In this thesis we examine some of the interesting aspects of the Wess-Zumino- Novikov-Witten model when this model has been gauged and its energy tensor twisted by the addition of the derivative of one of its Cartan subalgebra valued currents. Specifically we consider the group valued model with the group taken as 5^(211) which is the Lie super group used to describe N = 2 supersymmetry. This model is advocated as being a good and natural description of the N = 2 superstring (also known as the charged spinning string, or N = 2 fermionic string) when it tensors an additional topological system of ghosts. The evidence for this assertion is presented by gauging and twisting the model and then extracting the N = 2 super Liouville action by the method of Hamiltonian reduction. The connection between the 5L (2|1)/5L (2|1) Wess-Zumino-Novikov-Witten model and field theory is made through its current algebra. As is true of many super groups there exists more than one interpretation of the Dynkin diagram for the algebra of 5L(2|1) and this results in more that one set of currents for this model. The classical and quantum currents in free field form are found in both cases, as is the highly non-linear transformation by which the two sets of currents are related. An analysis of a section of the cohomology of physical states of the model is undertaken. It is shown that the additional topological ghost system that tensors the gauged, twisted SL (2\l) model when it describes the N = 2 string only contributes a vacuum state to the overall cohomology, so reducing the analysis. As the 5L(2|1)/5L(2|1) Wess-Zumino-Novikov-Witten model is a topological field theory its spectrum of physical states lie in the cohomology class defined with respect to the BRST charge. The spectrum formed from the free field currents composes the so called Wakimoto module and this is calculated via the BRST formalism.

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 Kähler 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.