Fluxes in Heterotic and Type II String Compactifications

We present a review of heterotic-type I string duality. In particular, we discuss the effective field theory of six- and four-dimensional compactifications with N>1 supersymmetries. We then describe various duality tests by comparing gauge couplings, N=2 prepotentials, as well as higher-derivative F-terms. Based on invited lectures delivered at: 33rd Karpacz Winter School of Theoretical Physics ``Duality, Strings and Fields,'' Przesieka, Poland, 13 - 22 February 1997; Trieste...

We investigate a simple class of type II string compactifications which incorporate nongeometric "fluxes" in addition to "geometric flux" and the usual H-field and R-R fluxes. These compactifications are nongeometric analogues of the twisted torus. We develop T-duality rules for NS-NS geometric and nongeometric fluxes, which we use to construct a superpotential for the dimensionally reduced four-dimensional theory. The resulting structure is invariant under T-duality, so that...

We present a new class of dualities relating non-geometric Calabi-Yau compactifications of type II string theory to T-fold compactifications of the heterotic string, both preserving four-dimensional $\mathcal{N}=2$ supersymmetry. The non-geometric Calabi-Yau space is a $K3$ fibration over $T^2$ with non-geometric monodromies in the duality group $O(\Gamma_{4,20})$; this is dual to a heterotic reduction on a $T^4$ fibration over $T^2$ with the $O(\Gamma_{4,20})$ monodromies no...

In this paper we study compactifications of the ${\cal N}=2$ heterotic $E_8\times E_8$ string on $(K3\times T^2)/\mathbb{Z}_3$ with various gauge backgrounds and calculate the topological couplings in the effective supergravity action that arise from one-loop amplitudes. We then identify candidates for dual type IIA compactifications on Calabi-Yau threefolds and compare the heterotic results with the corresponding topological string amplitudes. We find that the dual Calabi-Ya...

We study the duality between four-dimensional N=2 compactifications of heterotic and type IIA string theories. Via adiabatic fibration of the duality in six dimensions, type IIA string theory compactified on a K3-fibred Calabi-Yau threefold has a potential heterotic dual compactification. This adiabatic picture fails whenever the K3 fibre degenerates into multiple components over points in the base of the fibration. Guided by monodromy, we identify such degenerate K3 fibres a...

This is a very brief survey of some results in the geometry of string duality delivered at a lecture given at ICM 1998, Berlin. String Duality is the statement that one kind of string theory compactified on one space is equivalent in some sense to another string theory compactified on a second space. This draws a connection between two quite different spaces. Mirror symmetry is an example of this. Here we discuss mirror symmetry and another ``heterotic/type II'' duality which...

A type IIA string compactified on a Calabi-Yau manifold which admits a K3 fibration is believed to be equivalent to a heterotic string in four dimensions. We study cases where a Calabi-Yau manifold can have more than one such fibration leading to equivalences between perturbatively inequivalent heterotic strings. This allows an analysis of an example in six dimensions due to Duff, Minasian and Witten and enables us to go some way to prove a conjecture by Kachru and Vafa. The ...

We discuss type I -- heterotic duality in four-dimensional models obtained as a Coulomb phase of the six-dimensional U(16) orientifold model compactified on T^2 with arbitrary SU(16) Wilson lines. We show that Kahler potentials, gauge threshold corrections and the infinite tower of higher derivative F-terms agree in the limit that corresponds to weak coupling, large T^2 heterotic compactifications. On the type I side, all these quantities are completely determined by the spec...

We revisit the duality between heterotic string theory compactified on K3 x T^2 and type IIA compactified on a Calabi-Yau threefold X in the hypermultiplet sector. We derive an explicit map between the field variables of the respective moduli spaces at the level of the classical effective actions. We determine the parametrization of the K3 moduli space consistent with the Ferrara-Sabharwal form. From the expression of the holomorphic prepotential we are led to conjecture that...

In this paper we describe the heterotic dual of the type IIB theory compactified to four dimensions on a toroidal orientifold in the presence of fluxes. The type IIB background is most easily described in terms of an M-theory compactification on a four-fold. The heterotic dual is obtained by performing a series of U-dualities. We argue that these dualities preserve supersymmetry and that the supergravity description is valid after performing them. The heterotic string is comp...