Revisiting modular symmetry in magnetized torus and orbifold compactifications

We study the modular symmetry in T^2 and orbifold comfactifications with magnetic fluxes. There are |M| zero modes on T^2 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 (2, Z), SL (2, Z). Each wave function on T^2 with the magnetic flux M transforms under (2|M|), which is the normal subgroup of SL (2, Z). Then, |M| zero modes are representations of the quotient group {}₂|₌|^'/ (2|M|). We also study the modular symmetry on twisted and shifted orbifolds T^2/Z₍. Wave functions are decomposed into smaller representations by eigenvalues of twist and shift. They provide us with reduction of reducible representations on T^2.