This thesis is devoted to a discussion of various aspects of supersymmetric quantum field theories in four and two dimensions. In four dimensions, 𝒩 = 1 supersymmetric quantum gauge theories on various four-manifolds are constructed. Many of their properties, some of which are distinct to the theories on flat spacetime, are analyzed. In two dimensions, general 𝒩 = (2, 2) nonlinear sigma models on S² are constructed, both for chiral multiplets and twisted chiral multiplets. The explicit curvature coupling terms and their effects are discussed. Finally, 𝒩 = (0, 2) gauged linear sigma models with nonabelian gauge groups are analyzed. In particular, various dualities between these nonabelian gauge theories are discussed in a geometric content, based on their Higgs branch structure. / Ph. D.

In this thesis, we apply the methods of numerical differential geometry to several different problems in heterotic Calabi-Yau compactification. We review algorithms for computing both the Ricci-flat metric on Calabi-Yau manifolds and Hermitian Yang-Mills connections on poly-stable holomorphic vector bundles over those spaces. We apply the numerical techniques for obtaining Ricci-flat metrics to study hierarchies of curvature scales over Calabi-Yau manifolds as a function of their complex structure moduli. The work we present successfully finds known large curvature regions on these manifolds, and provides useful information about curvature variation at general points in moduli space. This research is important in determining the validity of the low energy effective theories used in the description of Calabi-Yau compactifications. The numerical techniques for obtaining Hermitian Yang-Mills connections are applied in two different fashions in this thesis. First, we demonstrate that they can be successfully used to numerically determine the stability of vector bundles with qualitatively different features to those that have appeared in the literature to date. Second, we use these methods to further develop some calculations of holomorphic Chern-Simons invariant contributions to the heterotic superpotential that have recently appeared in the literature. A complete understanding of these quantities requires explicit knowledge of the Hermitian Yang-Mills connections involved. This feature makes such investigations prohibitively hard to pursue analytically, and a natural target for numerical techniques. / Doctor of Philosophy / String theory is one of the most promising attempts to unify gravity with the other three fundamental interactions (electromagnetic, weak and strong) of nature. It is believed to give a self-consistent theory of quantum gravity, which, at low energy, could contain all of the physics that we known, from the Standard Model of particle physics to cosmology. String theories are often defined in nine spatial dimensions. To obtain a theory with three spatial dimensions one needs to hide, or ``compactify," six of the dimensions on a compact space which is small enough to have remained unobserved by our experiments. Unfortunately, the geometries of these spaces, called Calabi-Yau manifolds, and additional structures associated to them, called holomorphic vector bundles, turns out to be extremely complex. The equations determining the exact solutions of string theory for these quantities are highly non-linear partial differential equations (PDE's) which are simply impossible to solve analytically with currently known techniques. Nevertheless, knowledge of these solutions is critical in understanding much of the detailed physics that these theories imply. For example, to compute how the particles seen in three dimensions would interact with each other in a string theoretic model, the explicit form of these solutions would be required. Fortunately, numerical methods do exist for finding approximate solutions to the PDE's of interest. In this thesis we implement these algorithmic techniques and use them to study a variety of physical questions associated to the attempt to link string theory to the physics observed in our experiments.

This thesis is devoted to a discussion of two-dimensional theories with (0,2) supersymmetry. Examples of two-dimensional (0,2) gauged linear sigma models (GLSMs) are constructed for various spaces including Grassmannians, complete intersections in Grassmannians, and non-complete intersections such as Pfaffians. Generalizations of (2,2) Toda dual theories to (0,2) Toda-like theories are also discussed and some examples are given, including products of projective spaces and del Pezzo surfaces. Correlation functions are computed to show the examples are the correct mirror models. / Ph. D. / This thesis is devoted to a discussion in a specific branch of string theory in particle physics. String theory is a model of elementary particles, which are currently described by a theory called the Standard Model. String theory is a more fundamental theory than the Standard Model in two ways: it incorporates general relativity, <i>i.e.</i>, the theory of gravity; and it is a ultraviolet theory of the Standard Model, or equivalently, the Standard model is seen as a low energy approximation of string theory. This thesis is concerned with the quantum mechanics of string theory, described by quantum field theory along a two-dimensional worldsheet swept out by a one-dimensional string as it propagates in time. Specifically, this thesis explores examples of two-dimensional worldsheet theories with a technical property known as (0,2) supersymmetry.

The object of this work is to exploit various constructions of string theory and M-theory to yield new insights into supersymmetric theories in both four and three dimensions. In 4d, we extend work on Seiberg-Witten theory to study and compute BPS spectra of the class of complete N = 2 theories. The approach we take is based on the program of geometric engineering, in which 4d theories are constructed from compactifications of type IIB strings on Calabi-Yau manifolds. In this setup, the natural candidates for BPS states are D3 branes wrapped on supersymmetric 3-cycles in the Calabi-Yau. Our study makes use of the mathematical structure of quivers, whose representation theory encodes the notion of stability of BPS particles. Except for 11 exceptional cases, all complete theories can be constructed by wrapping stacks of two M5 branes on Riemann surfaces. By exploring the connection between quivers and M5 brane theories, we develop a powerful algorithm for computing BPS spectra, and give an in-depth study of its applications. In particular, we compute BPS spectra for all asymptotically free complete theories, as well as an infinite set of conformal \(SU(2)^k\) theories with certain matter content. From here, we go on to apply the insight gained from our 4d study to 3d gauge theories. We consider the analog of the M5 brane construction in the case of 3d N = 2 theories: pairs of M5 branes wrapped on a 3-manifold. Using the ansantz of R-flow, we study 3-manifolds consisting of Riemann surfaces fibered over R. When the construction is non-singular, the resulting IR physics is described by a free abelian Chern-Simons theory. The mathematical data of a tangle captures the data of the gauge theory, and the Reidemeister equivalances on tangles correspond to dualities of physical descriptions. To obtain interacting matter, we allow singularities in the construction. By extending the tangle description to these singular cases, we find a set of generalized Reidemeister moves that capture non-trivial mirror symmetries of 3d gauge theories. These results give a geometric origin to these well-known 3d dualities. / Physics

In this thesis we explore various aspects of Calabi-Yau (CY) manifolds and com- pactifications of the heterotic string over them. At first we focus on classifying symmetries and computing Hodge numbers of smooth CY quotients. Being non- simply connected, these quotients are an integral part of CY compactifications of the heterotic string, aimed at producing realistic string vacua. Discrete symmetries of such spaces that are generically present in the moduli space, are phenomenologically important since they may appear as symmetries of the associated low energy theory. We classify such symmetries for the class of smooth Complete Intersection CY (CICY) quotients, resulting in a large number of regular and R-symmetry examples. Our results strongly suggest that generic, non-freely acting symmetries for CY quotients arise relatively frequently. A large number of string derived Standard Models (SM) were recently obtained over this class of CY manifolds indicating that our results could be phenomenologically important. We also specialise to certain loci in the moduli space of a quintic quotient to produce highly symmetric CY quotients. Our computations thus far are the first steps towards constructing a sizeable class of highly symmetric smooth CY quotients. Knowledge of the topological properties of the internal space is vital in determining the suitability of the space for realistic string compactifications. Employing the tools of polynomial deformation and counting of invariant K&auml;hler classes, we compute the Hodge numbers of a large number of smooth CICY quotients. These were later verified by independent cohomology computations. We go on to develop the machinery to understand the geometry of CY manifolds embedded as hypersurfaces in a product of del Pezzo surfaces. This led to an interesting account of the quotient space geometry, enabling the computation of Hodge numbers of such CY quotients. Until recently only a handful of CY compactifications were known that yielded low energy theories with desirable MSSM features. The recent construction of rank 5 line bundle sums over smooth CY quotients has led to several SU(5) GUTs with the exact MSSM spectrum. We derive semi-analytic results on the finiteness of the number of such line bundle models, and study the relationship between the volume of the CY and the number of line bundle models over them. We also imply a possible correlation between the observed number of generations and the value of the gauge coupling constants of the corresponding GUTs. String compactifications with underlying SO(10) GUTs are theoretically attractive especially since the discovery that neutrinos have non-zero mass. With this in mind, we construct tens of thousands of rank 4 stable line bundle sums over smooth CY quotients leading to SO(10) GUTs.