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String theory leads to a flavor scheme where modular symmetries play a crucial role. Together with the traditional flavor symmetries they combine to an eclectic flavor group, which we determine via outer automorphisms of the Narain space group. Unbroken flavor symmetries are subgroups of this eclectic group and their size depends on the location in moduli space. This mechanism of local flavor unification allows a different flavor structure for different sectors of the theory (such as quarks and leptons) and also explains the spontaneous breakdown of flavor- and CP -symmetries (via a motion in moduli space). We derive the modular groups, including CP and R -symmetries, for different sub-sectors of six-dimensional string compactifications and determine the general properties of the allowed flavor groups from this top-down perspective. It leads to a very predictive flavor scheme that should be confronted with the variety of existing bottom-up constructions of flavor symmetry in order to clarify which of them could have a consistent top-down completion. [ABSTRACT FROM AUTHOR]

In a consistent top-down approach based on orbifold compactifications, modular and traditional flavor symmetries combine nontrivially to the so-called eclectic flavor symmetry. We extend this scheme from two extra dimensions, discussed previously, to the six extra dimensions of string theory. By doing so, new insights on the nature of CP and its spontaneous breaking emerge. Moreover, we identify a new interpretation of R -symmetries as unbroken remnants from modular symmetries that are associated with geometrically stabilized complex structure moduli. Hence, all symmetries (i.e. modular, traditional flavor, CP and R) share a common origin in string theory: on a technical level, they are given by outer automorphisms of the Narain space group. The eclectic top-down approach leads to a very restrictive scheme with high predictive power. It remains a challenge to connect this with existing bottom-up constructions of modular flavor symmetry. [ABSTRACT FROM AUTHOR]

A top-down approach to the flavor problem motivated from string theory leads to the concept of eclectic flavor groups that combine traditional and modular flavor symmetries. To make contact with models constructed in the bottom-up approach, we analyze a specific example based on the eclectic flavor group Ω (1) (a nontrivial combination of the traditional flavor group Δ (54) and the finite modular group T ′) in order to extract general lessons from the eclectic scheme. We observe that this scheme is highly predictive since it severely restricts the possible group representations and modular weights of matter fields. Thereby, it controls the structure of the Kähler potential and the superpotential, which we discuss explicitly. In particular, both Kähler potential and superpotential are shown to transform nontrivially, but combine to an invariant action. Finally, we find that discrete R -symmetries are intrinsic to eclectic flavor groups. [ABSTRACT FROM AUTHOR]

Flavor symmetry plays a crucial role in the standard model of particle physics but its origin is still unknown. We develop a new method (based on outer automorphisms of the Narain space group) to determine flavor symmetries within compactified string theory. A picture emerges where traditional (discrete) flavor symmetries, CP -like symmetries and modular symmetries (like T -duality) of string theory combine to unified flavor symmetries. The groups depend on the geometry of compact space and the geographical location of fields in the extra dimensions. We observe a phenomenon of "local flavor groups" with potentially different flavor symmetries for the various sectors of quarks and leptons. This should allow interesting connections to existing bottom-up attempts in flavor model building. [ABSTRACT FROM AUTHOR]

Abstract We identify a natural way to embed CP symmetry and its violation in string theory. The CP symmetry of the low energy effective theory is broken by the presence of heavy string modes. CP violation is the result of an interplay of CP and flavor symmetry. CP violating decays of the heavy modes could originate a cosmological matter-antimatter asymmetry. [ABSTRACT FROM AUTHOR]

We derive the potential modular symmetries of heterotic string theory. For a toroidal compactification with Wilson line modulus, we obtain the Siegel modular group Sp (4 , Z) that includes the modular symmetries SL (2 , Z) T and SL (2 , Z) U (of the "geometric" moduli T and U) as well as mirror symmetry. In addition, string theory provides a candidate for a CP -like symmetry that enhances the Siegel modular group to GSp (4 , Z). [ABSTRACT FROM AUTHOR]

Modular transformations of string theory (including the well-known stringy dualities) play a crucial role in the discussion of discrete flavor symmetries in the Standard Model. They are at the origin of CP -transformations and provide a unification of CP with traditional flavor symmetries. Here, we present a novel, fully systematic method to reliably compute the unified flavor symmetry of the low-energy effective theory, including enhancements from the modular transformations of string theory. The unified flavor group is non-universal in moduli space and exhibits the phenomenon of "Local Flavor Unification" where different sectors of the theory can be subject to different flavor structures. [ABSTRACT FROM AUTHOR]