On the quantum cohomology of the plane, old and new

TL;DR

This paper introduces a novel method for counting algebraic curves in the projective plane using a splitting approach, and relates quantum cohomology to mirror symmetry by comparing splitting techniques.

Contribution

It presents a new geometric splitting method for enumerating curves in b^2 and connects quantum cohomology to mirror symmetry through splitting b^1 versus b^2.

Findings

A splitting method for counting curves in b^2.

Quantum cohomology as a mirror to splitting b^2 by splitting b^1.

Indication of a proof for associativity of quantum multiplication.

Abstract

We describe a method for counting maps of curves of given genus (and variable moduli) to $\Bbb P^2$, essentially by splitting the $\Bbb P^2$ in two; then specialising to the case of genus 0 we show that the method of quantum cohomology may be viewed as the 'mirror' of the former method where one splits the $\Bbb P^1$ rather than the $\Bbb P^2$, and we indicate a proof of the associativity of quantum multiplication based on this idea.