Modular weights of wave functions on magnetized torus

TL;DR

The paper establishes a precise equivalence between the modular weight of wave functions on magnetized tori and their mass level, first for the T^2 case and then extended to T^{2g}. It constructs excited states using ladder operators that act simultaneously as modular-weight and mass-level raising operators, demonstrating h = n+1/2 = mass level for appropriate sectors. By embedding the analysis in both classical SL(2,R) and Siegel modular form frameworks, the work links modular transformations to the physical spectra and provides a structured method to obtain higher-genus wave functions with identical weight/mass properties. These results have potential implications for modular flavor symmetries in string-inspired models and for understanding the geometric origin of Yukawa couplings as modular forms.

Abstract

We study the origin of modular weights of wave functions in magnetized $T^{2}$ models. It is explicitly demonstrated that the modular weights of the wave functions on magnetized $T^2$ is equivalent to their mass level. We further extend this result to magnetized $T^{2g}$ models. As a result, we construct the wave functions of excited states in magnetized $T^{2g}$ models and show that their modular weights are likewise equivalent to the corresponding mass levels.