Flavor symmetries from modular subgroups in magnetized compactifications

A bstract We study the flavor structures of zero-modes, which are originated from the modular symmetry on $$ {T}_1^2\times {T}_2^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> and its orbifold with magnetic fluxes. We introduce the constraint on the moduli parameters by τ 2 = Nτ 1 , where τ i denotes the complex structure moduli on $$ {T}_i^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mi>i</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> . Such a constraint can be derived from the moduli stabilization. The modular symmetry of $$ {T}_1^2\times {T}_2^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> is $$ \textrm{SL}{\left(2,\mathbb{Z}\right)}_{\tau_1}\times \textrm{SL}{\left(2,\mathbb{Z}\right)}_{\tau_2}\subset \textrm{Sp}\left(4,\mathbb{Z}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>SL</mml:mi> <mml:msub> <mml:mfenced> <mml:mn>2</mml:mn> <mml:mi>ℤ</mml:mi> </mml:mfenced> <mml:msub> <mml:mi>τ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:msub> <mml:mo>×</mml:mo> <mml:mi>SL</mml:mi> <mml:msub> <mml:mfenced> <mml:mn>2</mml:mn> <mml:mi>ℤ</mml:mi> </mml:mfenced> <mml:msub> <mml:mi>τ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi>Sp</mml:mi> <mml:mfenced> <mml:mn>4</mml:mn> <mml:mi>ℤ</mml:mi> </mml:mfenced> </mml:math> and it is broken to Γ 0 ( N ) × Γ 0 ( N ) by the moduli constraint. The wave functions represent their covering groups. We obtain various flavor groups in these models.