Calabi-Yau manifolds

Abstract

Calabi-Yau manifolds are Ricci-flat K¨ahler manifolds that are vital to string theory and have many applications in math. The existence of Calabi-Yau metrics on a large class of K¨ahler manifolds is guaranteed by the Calabi-Yau theorem, due to E. Calabi and S.T. Yau, whose proof is by heavy analysis, which will be the first part of this thesis. Explicit examples of these metrics are very difficult to find, we will present a known construction due to E. Calabi in the second part of this thesis. The third part will consist of an overview of how Calabi-Yau manifolds appear in string theory. The definition of Calabi-Yau manifolds can be generalised to a space known as a Lie algebroid, which is a vector bundle equipped with a structure to make it look like the tangent bundle. In the final part of this thesis, we look at this generalisation and present some rather basic results. Moreover, we will look at Lie algebroids in string theory and present a possible application of Calabi-Yau Lie algebroids in string theory.