The effective action of type II Calabi-Yau orientifolds

TL;DR

This paper reviews the calculation of the N=1 effective action for type IIA and IIB Calabi-Yau orientifolds with fluxes, relating geometric data to superpotentials and exploring dual formulations and string theory origins.

Contribution

It introduces a direct relation between the chiral description and Hitchin's generalized geometry, and connects type IIA/IIB orientifolds to M-theory and F-theory frameworks.

Findings

Derived the Kähler potential, gauge kinetic functions, and flux superpotential from geometrical data.

Showed the relation of chiral description to Hitchin's generalized geometry.

Connected orientifolds to M-theory G_2 manifolds and F-theory fourfolds.

Abstract

This article first reviews the calculation of the N = 1 effective action for generic type IIA and type IIB Calabi-Yau orientifolds in the presence of background fluxes by using a Kaluza-Klein reduction. The Kahler potential, the gauge kinetic functions and the flux-induced superpotential are determined in terms of geometrical data of the Calabi-Yau orientifold and the background fluxes. As a new result, it is shown that the chiral description directly relates to Hitchin's generalized geometry encoded by special odd and even forms on a threefold, whereas a dual formulation with several linear multiplets makes contact to the underlying N = 2 special geometry. In type IIB setups, the flux-potentials can be expressed in terms of superpotentials, D-terms and, generically, a massive linear multiplet. The type IIA superpotential depends on all geometric moduli of the theory. It is reviewed, how type IIA orientifolds arise as a special limit of M-theory compactified on specific G_2 manifolds by matching the effective actions. In a similar spirit type IIB orientifolds are shown to descend from F-theory on a specific class of Calabi-Yau fourfolds. In addition, mirror symmetry for Calabi-Yau orientifolds is briefly discussed and it is shown that the N = 1 chiral coordinates linearize the appropriate instanton actions.