Symmetries and stabilisers in modular invariant flavour models

Abstract The idea of modular invariance provides a novel explanation of flavour mixing. Within the context of finite modular symmetries ΓN and for a given element γ ∈ ΓN, we present an algorithm for finding stabilisers (specific values for moduli fields τγ which remain unchanged under the action associated to γ). We then employ this algorithm to find all stabilisers for each element of finite modular groups for N = 2 to 5, namely, Γ2 ≃ S3, Γ3 ≃ A4, Γ4 ≃ S4 and Γ5 ≃ A5. These stabilisers then leave preserved a specific cyclic subgroup of ΓN. This is of interest to build models of fermionic mixing where each fermionic sector preserves a separate residual symmetry.

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ON THE SPECTRUM, NO-GHOST THEOREM AND MODULAR INVARIANCE OF W3 STRINGS

International Journal of Modern Physics A ◽

10.1142/s0217751x93001168 ◽

1993 ◽

Vol 08 (17) ◽

pp. 2875-2893 ◽

Cited By ~ 4

Author(s):

PETER WEST

Keyword(s):

Ising Model ◽

Partition Function ◽

Ghost Number ◽

Modular Invariance ◽

Modular Invariant

A spectrum-generating algebra is constructed and used to find all the physical states of the W3 string with standard ghost number. These states are shown to have positive norm and their partition function is found to involve the Ising model characters corresponding to the weights 0 and 1/16. The theory is found to be modular invariant if, in addition, one includes states that correspond to the Ising character of weight 1/2. It is shown that these additional states are indeed contained in the cohomology of Q.

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Worldline theories with towers of internal states

Journal of High Energy Physics ◽

10.1007/jhep12(2020)069 ◽

2020 ◽

Vol 2020 (12) ◽

Author(s):

Steven Abel ◽

Daniel Lewis

Keyword(s):

String Theory ◽

Saddle Point ◽

Invariant Theory ◽

Degrees Of Freedom ◽

Geometric Interpretation ◽

Modular Invariance ◽

Modular Invariant ◽

Internal States ◽

Non Local ◽

Internal Degrees Of Freedom

Abstract We study particle theories that have a tower of worldline internal degrees of freedom. Such a theory can arise when the worldsheet of closed strings is dimensionally reduced to a worldline, in which case the tower is infinite with regularly spaced masses. But our discussion is significantly more general than this, and there is scope to consider all kinds of internal degrees of freedom carried by the propagating particle. For example it is possible to consider towers corresponding to other geometries, or towers with no obvious geometric interpretation that still yield a modular invariant theory. Truncated towers generate non-local particle theories that share with string theory the property of having a Gross-Mende-like saddle point in their amplitudes. This provides a novel framework for constructing exotic theories which may have desirable properties such as finiteness and modular invariance.

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Degree of reductivity of a modular representation

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500231 ◽

2017 ◽

Vol 19 (03) ◽

pp. 1650023 ◽

Cited By ~ 1

Author(s):

Martin Kohls ◽

Müfi̇t Sezer

Keyword(s):

Fixed Point ◽

Lower Bound ◽

Cyclic Subgroup ◽

Representation Formula ◽

Modular Representation ◽

Dimensional Representation ◽

Finite Dimensional Representation ◽

Finite Dimensional ◽

Modular Groups ◽

Sharp Lower Bound

For a finite-dimensional representation [Formula: see text] of a group [Formula: see text] over a field [Formula: see text], the degree of reductivity [Formula: see text] is the smallest degree [Formula: see text] such that every nonzero fixed point [Formula: see text] can be separated from zero by a homogeneous invariant of degree at most [Formula: see text]. We compute [Formula: see text] explicitly for several classes of modular groups and representations. We also demonstrate that the maximal size of a cyclic subgroup is a sharp lower bound for this number in the case of modular abelian [Formula: see text]-groups.

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DIFFERENTIAL EQUATIONS, DUALITY AND MODULAR INVARIANCE

Communications in Contemporary Mathematics ◽

10.1142/s021919970500191x ◽

2005 ◽

Vol 07 (05) ◽

pp. 649-706 ◽

Cited By ~ 30

Author(s):

YI-ZHI HUANG

Keyword(s):

Differential Equations ◽

Vertex Operator ◽

Operator Algebra ◽

Correlation Functions ◽

Vertex Operator Algebra ◽

Modular Invariance ◽

Genus Zero ◽

Modular Invariant ◽

The One ◽

Genus One

We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0 and V(0) = ℂ1, (ii) every ℕ-gradable weak V-module is completely reducible and (iii) V is C2-cofinite. We establish the fundamental properties of these functions, including suitably formulated commutativity, associativity and modular invariance. The method we develop and use here is completely different from the one previously used by Zhu and others. In particular, we show that the q-traces of products of certain geometrically-modified intertwining operators satisfy modular invariant systems of differential equations which, for any fixed modular parameter, reduce to doubly-periodic systems with only regular singular points. Together with the results obtained by the author in the genus-zero case, the results of the present paper solves essentially the problem of constructing chiral genus-one weakly conformal field theories from the representations of a vertex operator algebra satisfying the conditions above.

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Modular symmetry at level 6 and a new route towards finite modular groups

Journal of High Energy Physics ◽

10.1007/jhep10(2021)238 ◽

2021 ◽

Vol 2021 (10) ◽

Author(s):

Cai-Chang Li ◽

Xiang-Gan Liu ◽

Gui-Jun Ding

Keyword(s):

Invariant Theory ◽

Modular Group ◽

Congruence Subgroup ◽

Principal Congruence ◽

Congruence Subgroups ◽

Modular Invariant ◽

Benchmark Model ◽

Modular Groups ◽

Principal Congruence Subgroup ◽

Comprehensive Study

Abstract We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as Γ(N′)/Γ(N″), and the modular group SL(2, ℤ) is ex- tended to a principal congruence subgroup Γ(N′). The original modular invariant theory is reproduced when N′ = 1. We perform a comprehensive study of $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry corresponding to N′ = 1 and N″ = 6, five types of models for lepton masses and mixing with $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry are discussed and some example models are studied numerically. The case of N′ = 2 and N″ = 6 is considered, the finite modular group is Γ(2)/Γ(6) ≅ T′, and a benchmark model is constructed.

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Modular invariance faces precision neutrino data

10.21468/scipostphys.5.5.042 ◽

2018 ◽

Vol 5 (5) ◽

Cited By ~ 46

Author(s):

Juan Carlos Criado ◽

Ferruccio Feruglio

Keyword(s):

Complex Modulus ◽

Real Field ◽

Neutrino Masses ◽

Modular Invariance ◽

Modular Invariant ◽

Normal Ordering ◽

Mixing Angles ◽

Invariant Model ◽

Higher Dimensional ◽

Group Evolution

We analyze a modular invariant model of lepton masses, with neutrino masses originating either from the Weinberg operator or from the seesaw. The constraint provided by modular invariance is so strong that neutrino mass ratios, lepton mixing angles and Dirac/Majorana phases do not depend on any Lagrangian parameter. They only depend on the vacuum of the theory, parametrized in terms of a complex modulus and a real field. Thus eight measurable quantities are described by the three vacuum parameters, whose optimization provides an excellent fit to data for the Weinberg operator and a good fit for the seesaw case. Neutrino masses from the Weinberg operator (seesaw) have inverted (normal) ordering. Several sources of potential corrections, such as higher dimensional operators, renormalization group evolution and supersymmetry breaking effects, are carefully discussed and shown not to affect the predictions under reasonable conditions.

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Fermion mass hierarchies, large lepton mixing and residual modular symmetries

Journal of High Energy Physics ◽

10.1007/jhep04(2021)206 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

P. P. Novichkov ◽

J. T. Penedo ◽

S. T. Petcov

Keyword(s):

Charged Lepton ◽

Hierarchical Structures ◽

Fine Tuning ◽

Fermion Mass ◽

Fermion Field ◽

Modular Invariant ◽

Field Representation ◽

Residual Symmetry ◽

Mixing Matrix ◽

Symmetric Points

Abstract In modular-invariant models of flavour, hierarchical fermion mass matrices may arise solely due to the proximity of the modulus τ to a point of residual symmetry. This mechanism does not require flavon fields, and modular weights are not analogous to Froggatt-Nielsen charges. Instead, we show that hierarchies depend on the decomposition of field representations under the residual symmetry group. We systematically go through the possible fermion field representation choices which may yield hierarchical structures in the vicinity of symmetric points, for the four smallest finite modular groups, isomorphic to S3, A4, S4, and A5, as well as for their double covers. We find a restricted set of pairs of representations for which the discussed mechanism may produce viable fermion (charged-lepton and quark) mass hierarchies. We present two lepton flavour models in which the charged-lepton mass hierarchies are naturally obtained, while lepton mixing is somewhat fine-tuned. After formulating the conditions for obtaining a viable lepton mixing matrix in the symmetric limit, we construct a model in which both the charged-lepton and neutrino sectors are free from fine-tuning.

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