In mathematics, specifically in symplectic geometry and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count the number of curves (pseudoholomorphic or algebraic) meeting prescribed conditions in a given ambient space (a symplectic manifold or a smooth projective variety). The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten. The rigorous mathematical definition of Gromov–Witten invariants is lengthy and difficult, so it is treated separately in the stable map article. This article attempts a more intuitive explanation of what the invariants mean, how they are computed, and why they are important.
Informal description Informally, for a fixed ambient space X {\displaystyle X} , the Gromov–Witten invariants count how many curves there are that pass through n {\displaystyle n} chosen submanifolds (or subvarieties) of X {\displaystyle X} . This is done using the technical machinery of cohomology and intersection theory in the moduli spaces of stable maps, which can be thought of as spaces parametrizing curves in X {\displaystyle X} . However, as these moduli spaces can have singularities, the resulting number is not always a natural number; it is sometimes described as a "virtual" count for this reason. This is because the singular points can contribute fractional values to the count. A classic example of a GW invariant is the number N d {\displaystyle N_{d}} of degree- d {\displaystyle d} rational plane curves passing through n = 3 d − 1 {\displaystyle n=3d-1} general points in the projective plane. (Note that 3 d − 1 {\displaystyle 3d-1} is in fact the dimension of the space of degree- d {\displaystyle d} rational plane curves, so we expect to get a finite number as the answer.) Starting from the base case N 1 = 1 {\displaystyle N_{1}=1} – there is exactly one line through any two points – the rest of these numbers can be calculated using a recursive formula, obtained through intersection theory in the moduli space of stable maps M ¯ 0 , 3 d ( X , d ) {\displaystyle {\overline {M}}_{0,3d}(X,d)} . We get N 1 = 1 , N 2 = 1 , N 3 = 12 , N 4 = 620 , N 5 = 87304 , … {\displaystyle N_{1}=1,N_{2}=1,N_{3}=12,N_{4}=620,N_{5}=87304,\ldots } .
Definition Consider the following:
X {\displaystyle X} : a closed symplectic manifold of dimension 2 k {\displaystyle 2k} ,
A {\displaystyle A} : a 2-dimensional homology class in X {\displaystyle X} ,
g {\displaystyle g} : a non-negative integer,
n {\displaystyle n} : a non-negative integer. Now we define the Gromov–Witten invariants associated to the 4-tuple: ( X , A , g , n ) {\displaystyle (X,A,g,n)} . Let M ¯ g , n {\displaystyle {\overline {\mathcal {M}}}_{g,n}} be the Deligne–Mumford moduli space of curves of genus g {\displaystyle g} with n {\displaystyle n} marked points and M ¯ g , n ( X , A ) {\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,A)} denote the moduli space of stable maps into X {\displaystyle X} of class A {\displaystyle A} , for some chosen almost complex structure J {\displaystyle J} on X {\displaystyle X} compatible with its symplectic form. The elements of M ¯ g , n ( X , A ) {\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,A)} are of the form:
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