We show how two topologically distinct spaces-the Kahler K3 x T^ 2 and the non-Kahler T^ 2 bundle over K3-can be smoothly connected in heterotic string theory. The transition occurs when the base K3 is deformed to the T^ 4/Z_2 orbifold limit. The orbifold theory can be mapped via duality to M-theory on K3 x K3 where the transition corresponds to an exchange of the two K3's.

Due to the orbifold singularities, the intersection numbers on the moduli space of curves Mg, n are in general rational numbers rather than integers. We study the properties of the denominators of these intersection numbers and their relationship with the orders of automorphism groups of stable curves. We also present a conjecture about a multinomial type numerical property for a general class of Hodge integrals.

We discuss conjectures following from the attractor mechanism in type II string theory about the possible Chern classes of stable holomorphic vector bundles on Calabi-Yau threefolds. In particular, we give sufficient conditions for Chern classes to correspond to stable bundles.

No abstract uploaded!

Using the celebrated Witten-Kontsevich theorem, we prove a recursive formula of the n -point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for Gromov-Witten invariants. The latter has been proved recently by X. Liu and R. Pandharipande. We also give a combinatorial interpretation of n -point functions in terms of summation over binary trees.