In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold.
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.
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.
denote the moduli space of stable maps into
The moduli space has real dimension Let denote the stabilization of the curve.
Put simply, a GW invariant counts how many curves there are that intersect
For the space of stable maps is an orbifold, whose points of isotropy can contribute noninteger values to the invariant.
There are numerous variations on this construction, in which cohomology is used instead of homology, integration replaces intersection, Chern classes pulled back from the Deligne–Mumford space are also integrated, etc.
operator is surjective, they must actually be computed with respect to a specific, chosen J.
It is most convenient to choose J with special properties, such as nongeneric symmetries or integrability.
Indeed, computations are often carried out on Kähler manifolds using the techniques of algebraic geometry.
However, a special J may induce a nonsurjective D and thus a moduli space of pseudoholomorphic curves that is larger than expected.
Making this idea precise requires significant technical arguments using Kuranishi structures.
Then one can use the Atiyah–Bott fixed-point theorem, of Michael Atiyah and Raoul Bott, to reduce, or localize, the computation of a GW invariant to an integration over the fixed-point locus of the action.
Another approach is to employ symplectic surgeries to relate X to one or more other spaces whose GW invariants are more easily computed.
For such applications one often uses the more elaborate relative GW invariants, which count curves with prescribed tangency conditions along a symplectic submanifold of X of real codimension two.
For algebraic threefolds, they are conjectured to contain the same information as integer valued Donaldson–Thomas invariants.
Physical considerations also give rise to Gopakumar–Vafa invariants, which are meant to give an underlying integer count to the typically rational Gromov-Witten theory.
The Gopakumar-Vafa invariants do not presently have a rigorous mathematical definition, and this is one of the major problems in the subject.
The Gromov-Witten invariants of smooth projective varieties can be defined entirely within algebraic geometry.
However, the major advantage that GW invariants have over the classical enumerative counts is that they are invariant under deformations of the complex structure of the target.
The GW invariants also furnish deformations of the product structure in the cohomology ring of a symplectic or projective manifold; they can be organized to construct the quantum cohomology ring of the manifold X, which is a deformation of the ordinary cohomology.
The associativity of the deformed product is essentially a consequence of the self-similar nature of the moduli space of stable maps that are used to define the invariants.
The quantum cohomology ring is known to be isomorphic to the symplectic Floer homology with its pair-of-pants product.
GW invariants are of interest in string theory, a branch of physics that attempts to unify general relativity and quantum mechanics.
In this theory, everything in the universe, beginning with the elementary particles, is made of tiny strings.
Unfortunately, the moduli space of such parametrized surfaces, at least a priori, is infinite-dimensional; no appropriate measure on this space is known, and thus the path integrals of the theory lack a rigorous definition.
Here there are six spacetime dimensions, which constitute a symplectic manifold, and it turns out that the worldsheets are necessarily parametrized by pseudoholomorphic curves, whose moduli spaces are only finite-dimensional.