Gromov–Witten Invariants and Quantum Cohomology: Using Symplectic and 2d Topological Field Theoretic Methods

Abstract

In this thesis we consider Gromov–Witten invariants and quantum cohomology, starting with the approach from symplectic geometry. We study pseudoholomorphic curves and their moduli spaces, whose compactifications incorporate the behaviour of bubbling. To define Gromov–Witten invariants we study pseudocycles and semipositivity. After seeing applications in symplectic geometry, we combine the Gromov–Witten invariants into the quantum cohomology ring. For tools to compute the Gromov–Witten invariants explicitly, we shift our focus to the physical approach. We study topological field theories as the correlators in A-twisted nonlinear sigma models correspond to Gromov–Witten invariants. Then we use nonlinear and gauged sigma models to compute the quantum cohomology of toric varieties, using the scale invariance of the correlation functions. By restricting to hypersurfaces in toric varieties, we formulate an explicit mirror map.