## Topological twisting and elliptically fibered Calabi-Yau spaces

##### Summary

In this joined thesis in theoretical physics and mathematics, we look at topological twisting and the construction of elliptically fibered Calabi-Yau spaces. These subjects are closely interlinked, since Calabi-Yau spaces and elliptically fibered Calabi-Yau spaces appear as mathematical tools in understanding topological twists.
In the setting of topological string theory, we study N=(2,2) SUSY models, where we perform a topological twist to obtain the A- and B-model. We perform a topological duality twist in the setting of N=4 SYM theory. Using these twists, we solve the problem of defining the theories on curved spaces, which is limited due to holonomy effects. In practice, this is done by changing the generators of the symmetry group which transform non-trivially under holonomy.
We also study the formation of Calabi-Yau manifolds following Batyrev's method. Using toric geometry and starting from a four-dimensional polyhedra or fan, we obtain a corresponding Calabi-Yau mirror pair. These spaces are seen to be interesting and relevant in physics on their own right, in our work we focus on its connection with topological string theory. Furthermore, starting from two two-dimensional polyhedra, we perform an operation to obtain a four-dimensional polyhedron, which will now correspond to a Calabi-Yau manifold with an elliptic fibration; Again, these appear in our study of N=4 SYM. We derive the hodge numbers and the elliptic structure of some examples obtained via this method.
Finally, we wrote a program in C++ that can construct the aforementioned spaces using this specific method; It calculates the Hodge numbers of all four-dimensional polyhedra with five vertices. Furthermore, toric geometric techniques can be performed in two dimensions. The code can be downloaded at: https://github.com/eriklumens/PolyTori .