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 vertical bar M vertical bar 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 Gamma equivalent to SL(2, Z), (Gamma) over tilde equivalent to o2, (SL) over tildeZ). Each wave function on T-2 with the magnetic flux M transforms under (Gamma) over tilde (2 vertical bar M vertical bar), which is the normal subgroup of (SL) over tilde (2, Z). Then, vertical bar M vertical bar zero modes are representations of the quotient group (Gamma) over tilde (2 vertical bar M vertical bar) equivalent to (Gamma) over tilde/(Gamma) over tilde2 vertical bar M vertical bar). 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.