Modular weights of wave functions on magnetized torus

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.

💡 Research Summary

The paper investigates the origin of modular weights of wave functions in magnetized torus compactifications, focusing first on the two‑dimensional torus T² and then extending the analysis to higher‑dimensional tori T^{2g}. The authors begin by reviewing the geometry of T², introducing the complex coordinate z, the complex structure modulus τ, and the SL(2,ℤ) modular symmetry that acts on (z,τ). They then discuss periodic functions associated with modular forms, defining a “periodic function” Y(τ,θ) that incorporates the automorphy factor j_h(g,τ₀) and satisfies an SL(2,ℝ) algebra generated by operators E, F, and H. In this algebra H measures the modular weight h, while E and F raise or lower the weight by two units.

Next, the authors introduce a constant U(1) magnetic flux M∈ℤ on T², together with possible Scherk–Schwarz phases (α_S,β_S). The covariant derivatives D_z and D_{\bar z} lead to an OSp(1|2) super‑Virasoro algebra involving the operators g_{±1/2} and ℓ_n (n=0,±1). The key observation is that the operator ℓ₀ coincides with the mass operator: its eigenvalue equals the mass level of scalar and spinor wave functions (m² ∝ ℓ₀). The ladder operators ℓ_{±1} and g_{±1/2} raise the mass level by two and one units, respectively.

In Section 2.4 the authors construct modified ladder operators \hat{E}, \hat{F}, and \hat{H} that act on the full wave functions Φ_{j}^{(n)} (including the dependence on the auxiliary angle θ). They prove that \hat{H}Φ_{j}^{(n)} = (n+½)Φ_{j}^{(n)}, showing that the modular weight h = n+½ is exactly the same as the mass level n obtained from ℓ₀. For spinor wave functions the relation becomes h_{±}=h∓½, consistent with the scalar result. Thus the modular weight of any wave function on a magnetized T² is precisely its mass level.

The analysis is then generalized to a magnetized T^{2g} (g≥1). The authors replace the complex modulus τ by a Siegel matrix Ω and the modular group by Sp(2g,ℤ). Siegel modular forms and their periodic functions are introduced, and a set of g independent SL(2,ℝ) sub‑algebras is identified, each with its own ladder operators \mathcal{g}{±1/2}^{(i)}, \mathcal{ℓ}{±1}^{(i)}, \mathcal{ℓ}_0^{(i)}. The total mass level is the sum n=∑_i n_i, and the modular weight becomes h = n + g/2. Modified operators \hat{E}^{(i)}, \hat{F}^{(i)}, \hat{H}^{(i)} satisfy the same algebraic relations, and \hat{H}^{(i)} acting on the excited wave functions yields the corresponding weight contribution (n_i+½). Consequently, the modular weight of any excited state on a magnetized T^{2g} coincides with its mass level, confirming the result found for T² in a higher‑dimensional setting.

The paper concludes that modular symmetry does not merely constrain the transformation properties of wave functions; it directly encodes their mass spectra. This insight provides a clear bridge between modular flavor models—where Yukawa couplings are identified with modular forms of specific weight—and the underlying higher‑dimensional field theory, where the same weight is dictated by the excitation level of the wave function. The authors also discuss potential applications to model building, suggesting that the explicit construction of excited wave functions and their modular properties can be used to generate realistic fermion mass hierarchies and mixing patterns. Appendices contain detailed proofs of the equivalence between ℓ_{-1} and \hat{E} in both T² and T^{2g} cases, as well as a brief review of two‑dimensional conformal field theory and its relation to modular forms. The work opens the way for systematic inclusion of higher‑excited states in modular flavor constructions and for exploring similar correspondences in orbifold and flux‑brane backgrounds.