Kähler versus Non-Kähler Compactifications

We review our present understanding of heterotic compactifications on non-Kähler complex manifolds with torsion. Most of these manifolds can be obtained by duality chasing a consistent F-theory compactification in the presence of fluxes. We show that the duality map generically leads to non-Kähler spaces on the heterotic side, although under some special conditions we recover Kähler compactifications. The dynamics of the heterotic theory is governed by a new superpotential and minimizing this superpotential reproduces all the torsional constraints. This superpotential also fixes most of the moduli, including the radial modulus. We discuss some new connections between Kähler and non-Kähler compactifications, including some phenomenological aspects of the latter compactifications. ∗ Based on the talks given at the QTS3 conference, University of Cincinnatti and SUSY 03. The Calabi-Yau (CY) compactifications of Candelas et al. [1] have led to some major progress in our understanding of string theory vacua. Compactifying the heterotic string on such manifolds results in four-dimensional models with minimal supersymmetry (susy). In terms of the corresponding two dimensional non-linear sigma model, we demand conformal invariance so that all the tadpoles vanish and the string equations of motion are satisfied. In this way we recover again CY spaces. For the bosonic case this will give us the model studied in [2]. By definition CY manifolds are Kähler and have a vanishing first Chern class. By Yau’s theorem therefore, for a given complex structure and a given cohomology class of the Kähler form there is a unique Ricci-flat metric with SU(3) holonomy. Generically, when considering ordinary CY compactifications of the heterotic string theory the three-form background fluxes and the dilaton are equal to zero. The fourdimensional spacetime is Minkowski and therefore has zero cosmological constant. Although susy would allow, in general, anti de-Sitter solutions only the Minkowski solution is realized. The cancellation of the two-loop sigma model beta function puts a strong constraint on the vector bundle, namely it has to be identified with the tangent bundle, implying that the three form is in the cohomology classes of the manifold . Further, by the Uhlenbeck-Yau theorem, there is an essentially unique choice of vector field for any given holomorphic stable vector bundle satisfying the Donaldson-Uhlenbeck-Yau (DUY) equations. This attractive scenario however is clouded by some inherent problems which are related to the degeneracy of string vacua. Essentially there are two different degeneracies appearing in string theory compactifications. First there are thousands of CY manifolds that could be potential solutions to the low energy effective theory. Second, once we choose a particular CY manifold, there are many different moduli associated with the complex structure and the Kähler structure deformations of the manifold. All these moduli are unstable at tree level and thus lead to a situation that is unattractive for phenomenology. It turns out that the radial modulus of the CY is one of the Kähler moduli. Therefore, when this field is not stabilized the CY will runaway to infinite size. This ruins the whole consistency of the compactification scenario [4]. 1 Ricci flatness is not an essential property of CY compactifications as has been demonstrated in [3]. We can restore Kählerity without having a Ricci flat metric. For the non-Kähler manifolds, however, we can never have a Ricci flat metric. We shall discuss this in more detail as we go along. 2 This is a sufficient condition, but not a necessary one.