Quantum Cohomology

Abstract

In this final chapter we will construct the Gromov-Witten potential, which is the generating function for the Gromov-Witten invariants, and use it to define a quantum product on $A^* (\mathbb{P}^r )$. Kontsevich’s formula and the other recursions we found in Chapter 4, are then interpreted as partial differential equations for the Gromov-Witten potential. The striking fact about all these equations is that they amount to the associativity of the quantum product! In particular, Kontsevich’s formula is equivalent to associativity of the quantum product of $\mathbb{P}^2$.