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We point out that kinetic and St\"uckelberg mixings that are generically present in the low energy effective action of axions can significantly widen the window of axion decay constants.
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Note however that string theory allows for the existence of U ( 1 ) U(1) -type D-brane instantons. Such instantons can contribute to the effective superpotential when a Euclidean D-brane wraps the same internal cycle as the D-brane, even if the D-brane supports a U ( 1 ) U(1) gauge group
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For Type IIB string theory, we considered the orientifold projection inverting the sign of the Calabi-Yau three-form. For an orientifold projection leaving the Calabi-Yau three-form completely invariant, the Hodge-numbers for the axions c a c^{a} and ρ α \rho^{\alpha} have to be swapped
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