---
tag: XVJA0JD79P
kind: concept
name: Gromov–Witten invariant
msc: [14N35, 53D45, 14N10]
sources:
  - url: https://en.wikipedia.org/wiki/Gromov%E2%80%93Witten_invariant
same_as:
  wikidata: Q1547297
  wikipedia: Gromov–Witten invariant
---

## Summary

A Gromov–Witten invariant is a rational number that counts curves in a smooth projective variety or a symplectic manifold.
The curves have a given genus and homology class, and they pass through given cycles.
It is defined by integrating cohomology classes over the virtual fundamental class of a [[moduli space|FJZRW9P0K3]] of stable maps.
These invariants solve classical questions of [[enumerative geometry|ZPZA1FKZZC]], they define quantum cohomology, and they are one side of [[mirror symmetry|YK3DMN2NSK]].

## Explanations

### L1 human

Ask how many curves of a fixed kind pass through a fixed set of points.
For lines in the plane through two points the answer is $1$, for conics through five points it is also $1$.
For rational plane cubics through eight general points the answer is $12$.
Gromov–Witten invariants give a uniform way to define and compute such numbers, by integrating over a space that parametrizes all the curves at once.
They stay the same when the variety is deformed, which makes them invariants.