Γ₂ ≃ S₃ Modular Flavor Symmetry

• Γ₂ ≃ S₃ modular flavor symmetry is defined as the quotient of PSL(2,ℤ) by its level-2 subgroup, resulting in a group isomorphic to the permutation group S₃.
• It utilizes modular forms—such as the weight-2 doublet Y^(2)—to build predictive fermion mass textures without additional flavon fields.
• The symmetry constrains Yukawa couplings and mass matrices, leading to precise predictions for neutrino mass orderings and CP violation, testable in future experiments.

The finite modular group $\Gamma_2\simeq S_3$ modular flavor symmetry arises by quotienting the full modular group $PSL(2,\mathbb{Z})$ by its level-2 principal congruence subgroup. The resulting group of order 6 is isomorphic to the symmetric group $S_3$, the permutation group of three objects. This symmetry acts on modular forms—a class of automorphic functions—whose properties are then used to constrain the flavor structure of fermion masses and mixings in both bottom-up and top-down model building. The $\Gamma_2 \simeq S_3$ modular symmetry provides a minimal, predictive framework in which hierarchical mass textures and realistic mixing patterns arise without recourse to flavon fields, as the entire structure is encoded in the dependence on the complex modulus $\tau$ and a finite ring of modular forms.

1. Algebraic Structure of $\Gamma_2\simeq S_3$

The group $PSL(2,\mathbb{Z})$ is generated by $S:\tau \mapsto -1/\tau$ and $T:\tau \mapsto \tau+1$, with $S^2=(ST)^3=I$ as defining relations. The level-2 principal congruence subgroup $PSL(2,\mathbb{Z})$0 consists of matrices $PSL(2,\mathbb{Z})$1 in $PSL(2,\mathbb{Z})$2, and the discrete quotient

$PSL(2,\mathbb{Z})$3

has order 6 and can be presented as

$PSL(2,\mathbb{Z})$4

In the standard 2-dimensional representation, the generators act as

$PSL(2,\mathbb{Z})$5

satisfying $PSL(2,\mathbb{Z})$6. There are three irreducible representations: the trivial singlet $PSL(2,\mathbb{Z})$7, the sign singlet $PSL(2,\mathbb{Z})$8, and the doublet $PSL(2,\mathbb{Z})$9.

2. Modular Forms and Their Properties

The lowest weight modular forms for $S_3$0 are two holomorphic weight-2 forms $S_3$1, transforming as the $S_3$2 doublet: \begin{align*} Y_1(\tau) &= \frac{1}{2}\left( \frac{\eta'(\tau/2)}{\eta(\tau/2)} + \frac{\eta'((\tau+1)/2)}{\eta((\tau+1)/2)} - 8 \frac{\eta'(2\tau)}{\eta(2\tau)} \right),\ Y_2(\tau) &= \frac{\sqrt{3}}{2}\left( \frac{\eta'(\tau/2)}{\eta(\tau/2)} - \frac{\eta'((\tau+1)/2)}{\eta((\tau+1)/2)} \right), \end{align*} with $S_3$3-expansions

$S_3$4

Higher-weight modular forms arise from $S_3$5-invariant tensor contractions of the weight-2 doublet: $S_3$6 Further products yield higher-weight singlets, doublets, and pseudo-singlets as detailed in (Okada et al., 2019, Belfkir et al., 2024).

3. Field Assignments and Modular Weights

Flavored matter multiplets (e.g., left-handed leptons $S_3$7, right-handed charged leptons $S_3$8, right-handed neutrinos, and Higgs doublets) are assigned to $S_3$9 irreducible representations and modular weights. Assignments may vary by model:

• In MSSM-like models, doublets $\Gamma_2 \simeq S_3$0, $\Gamma_2 \simeq S_3$1 are often assigned to $\Gamma_2 \simeq S_3$2, with one field (typically $\Gamma_2 \simeq S_3$3 or $\Gamma_2 \simeq S_3$4) as $\Gamma_2 \simeq S_3$5 or $\Gamma_2 \simeq S_3$6 (Behera et al., 17 Apr 2025, Meloni et al., 2023).
• Higgs fields are $\Gamma_2 \simeq S_3$7 singlets with modular weight $\Gamma_2 \simeq S_3$8.
• Dirac and Majorana mass terms for neutrinos employ modular forms according to allowed $\Gamma_2 \simeq S_3$9 tensor combinations.

The modular invariance of the superpotential requires all coupling terms to be $\tau$0 singlets and have total modular weight zero. This results in mass matrices constructed directly from modular forms $\tau$1 evaluated at the fixed modulus $\tau$2.

4. Phenomenological Consequences and Predictivity

The $\tau$3 modular flavor framework restricts Yukawa couplings and mass textures to be determined entirely by modular forms of $\tau$4 without the need for additional flavon fields. Benchmark models yield predictive patterns:

• Charged lepton and neutrino mass matrices are explicitly constructed from modular forms and $\tau$5 tensor products, with free parameters corresponding to a small set of complex structure modulus $\tau$6 and a limited number of order-one coupling constants and heavy mass scales (Okada et al., 2019, Meloni et al., 2023, Belfkir et al., 2024).
• Minimality: The number of free dimensionless parameters is reduced (9–12 in most models).
• Mass orderings: Depending on field assignment and mechanism (e.g., type-I seesaw, radiative seesaw, inverse seesaw), both normal and inverted orderings can be realized. Minimal models with two right-handed sterile neutrinos and doublet assignments tend to predict inverted ordering, with one massless neutrino and $\tau$7 in the $\tau$8–$\tau$9 meV range (Tavartkiladze, 31 Dec 2025, Behera et al., 17 Apr 2025).
• CP violation arises from the imaginary part of $\Gamma_2\simeq S_3$0 and is tightly linked to the fitted value of the modulus. Dirac and Majorana phases are strongly correlated and highly constrained in this framework.
• Predictive observables: Precise predictions are made for
  • sum of neutrino masses $\Gamma_2\simeq S_3$1,
  • effective 0$\Gamma_2\simeq S_3$2 mass $\Gamma_2\simeq S_3$3,
  • beta decay endpoint mass $\Gamma_2\simeq S_3$4,
  • mixing angles $\Gamma_2\simeq S_3$5, $\Gamma_2\simeq S_3$6, $\Gamma_2\simeq S_3$7,
  • lepton-flavor-violating decays,
  • all of which are testable in next-generation experiments (Behera et al., 17 Apr 2025, Tavartkiladze, 31 Dec 2025, Meloni et al., 2023, Belfkir et al., 2024).

5. Geometry, Residual Symmetries, and Stabilizers

The modular group acts on the complex modulus $\Gamma_2\simeq S_3$8 via fractional linear transformations. Fixed points (stabilizers) of group elements correspond to preserved cyclic subgroups, leading to residual discrete flavor symmetries at specific $\Gamma_2\simeq S_3$9. For $PSL(2,\mathbb{Z})$0, these include:

• $PSL(2,\mathbb{Z})$1 (cusp): preserves the $PSL(2,\mathbb{Z})$2-generated $PSL(2,\mathbb{Z})$3,
• $PSL(2,\mathbb{Z})$4: preserves $PSL(2,\mathbb{Z})$5; even-weight modular forms at this point are split into $PSL(2,\mathbb{Z})$6-invariant and $PSL(2,\mathbb{Z})$7-odd multiplets,
• $PSL(2,\mathbb{Z})$8: preserves $PSL(2,\mathbb{Z})$9.

At these stabilizers, modular forms align, enforcing texture zeros and rank-deficient blocks in mass matrices, enabling fully predictive mixing patterns (Varzielas et al., 2020, Tavartkiladze, 31 Dec 2025).

Element $S:\tau \mapsto -1/\tau$0	Order	Stabilizer $S:\tau \mapsto -1/\tau$1
$S:\tau \mapsto -1/\tau$2	2	$S:\tau \mapsto -1/\tau$3, $S:\tau \mapsto -1/\tau$4
$S:\tau \mapsto -1/\tau$5	2	$S:\tau \mapsto -1/\tau$6, $S:\tau \mapsto -1/\tau$7
$S:\tau \mapsto -1/\tau$8	3	$S:\tau \mapsto -1/\tau$9, $T:\tau \mapsto \tau+1$0

This structure allows precise control over the breaking pattern of flavor symmetries.

6. Embedding in String Theory and Top-Down Origin

Modular $T:\tau \mapsto \tau+1$1 emerges naturally from string compactifications, particularly from the automorphism group of Narain $T:\tau \mapsto \tau+1$2 tori in the presence of orbifold actions. In $T:\tau \mapsto \tau+1$3 orbifolds, $T:\tau \mapsto \tau+1$4 is the remnant of $T:\tau \mapsto \tau+1$5 acting trivially modulo 2 on the compactification lattice (Nilles et al., 2020, Kobayashi et al., 2024). The flavor group can be further enlarged by combining with generalized CP and R-symmetries as automorphisms or as outer automorphisms, yielding an eclectic flavor symmetry group structure.

Localized zero-modes at orbifold fixed points, under a certain ansatz on T-phases, naturally furnish $T:\tau \mapsto \tau+1$6 singlets and doublets, and their wavefunctions transform according to the irreducible representations. Yukawa couplings in such orbifold-based constructions are constrained to be modular forms of even weight and proper $T:\tau \mapsto \tau+1$7 covariant tensors (Kobayashi et al., 2024).

7. Extensions and Global Implications

Modular $T:\tau \mapsto \tau+1$8 symmetry applies not only to lepton flavor but, using appropriate assignments, to the full flavor structure of the Standard Model, including quarks. Grand-unified (e.g., Pati–Salam) models have been successfully constructed, fitting both quark and lepton observables with a minimal set of parameters (Belfkir et al., 2024). The predictive power, minimal parameter counting, and anomaly-freedom (due to $T:\tau \mapsto \tau+1$9) make $S^2=(ST)^3=I$0 an appealing symmetry in bottom-up and top-down flavor model-building, with direct testability via upcoming neutrino oscillation, $S^2=(ST)^3=I$1, and cosmological measurements.

• Modular $S^2=(ST)^3=I$2 symmetric radiative seesaw model: (Okada et al., 2019)
• Phenomenology of Inverse Seesaw Using $S^2=(ST)^3=I$3 Modular Symmetry: (Behera et al., 17 Apr 2025)
• A simplest modular $S^2=(ST)^3=I$4 model for leptons: (Meloni et al., 2023)
• Symmetries and stabilisers in modular invariant flavour models: (Varzielas et al., 2020)
• Minimal Modular Flavor Symmetry and Lepton Textures Near Fixed Points: (Tavartkiladze, 31 Dec 2025)
• Fermion Masses and Mixing in Pati-Salam Unification with $S^2=(ST)^3=I$5 Modular Symmetry: (Belfkir et al., 2024)
• Eclectic flavor scheme from ten-dimensional string theory -- II. Detailed technical analysis: (Nilles et al., 2020)
• Modular symmetry of localized modes: (Kobayashi et al., 2024)