Gromov–Witten Invariants, Topological Strings and Mirror Symmetry

Abstract

Gromov–Witten invariants count the number of holomorphic maps into some pro-jective variety. We introduce moduli stacks and follow Kontsevich’s construction of Gromov–Witten invariants by doing a Deligne–Mumford compactification on the relevant moduli space. We then define quantum cohomology and show how it gives the Witten–Dijkgraaf–Verlinde–Verlinde equations, and review contemporary research on finding Gromov–Witten invariants on products and blowups. In the second half, we consider the Landau–Ginzburg model, which, when twisted and promoted to a string theory, has correlators which correspond to Gromov–Witten invariants. Finally, we discuss mirror symmetry, a physical tool that can be used to compute Gromov–Witten invariants.