Non-Invertible Selection Rules on Heterotic Non-Abelian Orbifolds

Hajime Otsuka; Ryusei Nishida; Tatsuo Kobayashi

REVIEW 2 major objections 4 minor 73 references

Non-Invertible Selection Rules on Heterotic Non-Abelian Orbifolds

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read On non-Abelian orbifolds, heterotic string coupling selection rules are governed by conjugacy-class multiplication and are non-invertible, yielding a Yukawa texture on T6/T7 with zeros at the (1,1), (1,2), and (2,1) entries.

desk verdict Class-algebra selection rules are right and the T6/T7 texture is a nice payoff, but the non-Abelian fusion rule is asserted rather than derived and the T6/S3 example contains a concrete mistake. read the letter →

arxiv 2509.10019 v1 pith:J3RIRXIX submitted 2025-09-12 hep-th hep-ph

Tatsuo Kobayashi , Ryusei Nishida , Hajime Otsuka This is my paper

classification hep-thhep-ph PACS 11.25.-w11.25.Mj11.30.Hv

keywords non-invertibleselectionrulesheteroticorbifoldsnon-AbelianconjugacyclassesspacegroupYukawatextureT7orbifoldS3

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

This paper argues that on heterotic orbifolds with a non-Abelian point group, the allowed couplings among twisted string sectors are governed not by group-element multiplication but by the multiplication of conjugacy classes. Because a product of two conjugacy classes in a non-Abelian group generally expands into several classes, the selection rule dictates not one but a set of possible outcomes — the rule is non-invertible. The paper works this out for T^2/S3, T^6/S3, and T^6/T7, and shows that in the T^6/T7 example the non-invertible rule yields a 3x3 Yukawa texture with zeros at the (1,1), (1,2), and (2,1) entries. Such textures are of interest for flavor model building and for axion-less solutions to the strong CP problem, since they cannot be obtained from ordinary group symmetries alone.

What carries the argument

The central object is the conjugacy class of the space-group element (u,v), which labels a twisted sector, together with the class-algebra product σ_[g]σ_[g'] = Σ c_{ijk} σ_[g_k]. In an Abelian group each element is its own class, so the product is a single class and the rule is invertible; in a non-Abelian point group the product expands into a sum of classes, which is what makes the selection rule non-invertible. The same machinery — twist fields assigned to conjugacy classes — carries the argument from the point group to the full space group, including windings.

What would settle it

Compute the exact worldsheet three-point correlator on T^6/T7 for a coupling that the conjugacy-class rule allows (e.g., [(ω,0)][(ω,0)] → [(ω,0)]) but the Abelianization of the space group forbids, and check whether it vanishes; a nonzero value would confirm the non-invertible rule, while an exact zero would falsify it. Alternatively, compute the (1,1) entry of the Yukawa matrix (3.40) in a full heterotic T6/T7 model and check whether it is exactly zero.

Watch

Extended reading notes

Core claim

The central claim is that coupling selection rules in heterotic string theory on non-Abelian orbifolds are determined by the multiplication rules of conjugacy classes of the space group, [(u_i,0)][(u_j,0)] = c_{ijk}[(u_k,0)], where the coefficients c_{ijk} are non-negative integers and the right-hand side is not generally a single class. This makes the selection rules non-invertible: a pair of twist fields can fuse to any of several conjugacy classes, and in particular the product of two classes can include the untwisted sector. The paper verifies this structure concretely for T^2/S3, T^6/S3, and T^6/T7, and shows that on T^6/T7 the resulting allowed 3-point couplings produce a Yukawa matrix

Load-bearing premise

The load-bearing assumption is that twist-field OPEs follow the conjugacy-class multiplication of the space group, and not the Abelianization of the space group that a cited earlier work (footnote 4) suggests; if discrete torsion or worldsheet consistency modifies this rule, the non-invertible characterization and the T6/T7 texture would change.

Editorial extensions

If this is right

On non-Abelian orbifolds the point-group selection rules among twisted sectors are non-invertible; the paper gives explicit fusion rules for S3 and T7 (Eqs. (3.9), (3.22), (3.36)).
The T^6/T7 orbifold admits a Yukawa texture with zeros at (1,1), (1,2), (2,1) (Eq. (3.40)), a pattern unreachable by conventional group-theoretic selection rules.
Couplings including untwisted sectors are also non-invertible, because the product of twisted classes lands in an infinite tower of untwisted winding states; massive winding modes contribute via world-sheet instantons.
Non-invertible selection rules can also arise on Abelian orbifolds when twisted sectors have different sublattice structures (1-u)Λ, as shown for T^2/Z4.
Such textures are phenomenologically relevant: e.g., a texture where only the (3,3) entry carries a CP phase offers an axion-less solution to the strong CP problem (as cited in Refs. [11,12]).

Reading between the lines

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

If the class-algebra rule is confirmed by explicit correlator computations, the selection rules of every non-Abelian orbifold could be catalogued from the point group's class algebra, giving a systematic tool for flavor-model building.
The non-invertible fusion structure here resembles a fusion category without a fully defined categorical notion; it would be fruitful to see whether the T6/T7 rules satisfy associativity and whether they can be lifted to a genuine fusion category in the worldsheet theory.
The specific two-zero texture (3.40) could be used as a target for bottom-up model building; one could check whether a full heterotic T6/T7 model with Wilson lines and moduli can realize realistic quark masses and CP violation.
A direct test would be to compute the exact three-point correlator on T6/T7 for one of the 'allowed' couplings predicted by the class-algebra rule but forbidden by the Abelianization of the space group; a nonzero result would confirm the non-invertible assignment.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know before you read it. First, the central claim is sound: on non-Abelian orbifolds, twisted sectors are labelled by conjugacy classes of the space group, so coupling selection rules should be governed by the class algebra. Because the product of two classes is usually a sum of several classes, the rules are non-invertible. Second, the paper's T6/T7 example gives a concrete Yukawa texture with a strong-CP hook, but the text has a few warts that need cleaning.

What is actually new: explicit class-algebra fusion rules for T2/S3, T6/S3, and T6/T7 (Eqs. (3.9), (3.22), (3.36)) and the resulting T6/T7 texture (3.40) with zeros at (1,1), (1,2), (2,1). The ingredients are all old — conjugacy-class labelling goes back to [35,36], and the space-group selection rule to [49,50] — but nobody had written down the non-invertible point-group fusion in these heterotic models. The class products I checked are correct.

Now the soft spots, in proportion. Eq. (3.30) lists a coupling [(1,0)][(θ,m1,m2)][(ω,-m1,-m2)] that violates the paper's own rule: [(θ)][(ω)] = [(θ)] from (3.22), and that product never contains the identity class. That is a real internal inconsistency, though it sits in the T6/S3 section and does not affect the T6/T7 texture. Eq. (3.17) is garbled — a missing bracket and a dropped shift charge. More fundamentally, the step from the Abelian fusion rule (2.18) to the non-Abelian class product (3.2)-(3.3) is asserted, not derived. The paper cites Ref. [65] in a footnote for the "Abelianization of the space group" but never reconciles that with the non-Abelian picture. The stress-test worry that the abelianized rule would allow entries the paper forbids is, as far as I can tell, backwards: the abelianized condition is necessary but not sufficient, and the class-product rule is the correct stringy generalization. Still, a referee should ask the authors to justify or cite a derivation of that step.

Minor complaints: the claim that the T6/T7 texture "can not be derived from group theory" is overstated; a discrete flavour symmetry might well reproduce it. And the strong-CP significance depends on generation assignments and on the authors' earlier framework, which are not developed here.

Who should read this: people doing heterotic orbifold phenomenology and anyone following the non-invertible symmetry wave. The paper is short and mostly clean. I'd send it to peer review — the core idea is correct, the texture is interesting, and the flaws are fixable rather than fatal. Ask for a proper justification of the non-Abelian fusion rule and a cleanup of (3.30) and (3.17).

Referee Report

2 major / 4 minor

Summary. The paper studies coupling selection rules for closed strings on non-Abelian orbifolds in heterotic string theory. It argues that, because boundary conditions are classified by conjugacy classes of the space group, the point-group selection rule is governed by the multiplication of conjugacy classes of the point group. Since such class products generally yield a sum of several classes, the resulting selection rules are 'non-invertible.' The paper works out explicit class algebras for T^2/S3, T^6/S3, and T^6/T7, lists allowed 3-point couplings in each case, and derives a concrete Yukawa texture for T^6/T7 with zeros controlled by the class-product rule.

Significance. If the central assumption is correct, the paper provides a useful and clearly presented framework: non-Abelian point groups naturally give rise to non-invertible selection rules, extending the recent program on non-invertible flavor symmetries to heterotic non-Abelian orbifolds. The class-algebra computations in (3.9), (3.22), and (3.36) are explicit and correct, and the texture (3.40) is a concrete, in-principle testable consequence. The main weakness is that the load-bearing step—replacing group multiplication by conjugacy-class multiplication for twist fields—is assumed rather than derived from the worldsheet CFT, and the relation to the alternative 'Abelianization of the space group' rule cited in footnote 4 is not reconciled. The paper's phenomenological claims are therefore conditional on this rule being the correct one.

major comments (2)

[§3, Eqs. (3.2)–(3.3)] The central rule is assumed, not derived. In the Abelian case, Eq. (2.18) is supported by Refs. [35,36]; for non-Abelian point groups the replacement of group multiplication by class multiplication is introduced as 'Suppose that...' and no worldsheet derivation is provided. Footnote 4 cites Ref. [65] for the 'Abelianization of the space group' but does not reconcile that rule with the class-product rule. The difference is material for the paper's main example: in T^6/T7, [(ω,0)] and [(ω^3,0)] are both trivial in the abelianization of the space group, whereas the class-product rule gives [(ω,0)]^2 = [(ω,0)]+[(ω^3,0)] (3.38), forbidding [(1,0)][(ω,0)][(ω,0)]. This coupling controls the zeros at entries (1,1), (1,2), and (2,1) of the texture (3.40). Please derive the class-product fusion from the orbifold CFT, or state precisely how it follows from Ref. [65], and explain why the abelianized
[§3.2, Eq. (3.30)] The list of allowed 3-point couplings is inconsistent with the preceding class products. The third line reads [(1,0)][(θ,m1e1+m2e2)][(ω,−m1e1−m2e2)]. By (3.22) and the commutativity stated there, the point-group product of [(θ,·)] and [(ω,·)] is [(θ,0)], not [(1,0)], so under the class-product rule this coupling is forbidden. This appears to be a typo—perhaps the third factor should be [(θ,−m1e1−m2e2)]—but as written the example is incorrect and should be fixed.

minor comments (4)

[§3.3, around Eq. (3.40)] The statement that the texture 'can not be derived from group theory' is imprecise. The class-product rule is itself a finite algebraic structure; presumably the intended meaning is that it is not a group action of the point group. Please rephrase to avoid over-interpretation.
[§3.2, Eq. (3.29)] The notation [[(u_i,m1,m2,m3,m5)]] is introduced in (3.28), but the domain of summation and the precise meaning for u_i = 1, ω, θ should be stated explicitly. In particular, for u=θ the condition m5=0 is given, but the ranges for m1,m2,m3 and the definition of [[(1,...)]] should be spelled out.
[§2–§3, Eqs. (2.18), (3.3)] The equality σ_[g] σ_[g'] = c_{ijk} σ_[u_k] is presented as a multiplication rule; it may be helpful to clarify that this is a selection-rule/fusion-level statement, and that actual OPE coefficients are not computed in this paper.
[§3.3] The notation 'T7' for the non-Abelian group of order 21 is standard in the orbifold literature, but a brief parenthetical definition would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the group-theoretic class-product computations are self-contained; the non-Abelian selection rule itself is an explicit, unproven physical assumption, and self-citations are only contextual.

full rationale

The paper's derivation chain is a direct computation from explicitly stated conjugacy-class multiplication rules. In Sec. 2, the Abelian twist-field rule sigma_[g]sigma_[g'] = sigma_[gg'] (Eq. 2.18) is imported from Refs. [35,36] (Hamidi-Vafa; Dixon-Friedan-Martinec-Shenker), and Sec. 3 extends it to non-Abelian point groups by the conjugacy-class product (Eqs. 3.2-3.3). This extension is explicitly flagged as an assumption ('Suppose that the multiplication rules of conjugacy classes...'), not derived from the worldsheet CFT; therefore the non-invertible character of the resulting selection rules follows from the assumed class algebra, but the paper does not disguise this as an independent derivation. All subsequent results — the S3 products (3.9), (3.22), the T7 products (3.36), the shifted products (3.38), and the Yukawa texture (3.40) — are straightforward, parameter-free computations from the group presentations and lattice data. No fitted value is renamed as a prediction, and no result is imported from a same-author citation as the load-bearing premise. The self-citations in the introduction and in the strong-CP discussion are contextual. The alternative abelianized space-group selection rule of Ref. [65] is a genuine physical concern and is not reconciled, but that is a correctness/assumption risk, not a circularity. Accordingly the paper is self-contained against its own stated assumptions; score 2 reflects only minor contextual self-citation.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles, forces, or conserved quantities, and it fits no numbers to data. Its load-bearing costs are conceptual assumptions: the asserted extension of the Abelian twist-field fusion rule to non-Abelian conjugacy-class products (Eq. (3.3)), the restriction to massless (zero-winding) states, the geometric-only treatment that omits the gauge embedding, and the illustrative generation assignments in the Yukawa examples. The lattice arithmetic (sublattices (1-u)Lambda) is standard and verified in the main examples.

free parameters (2)

Massless-mode restriction (winding v = 0 along unrotated directions) = v = 0
Chosen by hand 'for generic moduli parameters' (Section 3, after Eq. (3.7)); the massless selection rules and all Yukawa patterns depend on dropping massive winding states.
Generation assignments in the Yukawa examples = T^6/T7: [(1,0)], [(ω,0)], [(ω^3,0)]; T^2/Z3: [(ω,0)], [(ω,e1)], [(ω,2e1)]
Illustrative choices ('suppose that three generations...' before Eqs. (2.32) and (3.39)); the zero pattern of texture (3.40) holds only for this assignment, and no concrete model realizing it is given.

assumptions (6)

domain assumption Closed-string boundary conditions on an orbifold are classified by conjugacy classes of the space group.
Invoked in Section 2 (Eqs. (2.7)-(2.9)) and extended to non-Abelian groups in Section 3 (Eq. (3.4)); imported from the orbifold CFT literature [35,36].
domain assumption A coupling is allowed iff the product of the corresponding space group (conjugacy class) elements contains the identity element, up to (1-u)Lambda identifications.
Eqs. (2.15) and (3.3); this is the space group selection rule, taken as the definition of allowed couplings; never derived from a string amplitude in this paper.
ad hoc to paper Twist fields multiply by the conjugacy class product: sigma_[g] sigma_[g'] = sigma_[gg'], including the non-Abelian case where [gg'] is a sum of classes.
Eqs. (2.18) (Abelian, citing [35,36]) and (3.2)-(3.3) (non-Abelian, asserted by analogy). The non-Abelian step is the load-bearing assumption that makes the rules non-invertible.
standard math The relevant shift identifications are the sublattices (1-u)Lambda for each point group element u.
Used throughout Section 3, e.g., (1-omega)Lambda_SU(3) and (1-theta)Lambda_SU(3) in Section 3.1; verified lattice arithmetic in the main examples.
domain assumption Winding modes on untwisted torus directions are massive for generic moduli and can be dropped from the massless selection rules.
Stated after Eq. (3.7) and used to justify focusing on v = 0 classes; standard in the orbifold literature, but it means the textures are statements about massless states only.
domain assumption The gauge embedding (shift vectors and Wilson lines) and modular invariance do not alter the geometric selection rules.
The paper never introduces the gauge sector; real heterotic models must satisfy level matching and gauge-invariance conditions, which can project out states or modify which classes are massless.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Non-Invertible Selection Rules on Heterotic Non-Abelian Orbifolds." pith.science (2026). https://pith.science/paper/J3RIRXIX

@misc{pith2026250910019,
  author       = {Pith},
  title        = {Pith review of: Non-Invertible Selection Rules on Heterotic Non-Abelian Orbifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J3RIRXIX}},
  note         = {Machine review of arXiv:2509.10019}
}

read the original abstract

We investigate coupling selection rules in heterotic string theory on non-Abelian orbifolds. Since boundary conditions on the orbifolds are classified by conjugacy classes of space group elements, non-Abelian orbifolds give rise to non-invertible selection rules on couplings among twisted sectors as well as ones including untwisted sectors. Furthermore, we find that non-invertible selection rules lead to characteristic patterns of Yukawa matrices.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

73 extracted references · 55 linked inside Pith

[65]

Ramos-Sánchez and P.K.S

S. Ramos-Sánchez and P.K.S. Vaudrevange,Note on the space group selection rule for closed strings on orbifolds,JHEP01(2019) 055 [1811.00580]

arXiv 2019
[1]

Kobayashi and H

T. Kobayashi and H. Otsuka,Non-invertible flavor symmetries in magnetized extra dimensions,JHEP11(2024) 120 [2408.13984]

arXiv 2024
[2]

Funakoshi, T

S. Funakoshi, T. Kobayashi and H. Otsuka,Quantum aspects of non-invertible flavor symmetries in intersecting/magnetized D-brane models,JHEP04(2025) 183 [2412.12524]

arXiv 2025
[3]

Kobayashi, H

T. Kobayashi, H. Otsuka and M. Tanimoto,Yukawa textures from non-invertible symmetries, JHEP12(2024) 117 [2409.05270]

arXiv 2024
[4]

Kobayashi, Y

T. Kobayashi, Y. Nishioka, H. Otsuka and M. Tanimoto,More about quark Yukawa textures from selection rules without group actions,JHEP05(2025) 177 [2503.09966]

arXiv 2025
[5]

Kobayashi, H

T. Kobayashi, H. Otsuka, M. Tanimoto and H. Uchida,Lepton mass textures from non-invertible multiplication rules,JHEP08(2025) 189 [2505.07262]

arXiv 2025
[6]

Kobayashi, H

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

arXiv
[7]

Nomura and H

T. Nomura and H. Okada,Radiative lepton seesaw model in a non-invertible fusion rule and gaugedB−Lsymmetry,2506.16706

arXiv

Show all 73 references

[8]

Chen, C.-Q

J. Chen, C.-Q. Geng, H. Okada and J.-J. Wu,A radiative lepton model in a non-invertible fusion rule,2507.11951. – 17 –

arXiv
[9]

Okada and Y

H. Okada and Y. Shigekami,Three-loop induced neutrino mass model in a non-invertible symmetry,2507.16198

arXiv
[10]

Jangid and H

S. Jangid and H. Okada,A natural realization of inverse seesaw model in a non-invertible selection rule,2508.16174

arXiv
[11]

Liang and T.T

Q. Liang and T.T. Yanagida,Non-invertible symmetry as an axion-less solution to the strong CP problem,Phys. Lett. B868(2025) 139706 [2505.05142]

2025 arXiv
[12]

Kobayashi, H

T. Kobayashi, H. Otsuka and T.T. Yanagida,Non-invertible Symmetry as a Solution to the Strong CP Problem in a GUT-inspired Standard Model,2508.12287

arXiv
[13]

Suzuki and L.-X

M. Suzuki and L.-X. Xu,Phenomenological implications of a class of non-invertible selection rules,2503.19964
[14]

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
[15]

Suzuki, L.-X

M. Suzuki, L.-X. Xu and H.Y. Zhang,Spurion Analysis for Non-Invertible Selection Rules from Near-Group Fusions,2508.14970

arXiv
[16]

Gomes,An introduction to higher-form symmetries,SciPost Phys

P.R.S. Gomes,An introduction to higher-form symmetries,SciPost Phys. Lect. Notes74 (2023) 1 [2303.01817]

2023 arXiv
[17]

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
[18]

Bhardwaj, L.E

L. Bhardwaj, L.E. Bottini, L. Fraser-Taliente, L. Gladden, D.S.W. Gould, A. Platschorre et al.,Lectures on generalized symmetries,Phys. Rept.1051(2024) 1 [2307.07547]

2024 arXiv
[19]

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
[20]

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
[21]

Cordova, S

C. Cordova, S. Hong, S. Koren and K. Ohmori,Neutrino Masses from Generalized Symmetry Breaking,Phys. Rev. X14(2024) 031033 [2211.07639]

2024 arXiv
[22]

Cordova and K

C. Cordova and K. Ohmori,Noninvertible Chiral Symmetry and Exponential Hierarchies, Phys. Rev. X13(2023) 011034 [2205.06243]

2023 arXiv
[23]

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
[24]

Delgado and S

A. Delgado and S. Koren,Non-invertible Peccei-Quinn symmetry, natural 2HDM alignment, and the visible axion,JHEP02(2025) 178 [2412.05362]

2025 arXiv
[25]

Dixon, J.A

L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten,Strings on Orbifolds,Nucl. Phys. B261 (1985) 678

1985
[26]

Dixon, J.A

L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten,Strings on Orbifolds. 2.,Nucl. Phys. B274 (1986) 285

1986
[27]

Ibanez, H.P

L.E. Ibanez, H.P. Nilles and F. Quevedo,Orbifolds and Wilson Lines,Phys. Lett. B187 (1987) 25

1987
[28]

Ibanez, J.E

L.E. Ibanez, J.E. Kim, H.P. Nilles and F. Quevedo,Orbifold Compactifications with Three Families of SU(3) x SU(2) x U(1)**n,Phys. Lett. B191(1987) 282. – 18 –

1987
[29]

Kobayashi, S

T. Kobayashi, S. Raby and R.-J. Zhang,Constructing 5-D orbifold grand unified theories from heterotic strings,Phys. Lett. B593(2004) 262 [hep-ph/0403065]

2004 arXiv
[30]

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. B704(2005) 3 [hep-ph/0409098]

2005 arXiv
[31]

Buchmuller, K

W. Buchmuller, K. Hamaguchi, O. Lebedev and M. Ratz,Supersymmetric standard model from the heterotic string,Phys. Rev. Lett.96(2006) 121602 [hep-ph/0511035]

2006 arXiv
[32]

Buchmuller, K

W. Buchmuller, K. Hamaguchi, O. Lebedev and M. Ratz,Supersymmetric Standard Model from the Heterotic String (II),Nucl. Phys. B785(2007) 149 [hep-th/0606187]

2007 arXiv
[33]

Lebedev, H.P

O. Lebedev, H.P. Nilles, S. Raby, S. Ramos-Sanchez, M. Ratz, P.K.S. Vaudrevange et al.,A Mini-landscape of exact MSSM spectra in heterotic orbifolds,Phys. Lett. B645(2007) 88 [hep-th/0611095]

2007 arXiv
[34]

Lebedev, H.P

O. Lebedev, H.P. Nilles, S. Raby, S. Ramos-Sanchez, M. Ratz, P.K.S. Vaudrevange et al., The Heterotic Road to the MSSM with R parity,Phys. Rev. D77(2008) 046013 [0708.2691]

2008 arXiv
[35]

Hamidi and C

S. Hamidi and C. Vafa,Interactions on Orbifolds,Nucl. Phys. B279(1987) 465

1987
[36]

Dixon, D

L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker,The Conformal Field Theory of Orbifolds,Nucl. Phys. B282(1987) 13

1987
[37]

Burwick, R.K

T.T. Burwick, R.K. Kaiser and H.F. Muller,General Yukawa couplings of strings on Z(N) orbifolds,Nucl. Phys. B355(1991) 689

1991
[38]

Choi and T

K.-S. Choi and T. Kobayashi,Higher order couplings from heterotic orbifold theory,Nucl. Phys. B797(2008) 295 [0711.4894]

2008 arXiv
[39]

Font, L.E

A. Font, L.E. Ibanez, H.P. Nilles and F. Quevedo,On the Concept of Naturalness in String Theories,Phys. Lett. B213(1988) 274

1988
[40]

Cabo Bizet, T

N.G. Cabo Bizet, T. Kobayashi, D.K. Mayorga Pena, S.L. Parameswaran, M. Schmitz and I. Zavala,R-charge Conservation and More in Factorizable and Non-Factorizable Orbifolds, JHEP05(2013) 076 [1301.2322]

2013 arXiv
[41]

Nilles, S

H.P. Nilles, S. Ramos-Sánchez, M. Ratz and P.K.S. Vaudrevange,A note on discreteR symmetries inZ 6-II orbifolds with Wilson lines,Phys. Lett. B726(2013) 876 [1308.3435]

2013 arXiv
[42]

Cabo Bizet, T

N.G. Cabo Bizet, T. Kobayashi, D.K. Mayorga Pena, S.L. Parameswaran, M. Schmitz and I. Zavala,Discrete R-symmetries and Anomaly Universality in Heterotic Orbifolds,JHEP02 (2014) 098 [1308.5669]

2014 arXiv
[43]

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

1988
[44]

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. B768(2007) 135 [hep-ph/0611020]

2007 arXiv
[45]

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

2014 arXiv
[46]

Thorngren and Y

R. Thorngren and Y. Wang,Fusion category symmetry. Part II. Categoriosities at c = 1 and beyond,JHEP07(2024) 051 [2106.12577]

2024 arXiv
[47]

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. – 19 –

arXiv
[48]

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]

2024 arXiv
[49]

Kobayashi and N

T. Kobayashi and N. Ohtsubo,Yukawa Coupling Condition ofZ(N) Orbifold Models,Phys. Lett. B245(1990) 441

1990
[50]

Kobayashi and N

T. Kobayashi and N. Ohtsubo,Geometrical aspects of Z(N) orbifold phenomenology,Int. J. Mod. Phys. A9(1994) 87

1994
[51]

Kobayashi,Selection rules for nonrenormalizable couplings in superstring theories,Phys

T. Kobayashi,Selection rules for nonrenormalizable couplings in superstring theories,Phys. Lett. B354(1995) 264 [hep-ph/9504371]

1995 arXiv
[52]

Cvetic,Suppression of Nonrenormalizable Terms in the Effective Superpotential for (Blownup) Orbifold Compactification,Phys

M. Cvetic,Suppression of Nonrenormalizable Terms in the Effective Superpotential for (Blownup) Orbifold Compactification,Phys. Rev. Lett.59(1987) 1795

1987
[53]

Kobayashi, S.L

T. Kobayashi, S.L. Parameswaran, S. Ramos-Sanchez and I. Zavala,Revisiting Coupling Selection Rules in Heterotic Orbifold Models,JHEP05(2012) 008 [1107.2137]

2012 arXiv
[54]

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

arXiv
[55]

Inoue, M

K. Inoue, M. Sakamoto and H. Takano,NONABELIAN ORBIFOLDS,Prog. Theor. Phys. 78(1987) 908

1987
[56]

Inoue, S

K. Inoue, S. Nima and H. Takano,Zero Mode and Modular Invariance in String on Nonabelian Orbifold,Prog. Theor. Phys.80(1988) 881

1988
[57]

Inoue and S

K. Inoue and S. Nima,String interactions on nonAbelian orbifold,Prog. Theor. Phys.84 (1990) 702

1990
[58]

Konopka,Non Abelian orbifold compactifications of the heterotic string,JHEP07 (2013) 023 [1210.5040]

S.J.H. Konopka,Non Abelian orbifold compactifications of the heterotic string,JHEP07 (2013) 023 [1210.5040]

2013 arXiv
[59]

Fischer, M

M. Fischer, M. Ratz, J. Torrado and P.K.S. Vaudrevange,Classification of symmetric toroidal orbifolds,JHEP01(2013) 084 [1209.3906]

2013 arXiv
[60]

Fischer, S

M. Fischer, S. Ramos-Sanchez and P.K.S. Vaudrevange,Heterotic non-Abelian orbifolds, JHEP07(2013) 080 [1304.7742]

2013 arXiv
[61]

Funakoshi, Y

S. Funakoshi, Y. Koga and H. Otsuka,Classification of Modular Symmetries in Non-Supersymmetric Heterotic String theories,2503.23741

arXiv
[62]

Hernandez-Segura and S

M. Hernandez-Segura and S. Ramos-Sanchez,Non-Abelian orbifolds of the SO(32) heterotic string,2506.08370

arXiv
[63]

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
[64]

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
[66]

J. Dong, T. Jeric, T. Kobayashi, R. Nishida and H. Otsuka,On discrete gauging and non-invertible selection rules,2507.02375. – 20 –

arXiv
[67]

Antusch, M

S. Antusch, M. Holthausen, M.A. Schmidt and M. Spinrath,Solving the Strong CP Problem with Discrete Symmetries and the Right Unitarity Triangle,Nucl. Phys. B877(2013) 752 [1307.0710]

2013 arXiv
[68]

Feruglio, A

F. Feruglio, A. Strumia and A. Titov,Modular invariance and the QCD angle,JHEP07 (2023) 027 [2305.08908]

2023 arXiv
[69]

Petcov and M

S.T. Petcov and M. Tanimoto,A 4 modular invariance and the strong CP problem,Eur. Phys. J. C84(2024) 914 [2404.00858]

2024 arXiv
[70]

Penedo and S.T

J.T. Penedo and S.T. Petcov,Finite modular symmetries and the strong CP problem,JHEP 10(2024) 172 [2404.08032]

2024 arXiv
[71]

Liang, R

Q. Liang, R. Okabe and T.T. Yanagida,Three-zero texture of quark-mass matrices as a solution to the strong CP problem,Phys. Lett. B859(2024) 139123 [2408.12146]

2024 arXiv
[72]

Frampton, S.L

P.H. Frampton, S.L. Glashow and D. Marfatia,Zeroes of the neutrino mass matrix,Phys. Lett. B536(2002) 79 [hep-ph/0201008]

2002 arXiv
[73]

Fritzsch, Z.-z

H. Fritzsch, Z.-z. Xing and S. Zhou,Two-zero Textures of the Majorana Neutrino Mass Matrix and Current Experimental Tests,JHEP09(2011) 083 [1108.4534]. – 21 –

2011 arXiv

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [65]

Ramos-Sánchez and P.K.S

S. Ramos-Sánchez and P.K.S. Vaudrevange,Note on the space group selection rule for closed strings on orbifolds,JHEP01(2019) 055 [1811.00580]

arXiv 2019

[2] [1]

Kobayashi and H

T. Kobayashi and H. Otsuka,Non-invertible flavor symmetries in magnetized extra dimensions,JHEP11(2024) 120 [2408.13984]

arXiv 2024

[3] [2]

Funakoshi, T

S. Funakoshi, T. Kobayashi and H. Otsuka,Quantum aspects of non-invertible flavor symmetries in intersecting/magnetized D-brane models,JHEP04(2025) 183 [2412.12524]

arXiv 2025

[4] [3]

Kobayashi, H

T. Kobayashi, H. Otsuka and M. Tanimoto,Yukawa textures from non-invertible symmetries, JHEP12(2024) 117 [2409.05270]

arXiv 2024

[5] [4]

Kobayashi, Y

T. Kobayashi, Y. Nishioka, H. Otsuka and M. Tanimoto,More about quark Yukawa textures from selection rules without group actions,JHEP05(2025) 177 [2503.09966]

arXiv 2025

[6] [5]

Kobayashi, H

T. Kobayashi, H. Otsuka, M. Tanimoto and H. Uchida,Lepton mass textures from non-invertible multiplication rules,JHEP08(2025) 189 [2505.07262]

arXiv 2025

[7] [6]

Kobayashi, H

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

arXiv

[8] [7]

Nomura and H

T. Nomura and H. Okada,Radiative lepton seesaw model in a non-invertible fusion rule and gaugedB−Lsymmetry,2506.16706

arXiv