Introduction to Gromov-Witten invariants

In his 1985-paper introducing J-holomorphic curves into symplectic topology, Gromov proposed, among other things, to define invariants of symplectic manifolds as bordism classes of spaces of J-holomorphic curves. Over the years, the idea got transformed, through the work of Kontsevich as well many others, into a more general construction of Gromov-Witten invariants as intersection numbers in moduli spaces of stable J-holomorphic maps. Furthermore, the initial intention to use the invariants for distinguishing symplectic structures got replaced with the goal of understanding the mysterious nature of numerous universal identities the invariants turned out to obey. The intricate structure of these identities reflects the topology of moduli spaces of abstract Riemann surfaces and is related to Conformal Field Theory, integrable hierarchies, loop groups and is a subject of active research.

Our plan for the lectures is to start with definitions and examples of stable maps, their moduli spaces, and Gromov-Witten invariants; then proceed to examining the well-understood structure of the universal identities satisfied by genus-0 Gromov-Witten invariants and uncovering the role of the so-called twisted loop group of hidden symmetries; the tentative plan for the third lecture is to give applications to enumerative algebraic geometry, such as Quantum-Riemann-Roch and Quantum Lefschetz Theorems.