Aspects of string theory compactifications

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Date

2004

Authors

Park, Hyukjae

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String Theory is defined consistently when the dimensionality of the spacetime is ten. To make contact with the apparent four-dimensional world that we live in, we need to “compactify” six of the dimensions of String Theory. This dissertation is dedicated to the study of various physical and mathematical aspects of this problem. In the first part of this dissertation, we construct the mirror of the Beauville manifold which is a Calabi-Yau 3-fold with non-abelian fundamental group. (The preservation of supersymmetry dictates that the internal space into which String Theory is compactified is a so-called Calabi-Yau manifold.) We use the conjecture of Batyrev and Borisov to find the previously misidentified mirror of its universal covering space, P 7 [2, 2, 2, 2]. The monomial-divisor mirror map is essential in identifying how the fundamental group of the Beauville manifold acts on the mirror of P 7 [2, 2, 2, 2]. Once we find the mirror of the Beauville manifold, we confirm the existence of the threshold bound state around the conifold point, which was originally conjectured in [12]. We also consider how the quantum symmetry group acts on the D-branes that become massless at the conifold point and show the action proposed in [11] is compatible with mirror symmetry. In the second part, we discuss an important subclass of D-branes on a Calabi-Yau manifold, X, which are in 1-1 correspondence with objects in D(X), the derived category of coherent sheaves on X. We study the action of the monodromies in Kahler moduli space on these D-branes. We refine and extend a conjecture of Kontsevich about the form of one of the generators of these monodromies (the monodromy about the “conifold” locus) and show that one can do quite explicit calculations of the monodromy action in many examples. As one application, we verify a prediction of Mayr about the action of the monodromy about the Landau-Ginsburg locus of the quintic. Prompted by the result of this calculation, we propose a modification of the derived category which implements the physical requirement that the shift-by-6 functor should be the identity. The last part of the dissertation is devoted to an F-theory compactification. We consider F-theory on an elliptically fibered Calabi-Yau 4-fold. We review the mechanisms used to stabilize various moduli in the theory. Especially, we take a closer look at Kahler moduli stabilization by the generation of non-perturbative superpotentials and argue that stabilization of all Khaler moduli in this way is non-generic. We consider an example where explicit analytic computation is possible and show that in this example, when all Kahler moduli are stabilized, the overall size is big enough for the supergravity approximation used here to be valid.

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