Modular origin of mass hierarchy: Froggatt-Nielsen like mechanism

We study Froggatt-Nielsen (FN) like flavor models with modular symmetry. The FN mechanism is a convincing solution to the flavor puzzle in the quark sector. The FN mechanism requires an extra U(1) gauge symmetry which is broken at high energies. Alternatively, in the framework of modular symmetry the modular weights can play the role of the FN charges of the extra U(1) symmetry. Based on the FN-like mechanism with modular symmetry we present new flavor models for the quark sector. Assuming that the three generations have a common representation under the modular symmetry, our models simply reproduce the FN-like Yukawa matrices. We also show that the realistic mass hierarchy and mixing angles, which are related to each other through the modular parameters and a scalar vev, can be realized in models with several finite modular groups (and their double covering groups) without unnatural hierarchical parameters.

Positive energy representations of Sobolev diffeomorphism groups of the circle

We show that any positive energy projective unitary representation of $$\mathrm{Diff}_+(S^1)$$ Diff + ( S 1 ) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms $$\mathcal {D}^s(S^1)$$ D s ( S 1 ) for any real $$s>3$$ s > 3 , and in particular to $$C^k$$ C k -diffeomorphisms $$\mathrm{Diff}_+^k(S^1)$$ Diff + k ( S 1 ) with $$k\ge 4$$ k ≥ 4 . A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on $$S^1$$ S 1 is covariant with respect to $$\mathcal {D}^s(S^1)$$ D s ( S 1 ) , $$s > 3$$ s > 3 . Moreover every direct sum of irreducible representations of a conformal net is also $$\mathcal {D}^s(S^1)$$ D s ( S 1 ) -covariant.

Modular symmetry by orbifolding magnetized T2 × T2: realization of double cover of ΓN

We study the modular symmetry of zero-modes on $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 and orbifold compactifications with magnetic fluxes, M1, M2, where modulus parameters are identified. This identification breaks the modular symmetry of $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 , SL(2, ℤ)1× SL(2, ℤ)2 to SL(2, ℤ) ≡ Γ. Each of the wavefunctions on $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 and orbifolds behaves as the modular forms of weight 1 for the principal congruence subgroup Γ(N), N being 2 times the least common multiple of M1 and M2. Then, zero-modes transform each other under the modular symmetry as multiplets of double covering groups of ΓN such as the double cover of S4.