In this thesis we probe various interactions between toric geometry and
string theory. First, the notion of a top was introduced by Candelas and Font
as a useful tool to investigate string dualities. These objects torically
encode the local geometry of a degeneration of an elliptic fibration. We
classify all tops and give a prescription for assigning an affine, possibly
twisted Kac-Moody algebra to any such top. Tops related to twisted Kac-Moody
algebras can be used to construct string compactifications with reduced rank of
the gauge group. Secondly, we compute all loop closed and open topological
string amplitudes on orientifolds of toric Calabi-Yau threefolds, by using
geometric transitions involving SO/Sp Chern-Simons theory, localization on the
moduli space of holomorphic maps with involution, and the topological vertex.
In particular, we count Klein bottles and projective planes with any number of
handles in some Calabi-Yau orientifolds. We determine the BPS structure of the
amplitudes, and illustrate our general results in various examples with and
without D-branes. We also present an application of our results to the BPS
structure of the coloured Kauffman polynomial of knots.
This thesis is based on hep-th/0303218 (with H. Skarke), hep-th/0405083 and
hep-th/0411227 (with B. Florea and M. Marino).Comment: Oxford University DPhil Thesis (Advisor: Philip Candelas), accepted
October 2005, 152 pp., 43 figure