Title:

CalabiYau threefolds and heterotic string compactification

This thesis is concerned with CalabiYau threefolds and vector bundles upon them, which are the basic mathematical objects at the centre of smooth supersymmetric compactifications of heterotic string theory. We begin by explaining how these objects arise in physics, and give a brief review of the techniques of algebraic geometry which are used to construct and study them. We then turn to studying multiplyconnected CalabiYau threefolds, which are of particular importance for realistic string compactifications. We construct a large number of new examples via free group actions on complete intersection CalabiYau manifolds (CICY's). For special values of the parameters, these group actions develop fixed points, and we show that, on the quotient spaces, this leads to a particular class of singularities, which are quotients of the conifold. We demonstrate that, in many cases at least, such a singularity can be resolved to yield another smooth CalabiYau threefold, with different Hodge numbers and fundamental group. This is a new example of the interconnectedness of the moduli spaces of distinct CalabiYau threefolds. In the second part of the thesis we turn to a study of two new `threegeneration' manifolds, constructed as quotients of a particular CICY, which can also be represented as a hypersurface in dP6 x dP6, where dP6 is the del Pezzo surface of degree six. After describing the geometry of this manifold, and especially its nonAbelian quotient, in detail, we show how to construct on the quotient manifolds vector bundles which lead to fourdimensional heterotic models with the standard model gauge group and three generations of particles. The example described in detail has the spectrum of the minimal supersymmetric standard model plus a single vectorlike pair of colour triplets.
