Title: Quantum geometry of the heterotic string

Abstract: Calabi-Yau geometry has proven to be a fruitful meeting-point for mathematicians and physicists: the mathematical tools have been invaluable in elucidating the properties of strings propagating on such non-trivial manifolds, and the physics perspective has uncovered surprising "quantum" properties such as mirror symmetry-- a precise correspondence linking the symplectic properties of a Calabi-Yau manifold to the complex properties of the (in general topologically distinct) mirror Calabi-Yau geometry.

I will present an important generalization of Calabi-Yau geometry that is currently under investigation by mathematicians and physicists alike. Motivated by heterotic string theory, this geometry involves a choice of a holomorphic vector bundle over a complex manifold obeying a generalization of the Calabi-Yau condition. I will discuss the recent progress, as well as important gaps in our understanding of both classical and quantum properties associated to these geometries.