ABSTRACT
This chapter describes in details the geometry of the conifold transition, because the local geometry is the key to the duality. It discusses transitions between Calabi-Yau threefolds and their significance in algebraic geometry and the physics of string theory. Greene and Plesser outlined an heuristic approach to ‘continuously’ extend mirror symmetry to all the Calabi-Yau threefolds belonging to the same connected component of the web generated by conifold transitions. Actually if transitions would connect each other to all Calabi-Yau threefolds, which is a rough version of the Reid’ s fantasy, then it could give an approach to establishing mirror symmetry for all of them. The chapter presents some background information on Chern-Simons theory and the evidence for the conjectures. It discusses the strategy of Atiyah Maldacena and Vafa and includes some basics on spaces with G2 holonomy.
