Abstract
We classify discrete modular symmetries in the effective action of Type IIB string on toroidal orientifolds with three-form fluxes, emphasizing on $T^6/\mathbb{Z}_2$ and $T^6/(\mathbb{Z}_2\times \mathbb{Z}_2^\prime)$ orientifold backgrounds. On the three-form flux background, the modular group is spontaneously broken down to its congruence subgroup whose pattern is severely constrained by a quantization of fluxes and tadpole cancellation conditions. We explicitly demonstrate that the congruence subgroups appearing in the effective action arise on magnetized D-branes wrapping certain cycles of tori.
Highlights
In the low-energy effective action of higher-dimensional theory, moduli are ubiquitous fields and they have certain symmetries, originating from the higher-dimensional gauge and/or Lorentz symmetries
The discrete modular symmetry arises in the flat direction of complex structure moduli whose moduli space has a congruence subgroup rather than SLð2; ZÞ
We argued that such a discrete modular group plays an important role of enlarging the axionic field range discussed in the context of swampland conjecture [64], and the flavor symmetry of quarks and leptons
Summary
In the low-energy effective action of higher-dimensional theory, moduli are ubiquitous fields and they have certain symmetries, originating from the higher-dimensional gauge and/or Lorentz symmetries. Number of zero modes appear on torus and orbifold compactifications with magnetic fluxes These zero modes transform under the modular group. We study a simple type IIB flux compactification on toroidal orientifolds with and without D-branes, where the modular symmetries associated with tori are partially broken into subgroups by the three-form fluxes. Subgroups of the modular group ⊗3i1⁄41 SLð2; ZÞi emerge in the flat directions of moduli fields such that the modular transformation is viable in the low-energy effective action. We briefly review the modular symmetry in four-dimensional (4D) effective action of type IIB string on T6=Z2 toroidal orientifold with three-form fluxes. After demonstrating the breaking mechanism of the modular symmetry discussed in [64], we extend their analysis and classify patterns of congruence subgroups in the lowenergy effective action
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