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.
Highlights
Modular symmetryThe modular group Γ is made of elements acting on the complex modulus τ (Im(τ ) > 0)
JHEP11(2020)085 fields as considered in the context of multiple modular symmetries [10]
We present the results in figures showing the stabilisers in the domains of the respective modular symmetries and we list them in tables displaying each group element and respective stabilisers
Summary
The modular group Γ is made of elements acting on the complex modulus τ (Im(τ ) > 0). The modular group Γ is isomorphic to the projective spacial linear group PSL(2, Z) = SL(2, Z)/Z2. It has two generators, Sτ and Tτ , satisfying Sτ2 = (Sτ Tτ )3 = e. D = 1 (mod N ) and b, c = 0 (mod N ) with N = 2, 3, 4, 5, · · · , we obtain a subset of Γ which is an infinite group and labelled as Γ(N ), Γ(N ) =. The quotient group Γ/Γ(N ) is labelled as ΓN It is finite and called finite modular group. The finite modular group ΓN for N ≤ 5 can be obtained by imposing conditions.
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