Incomplete zero-mode multiplets under Scherk-Schwarz phases in magnetized compactifications violate modular symmetry as a group but retain control over couplings via full modular forms.
Modular symmetry and non-Abelian discrete flavor symmetries in string compactification
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abstract
We study the modular symmetry in magnetized D-brane models on $T^2$. Non-Abelian flavor symmetry $D_4$ in the model with magnetic flux $M=2$ (in a certain unit) is a subgroup of the modular symmetry. We also study the modular symmetry in heterotic orbifold models. The $T^2/Z_4$ orbifold model has the same modular symmetry as the magnetized brane model with $M=2$, and its flavor symmetry $D_4$ is a subgroup of the modular symmetry.
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Blow-up of magnetized T²/Z_N preserves total magnetic flux, total curvature, and effective flux on connecting lines, while the number of localized modes at each singularity increases by one per mass level increment.
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More about modular symmetries and non-invertible properties in magnetized compactifications
Incomplete zero-mode multiplets under Scherk-Schwarz phases in magnetized compactifications violate modular symmetry as a group but retain control over couplings via full modular forms.
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Massive modes on magnetized blow-up manifold of $T^2/\mathbb{Z}_N$
Blow-up of magnetized T²/Z_N preserves total magnetic flux, total curvature, and effective flux on connecting lines, while the number of localized modes at each singularity increases by one per mass level increment.