Discrete flavor symmetries in D-brane models

TL;DR

The work addresses the origin and structure of discrete flavor symmetries in D-brane realizations of particle physics by tying them to Calabi-Yau discrete isometries and B-field transformations that leave both closed and open string backgrounds invariant. A geometric criterion is developed to identify exact versus approximate discrete flavor symmetries, applicable to toroidal, orbifold, and Calabi-Yau backgrounds, and is tested in intersecting and magnetized D-brane contexts. The authors classify the resulting non-Abelian flavor groups, notably dihedral groups like $D_4$ and their tensor products across torus factors, and show how different D-brane configurations (global and local) arrange chiral families into concrete representations, constraining Yukawa couplings beyond massive $U(1)$ D-brane remnants. Explicit Pati-Salam examples demonstrate how these flavor symmetries affect mass textures and couplings, with exactness depending on the brane content and tadpole consistency, offering a framework with potential phenomenological impact for Yukawa hierarchies in string-derived constructions.

Abstract

We study the presence of discrete flavor symmetries in D-brane models of particle physics. By analyzing the compact extra dimensions of these models one can determine when such symmetries exist both in the context of intersecting and magnetized D-brane constructions. Our approach allows to distinguish between approximate and exact discrete symmetries, and it can be applied to compactification manifolds with continuous isometries or to manifolds that only contain discrete isometries, like Calabi-Yau three-folds. We analyze in detail the class of rigid D-branes models based on a Z_2 x Z'_2 toroidal orientifold, for which the flavor symmetry group is either the dihedral group D_4 or tensor products of it. We construct explicit Pati-Salam examples in which families transform in non-Abelian representations of the flavor symmetry group, constraining Yukawa couplings beyond the effect of massive U(1) D-brane symmetries.

Figures (6)

• Figure 1: Branes with wrapping numbers $(n_a^i, m_a^i) = (1,1)\otimes(1,-1)\otimes(2,-1)$ (red) and $(n_b^i, m_b^i) = (-1,-3)\otimes(-1,-1)\otimes(2,1)$ (green) with $i)$ generic values of the position moduli and $ii)$ stuck at the fixed points in the $\mathbb{Z}_2 \times \mathbb{Z}_2'$ orbifold. The number of chiral families for $i)$ is $I_{ab}^{{\bf T}^6} = (-2)\times (-2)\times 4 = 16$ and for $ii)$ is $I_{ab} = 4$.
• Figure 2: Space inversion in one of the tori and D-branes with wrapping numbers $(1,3)$ and $(3,1)$. Some of the intersection points are invariant while others get exchanged. The even combination of points are $\{0,\,1+7,\,2+6,\,5+3,\,4\}$ and the odd ones $\{1-7,\, 2-6,\,5-3\}$. Notice that the number of even and odd points is in agreement with (\ref{['evenodd']}).
• Figure 3: ${\bf T}^2/\mathbb Z_2$ with a) no branes b) one brane on the cycle (1,3) c) two branes on (1,3) and (1,1) d) three branes on (1,3), (1,1) and (1,-1).
• Figure 4: Translation $z\rightarrow z+\frac{1+\tau}{2}$ in a square torus. This is the generator of the shift symmetry in the intersecting brane picture.
• Figure 5: Branes $a_1$ (red) and $a_2$ (blue) with labels for the different intersection points.