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Modular Zeros

Xiang-Gan Liu, Michael Ratz

TL;DR

The work identifies modular zeros as intrinsic gaps in the spaces of vector-valued modular forms (VVMFs), which structurally forbid certain couplings beyond conventional symmetry arguments, and connects these gaps to stringy zeros and flavor-texture phenomena.By introducing modular spin, the authors relate the weight of VVMFs to the action of the central element $\mathsf{S}^2$ and show how this underpins enhanced R-symmetries in heterotic orbifolds, with implications for the transformation properties of the superpotential and fermionic coordinates.The paper demonstrates, through concrete examples in $\Gamma'_3\cong T'$ models and orbifold setups, that the absence of certain Yukawa terms can be explained by $k_{\\mathcal{Y}}$- and representation-dependent gaps, yielding a unified picture of modular, stringy, and texture zeros.These modular zeros offer a promising framework to address the $\mu$ problem and reproduce Weinberg-type mass textures, linking the Cabibbo angle to hierarchical down-type quark masses via the Fourier structure of modular forms in the large-\Im\tau regime.

Abstract

Modular symmetries are known to be powerful and have various remarkable properties. We point out that the structure of vector-valued modular forms (VVMFs) space leads to the absence of couplings which cannot be explained in terms of the usual symmetries. These modular zeros, which correspond to gaps in spaces of VVMFs, have the power of explaining certain stringy zeros, and to explain the renowned Weinberg texture that relates the Cabibbo angle to the hierarchies of the light down and strange quarks.

Modular Zeros

TL;DR

The work identifies modular zeros as intrinsic gaps in the spaces of vector-valued modular forms (VVMFs), which structurally forbid certain couplings beyond conventional symmetry arguments, and connects these gaps to stringy zeros and flavor-texture phenomena.By introducing modular spin, the authors relate the weight of VVMFs to the action of the central element $\mathsf{S}^2$ and show how this underpins enhanced R-symmetries in heterotic orbifolds, with implications for the transformation properties of the superpotential and fermionic coordinates.The paper demonstrates, through concrete examples in $\Gamma'_3\cong T'$ models and orbifold setups, that the absence of certain Yukawa terms can be explained by $k_{\\mathcal{Y}}$- and representation-dependent gaps, yielding a unified picture of modular, stringy, and texture zeros.These modular zeros offer a promising framework to address the $\mu$ problem and reproduce Weinberg-type mass textures, linking the Cabibbo angle to hierarchical down-type quark masses via the Fourier structure of modular forms in the large-\Im\tau regime.

Abstract

Modular symmetries are known to be powerful and have various remarkable properties. We point out that the structure of vector-valued modular forms (VVMFs) space leads to the absence of couplings which cannot be explained in terms of the usual symmetries. These modular zeros, which correspond to gaps in spaces of VVMFs, have the power of explaining certain stringy zeros, and to explain the renowned Weinberg texture that relates the Cabibbo angle to the hierarchies of the light down and strange quarks.
Paper Structure (12 sections, 36 equations, 1 figure, 1 table)

This paper contains 12 sections, 36 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Modular Symmetries
  3. Modular Zeros: Gaps in the Modular Forms Space
  4. Definition of Modular Zeros
  5. An Example
  6. Modular Spin
  7. Modular Spin and Discrete R-Symmetries
  8. Stringy Zeros
  9. Modular Texture Zeros
  10. Summary
  11. The Group T' and Gaps in Its Modular Forms Space
  12. Holomorphic Zeros

Figures (1)

  • Figure 3.1: Cartoon of the $\mathds{T}^2/\mathds{Z}_{3}$ orbifold pillow.