Revisiting modular symmetry in magnetized torus and orbifold compactifications

We study the modular symmetry in T² and orbifold comfactifications with magnetic fluxes. There are $|M|$ zero modes on T² with the magnetic flux M. Their wave functions as well as massive modes behave as modular forms of weight $1/2$ and represent the double covering group of Γ≡SL(2,Z), Γ≡SL(2,Z). Each wave function on T² with the magnetic flux M transforms under Γ(2|M|), which is the normal subgroup of SL(2,Z). Then, $|M|$ zero modes are representations of the quotient group ${{{}}{{{Γ}}}}2|M|^{{'}}{≡}{{}}{{{Γ}}}/{{}}{{{Γ}}}(2|M|)$. We also study the modular symmetry on twisted and shifted orbifolds ${T}²/{Z}N$. Wave functions are decomposed into smaller representations by eigenvalues of twist and shift. They provide us with reduction of reducible representations on ${T}²$.