The identification of new symmetries, such as the Z3 grading in Eq. (4.18) and the S4 invariance of the Delta(54) class algebra in Sec. 6.2, is a genuine extension of earlier results and is potentially useful for model building. The main weakness is that the step from class multiplication tables to physical coupling selection rules, and especially the claim of loop-level protection, is assumed rather than derived for the new Z3 and S3 gaugings; this limits the currently established phenomenological reach of the paper.

major comments (3)
  1. [Sec. 7.1, Eqs. (7.2)-(7.3), Table 14] The claim that the ZM gradings and permutation symmetries of the multiplication rules are "not violated by loop effects" is not established. A grading of a fusion ring is a statement about the multiplication table; it does not automatically imply that the corresponding charge assignment is a symmetry of the Lagrangian, of the path-integral measure, or of the higher OPE data of the underlying quantum theory. For the Z2 case one can invoke the orbifold quantum symmetry, but for the Z3 and S3 cases (e.g., the Z3 grading of Eq. (4.18) and the S2 symmetry in Table 14) no such symmetry is constructed. Please either construct the relevant symmetry explicitly or weaken the statement to a tree-level/fusion-algebra statement.
  2. [Sec. 2.1, Eqs. (2.11)-(2.12); Secs. 3-6] The physical selection-rule criterion, namely that an n-point coupling is allowed iff the class product of the assigned classes contains the identity, is assumed rather than derived for the new gaugings. For Z2, Eq. (2.10) provides an explicit invariant-mode construction. In Secs. 3, 4.1, and 5, the analogous statements are made by writing invariant sums such as Eqs. (3.18), (4.22), and (5.15), and then asserting that the selection rules "can be derived in a way similar" to the Z2 case. No computation of three-point or higher couplings, and no argument that the class-product criterion (rather than some finer invariant data) controls them, is provided. The paper also does not derive selection rules for fields in the B-type classes (e.g., the twist-field classes in Tables 8, 10, and 13), beyond stating that twist fields may represent them. Since Sec. 7.2 uses these rules to produce neutrino textures, this missing step is load-bearing for the phenomenological part.
  3. [Sec. 3, Eq. (3.16)] The general formula for C^k_3 . C^l_3 is incomplete as written because it does not specify how to interpret terms whose index labels the identity class. In the worked N=7, m=2 example, the product C^1_3 . C^2_3 produces the term C^0_3, which must be read as 3C1 to reproduce Table 3; without this replacement, Eq. (3.16) gives a different result. Analogous replacement rules are stated for other cases, e.g., Eq. (4.17) and Eq. (6.13), but not for Eq. (3.16). The formula is therefore not well-defined for non-prime N and is inconsistent with the paper's own table unless an implicit convention is supplied. Please add the missing replacement rules, e.g., C^0_3 => 3C1, and mention the convention at the point of Eq. (3.16).
minor comments (5)
  1. [Eq. (4.10)] In the N/3-integer case, the list "(s = N/2, 2N/3)" should presumably read "(s = N/3, 2N/3)"; as written it is inconsistent with the premise N/3 is an integer.
  2. [After Eq. (4.9)] The replacement-rule sentence contains a typo: "C(k+m,-l+n)_3" should be "C(k+m,l+n)_3".
  3. [Sec. 4.1, after Eq. (4.10)] The definition of C^(k,l)_3 does not explicitly exclude (k,l)=(0,0), although C_1 is already defined separately; please add the exclusion for clarity.
  4. [Table 14] The summary table lists only a few of the symmetries found in the text; the larger symmetries of Delta(27) and Delta(54) mentioned in Secs. 4.2 and 6.2 are absent. Adding rows or a remark that the table is not exhaustive would avoid confusion.
  5. [Sec. 3, first use of "Z3 gauging of ZN"] The phrase "Z3 gauging of ZN" may be confusing because the Z3 that is gauged is an outer automorphism rather than a subgroup of ZN; a sentence clarifying this at first use would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: fusion rules and residual symmetries are computed from group data, not fitted or assumed.

full rationale

The paper's derivation chain is self-contained at the algebraic level. The class multiplication rules in Secs. 2–6 are obtained by explicit multiplication of the conjugacy classes defined through the stated automorphism actions (e.g., Eq. (2.4)->(2.5), Eq. (4.2)->(4.4), Eq. (6.1)->(6.3)), not by fitting or by citing the target result. The residual symmetries collected in Table 14 (Z2, S2, Z3, S3, S4) are read off from the resulting multiplication tables or are the automorphism groups of the finite groups (GL(2,3) for Delta(27), S4 for Delta(54)), imported from non-overlapping references [41,42] and then checked against the tables; they are consequences, not inputs. The physical criterion that an n-point coupling is allowed iff the product of the corresponding classes contains the identity is stated explicitly in Sec. 2.1 (Eqs. (2.11)-(2.12)) and used consistently; it is an interpretive assumption connecting the algebra to field theory, not a quantity that is being 'predicted' from fitted data. The review of Z2 gauging cites prior work by the same authors (Refs. [8,22]), but the new Z3 and S3 gauging rules are derived in this paper from the group axioms and do not depend on those citations. Limitations exist but are not circularity: Sec. 7.1 asserts without proof that ZM-graded selection rules are not violated by loop effects, and the presentation in Sec. 3 omits the replacement rule (C(0)=>3C1) that is later stated in Sec. 4.1; these are completeness and correctness issues, not reductions of a result to its own input. No equation is reused as its own prediction, and no parameter is fitted to data. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

No free parameters are fitted to data. The only input choices are the discrete groups and the automorphisms, which are fixed by group theory. The central domain assumption is that class products of conjugacy classes determine coupling selection rules, inherited from the prior non-invertible symmetry program.

assumptions (2)
  • domain assumption Fields can be labeled by conjugacy classes of a gauged discrete group G, and an n-point coupling is allowed iff the class product contains the identity.
    Used throughout Secs. 2, 4-7 to translate the computed fusion rules into coupling selection rules; e.g., Eq. (2.12) and the discussion in Sec. 7.1. Not proven in this paper; taken from Refs. [8,22].
  • domain assumption The Z3 and S3 automorphisms used for gauging correspond to actual orbifold or gauging procedures in string or higher-dimensional field theory.
    Invoked in Secs. 3-6 to argue that the classes are realized by invariant modes such as phi_{k,l}; the geometric realization is cited to prior orbifold literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On discrete gauging and non-invertible selection rules." pith.science (2026). https://pith.science/paper/FQWZC5RM

@misc{pith2026250702375,
  author       = {Pith},
  title        = {Pith review of: On discrete gauging and non-invertible selection rules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FQWZC5RM}},
  note         = {Machine review of arXiv:2507.02375}
}
read the original abstract

We clarify selection rules of conjugacy classes of several finite discrete groups where we deal with both gauged and ungauged cases. We find that the selection rules enjoy finite Abelian or non-Abelian discrete symmetries originating from the inner and/or outer automorphism of underlying discrete groups. Since the selection rules of conjugacy classes do not obey conventional group-like selection rules, they open up new coupling selection rules of fields which are labeled by the conjugacy classes.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Radiative lepton model in a non-invertible fusion rule

    hep-ph 2025-10 conditional novelty 6.0 of 10

    A radiative lepton mass model with a Z2-gauged Z5 non-invertible fusion rule can fit neutrino oscillation data and predicts charged-lepton EDMs that indirectly bound 0νββ to ≲28–33.5 meV.

  2. Non-Invertible Selection Rules on Heterotic Non-Abelian Orbifolds

    hep-th 2025-09 conditional novelty 6.0 of 10

    Non-Abelian orbifolds in heterotic string theory produce non-invertible coupling selection rules, since twisted sectors are labeled by conjugacy classes whose products contain multiple classes and yield characteristic...

  3. A natural realization of inverse seesaw model in a non-invertible selection rule

    hep-ph 2025-08 unverdicted novelty 6.0 of 10

    A Z3 Tambara-Yamagami fusion rule is used to build an inverse seesaw model with radiatively generated Majorana masses and a dark matter candidate, fitted to neutrino oscillation data.

  4. Non-invertible Symmetry as a Solution to the Strong CP Problem in a GUT-inspired Standard Model

    hep-ph 2025-08 unverdicted novelty 6.0 of 10

    A non-invertible symmetry imposes a three-zero down-quark texture so spontaneous CP breaking gives the CKM phase without producing a QCD theta term.

  5. A radiative lepton model in a non-invertible fusion rule

    hep-ph 2025-07 conditional novelty 6.0 of 10

    Electron and muon masses are generated at one loop through loop-level breaking of the Ising fusion rule, while neutrino masses arise from an unbroken Z2-like sector that stabilizes loop particles.

  6. Revisiting One-Zero and Two-Zero Neutrino Mass Textures in Light of Recent Oscillation and Cosmological Data

    hep-ph 2026-07 conditional novelty 5.0 of 10

    With CMB+BAO mass-sum bounds only A1/A2 two-zero textures survive; viable one-zero textures (via flow matching) predict distinct Σmi, ⟨mee⟩, and δCP patterns, realizable by non-invertible selection rules.

  7. Three-loop induced neutrino mass model in a non-invertible symmetry

    hep-ph 2025-07 reject novelty 4.0 of 10

    A Ma-model extension with a non-invertible symmetry generates neutrino masses at three loops and yields S0 or eta_R dark matter candidates, with scanned parameter regions shown.

Reference graph

Works this paper leans on

50 extracted references · 12 canonical work pages · cited by 7 Pith papers

  1. [1]

    Dijkgraaf, E.P

    R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, C = 1 Conformal Field Theories on Riemann Surfaces, Commun. Math. Phys.115 (1988) 649

    work page 1988

  2. [2]

    Kobayashi, S

    T. Kobayashi, S. Raby and R.-J. Zhang, Searching for realistic 4d string models with a Pati-Salam symmetry: Orbifold grand unified theories from heterotic string compactification on a Z(6) orbifold, Nucl. Phys. B 704 (2005) 3 [ hep-ph/0409098]

    arXiv 2005

  3. [3]

    Kobayashi, H.P

    T. Kobayashi, H.P. Nilles, F. Ploger, S. Raby and M. Ratz, Stringy origin of non-Abelian discrete flavor symmetries, Nucl. Phys. B 768 (2007) 135 [ hep-ph/0611020]

    arXiv 2007

  4. [4]

    F. Beye, T. Kobayashi and S. Kuwakino, Gauge Origin of Discrete Flavor Symmetries in Heterotic Orbifolds, Phys. Lett. B736 (2014) 433 [ 1406.4660]

    arXiv 2014

  5. [5]

    Thorngren and Y

    R. Thorngren and Y. Wang, Fusion category symmetry. Part II. Categoriosities at c = 1 and beyond, JHEP 07 (2024) 051 [ 2106.12577]

    arXiv 2024

  6. [6]

    Heckman, J

    J.J. Heckman, J. McNamara, M. Montero, A. Sharon, C. Vafa and I. Valenzuela, On the Fate of Stringy Non-Invertible Symmetries, 2402.00118

    arXiv

  7. [7]

    Kaidi, Y

    J. Kaidi, Y. Tachikawa and H.Y. Zhang, On a class of selection rules without group actions in field theory and string theory, SciPost Phys. 17 (2024) 169 [ 2402.00105]. – 29 –

    arXiv 2024

  8. [8]

    Kobayashi and H

    T. Kobayashi and H. Otsuka, Non-invertible flavor symmetries in magnetized extra dimensions, JHEP 11 (2024) 120 [ 2408.13984]

    arXiv 2024

Show all 50 references
  1. [9]

    Funakoshi, T

    S. Funakoshi, T. Kobayashi and H. Otsuka, Quantum aspects of non-invertible flavor symmetries in intersecting/magnetized D-brane models, JHEP 04 (2025) 183 [ 2412.12524]

    2025 arXiv

  2. [10]

    Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl

    E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360

    1988

  3. [11]

    Moore and N

    G.W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177

    1989

  4. [12]

    Moore and N

    G.W. Moore and N. Seiberg, Taming the Conformal Zoo, Phys. Lett. B220 (1989) 422

    1989

  5. [13]

    Fuchs, Fusion rules in conformal field theory, Fortsch

    J. Fuchs, Fusion rules in conformal field theory, Fortsch. Phys. 42 (1994) 1 [hep-th/9306162]

    1994 arXiv

  6. [14]

    Bhardwaj and Y

    L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, JHEP 03 (2018) 189 [ 1704.02330]

    2018 arXiv

  7. [15]

    J. Dong, T. Kobayashi, R. Nishida, S. Nishimura and H. Otsuka, Coupling Selection Rules in Heterotic Calabi-Yau Compactifications, 2504.09773

    arXiv

  8. [16]

    Schafer-Nameki, ICTP lectures on (non-)invertible generalized symmetries, Phys

    S. Schafer-Nameki, ICTP lectures on (non-)invertible generalized symmetries, Phys. Rept. 1063 (2024) 1 [ 2305.18296]

    2024 arXiv

  9. [17]

    Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries, 2308.00747

    S.-H. Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries, 2308.00747

    arXiv

  10. [18]

    Choi, H.T

    Y. Choi, H.T. Lam and S.-H. Shao, Noninvertible Global Symmetries in the Standard Model, Phys. Rev. Lett.129 (2022) 161601 [ 2205.05086]

    2022 arXiv

  11. [19]

    Cordova, S

    C. Cordova, S. Hong, S. Koren and K. Ohmori, Neutrino Masses from Generalized Symmetry Breaking, Phys. Rev. X 14 (2024) 031033 [ 2211.07639]

    2024 arXiv

  12. [20]

    Cordova and K

    C. Cordova and K. Ohmori, Noninvertible Chiral Symmetry and Exponential Hierarchies, Phys. Rev. X 13 (2023) 011034 [ 2205.06243]

    2023 arXiv

  13. [21]

    Cordova, S

    C. Cordova, S. Hong and S. Koren, Non-Invertible Peccei-Quinn Symmetry and the Massless Quark Solution to the Strong CP Problem, 2402.12453

    arXiv

  14. [22]

    Kobayashi, H

    T. Kobayashi, H. Otsuka and M. Tanimoto, Yukawa textures from non-invertible symmetries, JHEP 12 (2024) 117 [ 2409.05270]

    2024 arXiv

  15. [23]

    Kobayashi, Y

    T. Kobayashi, Y. Nishioka, H. Otsuka and M. Tanimoto, More about quark Yukawa textures from selection rules without group actions, 2503.09966

    arXiv

  16. [24]

    Suzuki and L.-X

    M. Suzuki and L.-X. Xu, Phenomenological implications of a class of non-invertible selection rules, 2503.19964

  17. [25]

    Liang and T.T

    Q. Liang and T.T. Yanagida, Non-invertible symmetry as an axion-less solution to the strong CP problem, 2505.05142

    arXiv

  18. [26]

    Kobayashi, H

    T. Kobayashi, H. Otsuka, M. Tanimoto and H. Uchida, Lepton mass textures from non-invertible multiplication rules, 2505.07262

    arXiv

  19. [27]

    Kobayashi, H

    T. Kobayashi, H. Okada and H. Otsuka, Radiative neutrino mass models from non-invertible selection rules, 2505.14878

  20. [28]

    Nomura and H

    T. Nomura and H. Okada, Radiative lepton seesaw model in a non-invertible fusion rule and gauged B − L symmetry, 2506.16706. – 30 –

    arXiv

  21. [29]

    Kobayashi, H

    T. Kobayashi, H. Mita, H. Otsuka and R. Sakuma, Matter symmetries in supersymmetric standard models from non-invertible selection rules, 2506.10241

    arXiv

  22. [30]

    Abe, K.-S

    H. Abe, K.-S. Choi, T. Kobayashi and H. Ohki, Non-Abelian Discrete Flavor Symmetries from Magnetized/Intersecting Brane Models, Nucl. Phys. B 820 (2009) 317 [ 0904.2631]

    2009 arXiv

  23. [31]

    Berasaluce-Gonzalez, P.G

    M. Berasaluce-Gonzalez, P.G. Camara, F. Marchesano, D. Regalado and A.M. Uranga, Non-Abelian discrete gauge symmetries in 4d string models, JHEP 09 (2012) 059 [1206.2383]

    2012 arXiv

  24. [32]

    Marchesano, D

    F. Marchesano, D. Regalado and L. Vazquez-Mercado, Discrete flavor symmetries in D-brane models, JHEP 09 (2013) 028 [ 1306.1284]

    2013 arXiv

  25. [33]

    H. Abe, T. Kobayashi and H. Ohki, Magnetized orbifold models, JHEP 09 (2008) 043 [0806.4748]

    2008 arXiv

  26. [34]

    Ishimori, T

    H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto, Non-Abelian Discrete Symmetries in Particle Physics, Prog. Theor. Phys. Suppl.183 (2010) 1 [1003.3552]

    2010 arXiv

  27. [35]

    Kobayashi, H

    T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto, An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists(1, 2022), 10.1007/978-3-662-64679-3

    2022 doi

  28. [36]

    Hamidi and C

    S. Hamidi and C. Vafa, Interactions on Orbifolds, Nucl. Phys. B 279 (1987) 465

    1987

  29. [37]

    Dixon, D

    L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker, The Conformal Field Theory of Orbifolds, Nucl. Phys. B 282 (1987) 13

    1987

  30. [38]

    T.-H. Abe, Y. Fujimoto, T. Kobayashi, T. Miura, K. Nishiwaki and M. Sakamoto, ZN twisted orbifold models with magnetic flux, JHEP 01 (2014) 065 [ 1309.4925]

    2014 arXiv

  31. [39]

    T.-h. Abe, Y. Fujimoto, T. Kobayashi, T. Miura, K. Nishiwaki and M. Sakamoto, Operator analysis of physical states on magnetizedT 2/ZN orbifolds, Nucl. Phys. B 890 (2014) 442 [1409.5421]

    2014 arXiv

  32. [40]

    Kobayashi and S

    T. Kobayashi and S. Nagamoto, Zero-modes on orbifolds : magnetized orbifold models by modular transformation, Phys. Rev. D96 (2017) 096011 [ 1709.09784]

    2017 arXiv

  33. [41]

    Holthausen, M

    M. Holthausen, M. Lindner and M.A. Schmidt, CP and Discrete Flavour Symmetries, JHEP 04 (2013) 122 [ 1211.6953]

    2013 arXiv

  34. [42]

    Fallbacher and A

    M. Fallbacher and A. Trautner, Symmetries of symmetries and geometrical CP violation, Nucl. Phys. B 894 (2015) 136 [ 1502.01829]

    2015 arXiv

  35. [43]

    Krauss and F

    L.M. Krauss and F. Wilczek, Discrete Gauge Symmetry in Continuum Theories, Phys. Rev. Lett. 62 (1989) 1221

    1989

  36. [44]

    Ibanez and G.G

    L.E. Ibanez and G.G. Ross, Discrete gauge symmetry anomalies, Phys. Lett. B260 (1991) 291

    1991

  37. [45]

    Banks and M

    T. Banks and M. Dine, Note on discrete gauge anomalies, Phys. Rev. D45 (1992) 1424 [hep-th/9109045]

    1992 arXiv

  38. [46]

    Araki, T

    T. Araki, T. Kobayashi, J. Kubo, S. Ramos-Sanchez, M. Ratz and P.K.S. Vaudrevange, (Non-)Abelian discrete anomalies, Nucl. Phys. B 805 (2008) 124 [ 0805.0207]

    2008 arXiv

  39. [47]

    M.-C. Chen, M. Fallbacher, M. Ratz, A. Trautner and P.K.S. Vaudrevange, Anomaly-safe discrete groups, Phys. Lett. B747 (2015) 22 [ 1504.03470]. – 31 –

    2015 arXiv

  40. [48]

    Kobayashi and H

    T. Kobayashi and H. Uchida, Anomaly of non-Abelian discrete symmetries, Phys. Rev. D 105 (2022) 036018 [ 2111.10811]

    2022 arXiv

  41. [49]

    Frampton, S.L

    P.H. Frampton, S.L. Glashow and D. Marfatia, Zeroes of the neutrino mass matrix, Phys. Lett. B 536 (2002) 79 [ hep-ph/0201008]

    2002 arXiv

  42. [50]

    Fritzsch, Z.-z

    H. Fritzsch, Z.-z. Xing and S. Zhou, Two-zero Textures of the Majorana Neutrino Mass Matrix and Current Experimental Tests, JHEP 09 (2011) 083 [ 1108.4534]. – 32 –

    2011 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.