We begin developing a theory of morphisms of moduli spaces of pseudoholomorphic curves and discs with Lagrangian boundary conditions as Kuranishi spaces, using a modification of the procedure of Fukaya-Oh-Ohta-Ono. As an example, we consider the total space of the line bundles O(−n) and O on P1 as toric Kähler manifolds, and we construct isomorphic Kuranishi structures on the moduli space of holomorphic discs in O(−n) on P1 with boundary on a moment map fiber Lagrangian L and on a moduli space of holomorphic discs subject to appropriate tangency conditions in O. We then deform this latter Kuranishi space and use this deformation to define a Lagrangian potential for L in O(−n), and hence a superpotential for O(−n). With some conjectural assumptions regarding scattering diagrams in P1 × P, this superpotential can then be calculated tropically analogously to a bulk-deformed potential of a Lagrangian in P1 × P1.

In this dissertation we study several topics related to the geometry and physics of heterotic string compactification. After an introduction to some of the basic ideas of this field, we review the heterotic line bundle standard model construction and a complex structure mod- uli stabilization mechanism associated to certain hidden sector gauge bundles. Once this foundational material has been presented, we move on to the original research of this disser- tation. We present a scan over all known heterotic line bundle standard models to examine the frequency with which the particle spectrum is forced to change, or "jump," by the hidden sector moduli stabilization mechanism just mentioned. We find a significant percentage of forced spectrum jumping in those models where such a change of particle content is possible. This result suggests that one should consider moduli stabilization concurrently with model building, and that failing to do so could lead to misleading results. We also use state of the art techniques to study Yukawa couplings in these models. We find that a large portion of Yukawa couplings which naively would be expected to be non-zero actually vanish due to certain topological selection rules. There is no known symmetry which is responsible for this vanishing. In the final part of this dissertation, we study the Chern-Simons contribution to the superpotential of heterotic theories. This quantity is very important in determining the vacuum stability of these models. By explicitly building real bundle morphisms between vec- tor bundles over Calabi-Yau manifolds, we show that this contribution to the superpotential vanishes in many cases. However, by working with more complicated, and realistic geome- tries, we also present examples where the Chern-Simons contribution to the superpotential is non-zero, and indeed fractional. / Doctor of Philosophy / String theory is a candidate for a unified theory of all of the known interactions of nature. To be consistent, the theory needs to be formulated in 9 spatial dimensions, rather than the 3 of everyday experience. To connect string theory with reality, we need to reproduce the known physics of 3 dimensions from the 9 dimensional theory by hiding, or "compactifying," 6 directions on a compact internal space. The most common choice for such an internal space is called a Calabi-Yau manifold. In this dissertation, we study how the geometry of the Calabi-Yau manifold determines physical quantities seen in 3 dimensions such as the number of particle families, particle interactions and potential energy. The first project in this dissertation studies to what extent the process of making the Calabi-Yau manifold rigid, something which is required observationally, affects the particle spectrum seen in 3 dimensions. By scanning over a large model set, we conclude that computation of the particle spectrum and such "moduli stabilization" issues should be considered in concert, and not in isolation. We also showed that a large portion of the interactions that one would naively expect between the particles in such string models are actually absent. There is no known symmetry of the theory that accounts for this structure, which is linked to the topology of the extra spatial dimensions. In the final part of the dissertation, we show how to calculate previously unknown contributions to the potential energy of these string theory models. By linking to results from the mathematics literature, we show that these contributions vanish in many cases. However, we present examples where it is non-zero, a fact of crucial importance in understanding the vacua of heterotic string theories.

This thesis studies derived equivalences between total spaces of vector bundles and dg-quivers. A dg-quiver is a graded quiver whose path algebra is a dg-algebra. A quiver with superpotential is a dg-quiver whose differential is determined by a "function" Φ. It is known that the bounded derived category of representations of quivers with superpotential with finite dimensional cohomology is a Calabi- Yau triangulated category. Hence quivers with superpotential can be viewed as noncommutative Calabi- Yau manifolds. One might then ask if there are derived equivalences between Calabi-Yau manifolds and quivers with superpotential. In this thesis, we answer this question and, generalizing Bridgeland [15], give a recipe on how to construct such derived equivalences.