Abstract
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).
Highlights
Modular symmetries might play an important role for a description of the flavor structure in particle physics [1]
In the top-down discussion, this included i) the T2/Z3 orbifold with Kahler modulus T [3,4,5] subject to the modular group SL(2, Z)T and ii) the T2/Z2 orbifold with T and U moduli with a corresponding modular group SL(2, Z)T × SL(2, Z)U combined with a mirror symmetry that interchanges T and U [6]
Our results are based on the observation that string theory includes more moduli beyond the T - and U -moduli in form of Wilson lines connected to gauge symmetries in extra dimensions
Summary
Modular symmetries might play an important role for a description of the flavor structure in particle physics [1]. It is convenient to define the so-called generalized metric of the Narain lattice in terms of the metric G := eTe (of the D-dimensional torus spanned by the geometrical vielbein e), the anti-symmetric B-field B and the Wilson lines A,. We discuss various subgroups and generators of Oη(2, 3, Z) ⊂ Oη(2, 2 + 16, Z), derive their actions on the moduli (T, U, Z) and compare them to the Siegel modular group. Due to the 16 extra left-moving degrees of freedom of the heterotic string, the general modular group Oη(2, 2 + 16, Z) has additional elements called “Wilson line shifts”. There exist further Oη(2, 2 + 16, Z) transformations not present in Sp(4, Z): One can perform Weyl reflections in the 16-dimensional lattice of E8 × E8 (or Spin(32)/Z2), see for example MW(∆W ) in ref. [12]
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