| Summary: | In this work, we extend the well-known spectral cover construction first developed by Friedman, Morgan, and Witten to describe more general vector bundles on elliptically fibered Calabi-Yau geometries. In particular, we consider the case in which the Calabi-Yau fibration is not in Weierstrass form but can rather contain fibral divisors or multiple sections (i.e., a higher rank Mordell-Weil group). In these cases, general vector bundles defined over such Calabi-Yau manifolds cannot be described by ordinary spectral data. To accomplish this, we employ well-established tools from the mathematics literature of Fourier-Mukai functors. We also generalize existing tools for explicitly computing Fourier-Mukai transforms of stable bundles on elliptic Calabi-Yau manifolds. As an example of these new tools, we produce novel examples of chirality changing small instanton transitions. Next, we provide a geometric formalism that can substantially increase the understood regimes of heterotic/F-theory duality.
We consider heterotic target space dual (0,2) GLSMs on elliptically fibered Calabi-Yau manifolds. In this context, each half of the ``dual" heterotic theories must, in turn, have an F-theory dual. Moreover, the apparent relationship between two heterotic compactifications seen in (0,2) heterotic target space dual pairs should, in principle, induce some putative correspondence between the dual F-theory geometries. It has previously been conjectured in the literature that (0,2) target space duality might manifest in F-theory as multiple $K3$-fibrations of the same elliptically fibered Calabi-Yau manifold. We investigate this conjecture in the context of both 6-dimensional and 4-dimensional effective theories and demonstrate that in general, (0,2) target space duality cannot be explained by such a simple phenomenon alone. In all cases, we provide evidence that non-geometric data in F-theory must play at least some role in the induced F-theory correspondence while leaving the full determination of the putative new F-theory duality to the future work. Finally, we consider F-theory over elliptically fibered manifolds, with a general conic base. Such manifolds are quite standard in F-theory sense, but our goal is to explore the extent of the heterotic/F-theory duality over such manifolds. We consider heterotic target space dual (0,2) GLSMs on elliptically fibered Calabi-Yau manifolds. In this context, each half of the ``dual" heterotic theories must, in turn, have an F-theory dual. Moreover, the apparent relationship between two heterotic compactifications seen in (0,2) heterotic target space dual pairs should, in principle, induce some putative correspondence between the dual F-theory geometries. It has previously been conjectured in the literature that (0,2) target space duality might manifest in F-theory as multiple $K3$-fibrations of the same elliptically fibered Calabi-Yau manifold. We investigate this conjecture in the context of both 6-dimensional and 4-dimensional effective theories and demonstrate that in general, (0,2) target space duality cannot be explained by such a simple phenomenon alone. In all cases, we provide evidence that non-geometric data in F-theory must play at least some role in the induced F-theory correspondence while leaving the full determination of the putative new F-theory duality to the future work. Finally, we consider F-theory over elliptically fibered manifolds, with a general conic base. Such manifolds are quite standard in F-theory sense, but our goal is to explore the extent of the heterotic/F-theory duality over such manifolds. === Doctor of Philosophy === String theory is the only physical theory that can lead to self-consistent, effective quantum gravity theories. However, quantum mechanics restricts the dimension of the effective spacetime to ten (and eleven) dimensions. Hence, to study the consequences of string theory in four dimensions, one needs to assume the extra six dimensions are curled into small compact dimensions.
Upon this ``compactification," it has been shown (mainly in the 1990s) that different classes of string theories can have equivalent four-dimensional physics. Such classes are called dual. The advantage of these dualities is that often they can map perturbative and non-perturbative limits of these theories.
The goal of this dissertation is to explore and extend the geometric limitations of the duality between heterotic string theory and F-theory. One of the main tools in this particular duality is the Fourier-Mukai transformation. In particular, we consider Fourier-Mukai transformations over non-standard geometries. As an application, we study the F-theory dual of a heterotic/heterotic duality known as target space duality. As another side application, we derive new types of small instanton transitions in heterotic strings. In the end, we consider F-theory compactified over particular manifolds that if we consider them as a geometry dual to a heterotic string, can lead to unexpected consequences.
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