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.
$Z_N$ twisted orbifold models with magnetic flux
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abstract
We propose new backgrounds of extra dimensions to lead to four-dimensional chiral models with three generations of matter fermions, that is $T^2/Z_N$ twisted orbifolds with magnetic fluxes. We consider gauge theory on six-dimensional space-time, which contains the $T^2/Z_N$ orbifold with magnetic flux, Scherk-Schwarz phases and Wilson line phases. We classify all the possible Scherk-Schwarz and Wilson line phases on $T^2/Z_N$ orbifolds with magnetic fluxes. The behavior of zero modes is studied. We derive the number of zero modes for each eigenvalue of the $Z_N$ twist, showing explicitly examples of wave functions. We also investigate Kaluza-Klein mode functions and mass spectra.
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2026 3verdicts
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Magnetized SYM on (T^2)^3/(Z_2 x Z'_2) with 7-branes, fluxes, and Wilson lines produces MSSM chiral fields and semi-realistic hierarchical Yukawas for quarks and leptons.
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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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.