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Geometry and Connectedness of Heterotic String Compactifications with Fluxes

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I will discuss the geometry of heterotic string compactifications with fluxes. The compactifications on 6 dimensional manifolds which preserve N=1 supersymmetry in 4 dimensions must be complex manifolds with vanishing first Chern class, but which are not in general Kahler (and therefore not Calabi-Yau manifolds) together with a vector bundle on the manifold which must satisfy a complicated differential equation. The flux, which can be viewed as a torsion, is the obstruction to the manifold being Kahler. I will describe how these compactifications are connected to the more traditional compactifications on Calabi-Yau manifolds through geometric transitions like flops and conifold transitions. For instance, one can construct solutions by flopping rational curves in a Calabi-Yau manifold in such a way that the resulting manifold is no longer Kahler. Time permitting, I will discuss open problems, for example the understanding of the the moduli space of heterotic compactifications and the related problem of determining the massless spectrum in the effective 4 dimensional supersymmetric field theory. The study of these compactifications is interesting on its own right both in string theory, in order to understand more generally the degrees of freedom of these theories, and also in mathematics. For instance, the connectedness between the solutions is related to problems in mathematics like the conjecture by Mile Reid that complex manifolds with trivial canonical bundle are all connected through geometric transitions.