Conifold transitions and aspects of duality

Recent work by Greene, Morrison and Strominger has lead to a consistent physical interpretation of certain types of transitions between different string vacua. These transitions, discovered several years ago, involve singular conifold configurations which connect distinct Calabi-Yau manifolds. In this paper we generalize the splitting conifold transition to weighted manifolds and describe a class of connections between the webs of ordinary and weighted projective Calabi-Yau spaces. Combining these two constructions we find evidence for a dual analog of conifold transitions in heterotic N = 2 compactifications on K3 x T-2 and describe the first conifold transition of a Calabi-Yau manifold whose heterotic dual has been identified by Kachru and Vafa. We furthermore present a special type of conifold transition which, when applied to certain classes of Calabi-Yau K3 fibrations, preserves the fiber structure. Along the way we describe a new phenomenon which occurs when hypersurface singularities 'collide' with orbifold singularities. Finally we point out the importance of weighted conifold transitions which are not of splitting type. Such non-splitting conifold transitions turn out to connect the known web of Calabi-Yau spaces to new regions of the collective moduli space.