Toric Geometry and String Theory

Acknowledgements First and foremost, I would like to thank Philip Candelas for his invaluable help throughout my four years in Oxford. I also owe special thanks to Bogdan Florea, Marcos Mariño and Harald Skarke for stimulating and enjoyable collaborations on the research topics discussed in this thesis. many others for interesting mathematical physics discussions. Many thanks to my parents, who have always supported me in my choices of life and encouraged me to be true and honest with myself. Alexandra, thank you so much for your love and patience; distance relationships are difficult, but you showed me that no matter what the distance is love always exists. Thanks also to my sister Maryse, for your continual support and friendship; and for your invitation to discover Burmese and Malian cultures! I would like to thank all my friends in Oxford, you showed me that better worlds exist, and most importantly that we can start creating them ourselves, right here and right now. Thanks to the OSSTW, OSAN, Oxford and UK Indymedia, ZOMBIE, PGA, OCSET and OARC crews, for the inspiration and the fantastic work to make our communities a better place to live. Thanks also to Maarit, Will and Ralph, for the wonderful Sunday nights spent in your good company. Thanks to all my friends from Québec, it is amazing that our friendship is still as intense as ever! Finally, I must not forget to thank all the people I have played music with, for all these special moments where collective creation of music engendered unity through diversity... 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 …