In modular invariant models of flavor, observables must be modular invariant. The observables discussed so far in the literature are functions of the modulus $τ$ and its conjugate, $\barτ$. We point out that certain combinations of observables depend only on $τ$, i.e. are meromorphic, and in some cases even holomorphic functions of $τ$. These functions, which we dub ``invariants'' in this Letter, are highly constrained, renormalization group invariant, and allow us to derive many of the models' features without the need for extensive parameter scans. We illustrate the robustness of these invariants in two existing models in the literature based on modular symmetries, $Γ_{3}$ and $Γ_{5}$. We find that, in some cases, the invariants give rise to robust relations among physical observables that are independent of $τ$. Furthermore, there are instances where additional symmetries exist among the invariants. These symmetries are relevant phenomenologically and may provide a dynamical way to realize symmetries of mass matrices.
Minimal lepton models with non-holomorphic modular $A_{4}$ symmetry
Northwest University（西北大学）；Shaanxi Key Laboratory for Theoretical Physics Frontiers（陕西省理论物理前沿重点实验室）；NSFC-SPTP Peng Huanwu Center for Fundamental Theory（国家自然科学基金委-中子科学合作彭桓武基础理论中心）
Complete Operator Basis for the modular invariant SMEFT
Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院，中国科学院大学）；University of Chinese Academy of Sciences（中国科学院大学）；Chinese Academy of Sciences（中国科学院）；International Centre for Theoretical Physics Asia-Pacific（亚太理论物理中心）
