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. 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.
Definition
Consider the following: X: a closed symplectic manifold of dimension 2k, A: a 2-dimensional homology class in X, g: a non-negative integer, n: a non-negative integer. Now we define the Gromov–Witten invariants associated to the 4-tuple: (X, A, g, n). Let be the Deligne–Mumford moduli space of curves of genus g with n marked points and denote the moduli space of stable maps into X of class A, for some chosen almost complex structure J on X compatible with its symplectic form. The elements of are of the form: , where C is a (not necessarily stable) curve with n marked points x1, ..., xn and f : C → X is pseudoholomorphic. The moduli space has real dimension
Let
denote the stabilization of the curve. Let
which has real dimension . There is an evaluation map
The evaluation map sends the fundamental class of to a d-dimensional rational homology class in Y, denoted
In a sense, this homology class is the Gromov–Witten invariant of X for the data g, n, and A. It is an invariant of the symplectic isotopy class of the symplectic manifold X.
To interpret the Gromov–Witten invariant geometrically, let β be a homology class in and homology classes in X, such that the sum of the codimensions of equals d. These induce homology classes in Y by the Künneth formula. Let
where denotes the intersection product in the rational homology of Y. This is a rational number, the Gromov–Witten invariant for the given classes. This number gives a "virtual" count of the number of pseudoholomorphic curves (in the class A, of genus g, with domain in the β-part of the Deligne–Mumford space) whose n marked points are mapped to cycles representing the .
Put simply, a GW invariant counts how many curves there are that intersect n chosen submanifolds of X. However, due to the "virtual" nature of the count, it need not be a natural number, as one might expect a count to be. 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.
Computational techniques
Gromov–Witten invariants are generally difficult to compute. While they are defined for any generic almost complex structure J, for which the linearization D of the 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. Loosely speaking, one corrects for this effect by forming from the cokernel of D a vector bundle, called the obstruction bundle, and then realizing the GW invariant as the integral of the Euler class of the obstruction bundle. Making this idea precise requires significant technical arguments using Kuranishi structures.
The main computational technique is localization. This applies when X is toric, meaning that it is acted upon by a complex torus, or at least locally toric. 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.