Gromov–Witten invariants and topological strings: a progress report

Abstract

In this talk I summarize recent progress in the theory of Gromov–Witten invariants from topological string theory and string dualities. On the one hand, large N dualities have led to the theory of the topological vertex, which solves Gromov–Witten theory to all genera on toric, noncompact Calabi–Yau threefolds. On the other hand, heterotic/type II duality and the holomorphic anomaly equations can be used to analyze Gromov–Witten theory in some simple compact examples. I sketch the physical ideas behind these results and connect the results obtained in physics with the results obtained in algebraic geometry.