Exploring types I and IIA effective actions through T-duality

Abstract: It is well-established that compactifying type I string theory on a circle ( S^{(1)} ) transforms the theory under T-duality into type I' theory, the compactification of type IIA string theory on the orbifold ( \tilde{S}^{(1)}/\mathbb{Z}_2 ), where the ( \mathbb{Z}_2 ) action combines worldsheet parity with spacetime reflection along the dual circle ( \tilde{S}^{(1)} ). We propose that, upon compactification, the untwisted (twisted) sector of the type I effective action should map under the Buscher rules to the untwisted (twisted) sector of the type I' effective action. This T-duality constraint offers significant insight into the determination of bosonic couplings in the effective action of type IIA theories, specifically those that remain after orbifold reduction, as well as in the untwisted sector of the type I effective action. However, its scope is limited and insufficient to fully determine the couplings within the twisted sectors of type I and type I' theories. Within this framework, we demonstrate that the leading 2-derivative couplings in untwisted sector of type I and the 2-derivative couplings in type IIA theory are uniquely determined, except for the Chern-Simons term in type IIA, which is absent in the orbifold reduction.