Abstract
Modular symmetries naturally combine with traditional flavor symmetries and mathcal{CP} , giving rise to the so-called eclectic flavor symmetry. We apply this scheme to the two-dimensional ℤ2 orbifold, which is equipped with two modular symmetries SL(2, ℤ)T and SL(2, ℤ)U associated with two moduli: the Kähler modulus T and the complex structure modulus U. The resulting finite modular group is ((S3× S3) ⋊ ℤ4) × ℤ2 including mirror symmetry (that exchanges T and U) and a generalized mathcal{CP} -transformation. Together with the traditional flavor symmetry (D8× D8)/ℤ2, this leads to a huge eclectic flavor group with 4608 elements. At specific regions in moduli space we observe enhanced unified flavor symmetries with as many as 1152 elements for the tetrahedral shaped orbifold and leftlangle Trightrangle =leftlangle Urightrangle =exp left(frac{pi mathrm{i}}{3}right) . This rich eclectic structure implies interesting (modular) flavor groups for particle physics models derived form string theory.
Highlights
CP, giving rise to the so-called eclectic flavor symmetry. We apply this scheme to the twodimensional Z2 orbifold, which is equipped with two modular symmetries SL(2, Z)T and SL(2, Z)U associated with two moduli: the Kähler modulus T and the complex structure modulus U
The finite modular group of the T2/Z2 orbifold is derived in section 3 and turns out to be [144, 115], the multiplicative closure of mirror symmetry and the S3 × S3 finite modular groups arising from SL(2, Z)T and SL(2, Z)U
The full Narain space group and its outer automorphisms reveal a common origin of all discrete symmetries for strings on orbifolds, giving rise to the eclectic flavor symmetry that consists of traditional flavor, modular, CP and R-symmetries
Summary
In order to specify the two-dimensional T2/Z2 orbifold, we first define the geometrical space group S. Discrete flavor symmetries of the effective four-dimensional field theory from strings on orbifolds find their origin in the outer automorphisms of the so-called Narain space group [10, 11]. Combined with the geometrical transformations h1 and h2, which exchange orbifold fixed points pairwise, these string selection rules yield a non-Abelian flavor symmetry as follows: in the T2/Z2 orbifold, the string selection rules give rise to a Z2 × Z2 × Z2 symmetry, under which twisted matter fields transform as φ(n1,n2) −h→3 (−1)n1 φ(n1,n2). The full Narain space group and its outer automorphisms reveal a common origin of all discrete symmetries for strings on orbifolds, giving rise to the eclectic flavor symmetry that consists of traditional flavor, modular, CP and R-symmetries. We will analyze in detail the modular symmetries and their finite modular groups that arise in the T2/Z2 orbifold
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