Modular forms of real weights and generalised Dedekind symbols

Manin, Yuri I.

doi:10.1007/s40687-018-0120-x

Cited by 3 publications

(2 citation statements)

References 18 publications

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“…The theory of modular forms with real weight and even complex weight has been developed as well [188], and then the modular group SL(2, Z) should be extended to the universal covering groups. Some concrete examples are given in [189,190].…”

Section: Multiplier System and Explicit Form Of The Rational Weight M...mentioning

confidence: 99%

Neutrino mass and mixing with modular symmetry

Ding,

King

2024

Rep. Prog. Phys.

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This is a review article about neutrino mass and mixing and flavour model building strategies based on modular symmetry. After an introduction to neutrino mass and lepton mixing, we then turn to the main subject of this review, namely a pedagogical introduction to modular symmetry as a candidate for family symmetry, from the bottom-up point of view. After an informal introduction to modular symmetry, we introduce the modular group, and discuss its fixed points and residual symmetry, assuming supersymmetry throughout. We then introduce finite modular groups of level N and modular forms with integer or rational modular weights, corresponding to simple geometric groups or their double or metaplectic covers, including the most general finite modular groups and vector-valued modular forms, with detailed results for N=2,3,4,5. The interplay between modular symmetry and generalized CP symmetry is discussed, deriving CP transformations on matter multiplets and modular forms, highlighting the CP fixed points and their implications. In general, compactification of extra dimensions generally leads to a number of moduli, and modular invariance with factorizable and non-factorizable multiple moduli based on symplectic modular invariance and automorphic forms is reviewed. Modular strategies for understanding fermion mass hierarchies are discussed, including the weighton mechanism, small deviations from fixed points, and texture zeroes. Then examples of modular models are discussed based on single modulus A4 models, a minimal S′4 model of leptons (and quarks), and a multiple moduli model based on three S4 groups capable of reproducing the Littlest Seesaw model. We then extend the discussion to include Grand Unified Theories (GUTs) based on modular (flipped) SU(5) and SO(10). Finally we discuss top-down approaches, including eclectic flavour symmetry and moduli stabilisation.

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Section: Multiplier System and Explicit Form Of The Rational Weight M...mentioning

confidence: 99%

Neutrino mass and mixing with modular symmetry

Ding,

King

2024

Rep. Prog. Phys.

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show abstract

“…We would like to mention that the theory of modular forms with real weight and even complex weight are also developed [62], and then the modular group SL 2 (Z) should be extended to the universal covering groups. Some concrete examples are given in [63,64].…”

Section: Rational Weight Modular Formsmentioning

confidence: 99%

Half-integral weight modular forms and application to neutrino mass models

Liu,

Yao,

et al. 2020

Preprint

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We generalize the modular invariance approach to include the half-integral weight modular forms. Accordingly the modular group should be extended to its metaplectic covering group for consistency. We introduce the well-defined half-integral weight modular forms for congruence subgroup Γ(4N ) and show that they can be decomposed into the irreducible multiplets of finite metaplectic group Γ 4N . We construct concrete expressions of the half-integral/integral modular forms for Γ(4) up to weight 6 and arrange them into the irreducible representations of Γ 4 . We present three typical models with Γ 4 modular symmetry for neutrino masses and mixing, and the phenomenological predictions of each model are analyzed numerically.

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Half-integral weight modular forms and application to neutrino mass models

Liu

Yao

et al. 2020

Phys. Rev. D

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Modular forms of real weights and generalised Dedekind symbols

Cited by 3 publications

References 18 publications

Neutrino mass and mixing with modular symmetry

Neutrino mass and mixing with modular symmetry

Half-integral weight modular forms and application to neutrino mass models

Half-integral weight modular forms and application to neutrino mass models

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