Gromov–Witten invariant

In mathematics, specifically in symplectic topology and algebraic geometry, Gromov-Witten (GW) invariants are rational numbers that 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; they also play a crucial role in 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.

Formal definition

Fix a closed symplectic manifold ${\displaystyle X}$ . Let ${\displaystyle A}$  be a two-dimensional homology class in ${\displaystyle X}$ , ${\displaystyle g}$  and ${\displaystyle n}$  any natural numbers (including zero), and

${\displaystyle {\bar {M}}_{g,n}}$ 

the Deligne-Mumford moduli space of curves. Let

${\displaystyle Y:={\bar {M}}_{g,n}\times X^{n}}$ ,

and let

${\displaystyle GW_{g,n}^{X,A}\in H_{d}(Y)=\bigoplus _{i+j_{1}+\cdots j_{n}=d}H_{i}({\bar {M}}_{g,n})\otimes H_{j_{1}}(X)\otimes \cdots \otimes H_{j_{n}}(X)}$ 

be the image, under the evaluation map, of the moduli space of stable maps

${\displaystyle {\bar {M}}_{g,n}(X,A).}$ 

(Here all homology is taken with rational coefficients.) In a sense, this homology class is the Gromov-Witten invariant of ${\displaystyle X}$  for the data ${\displaystyle g}$ , ${\displaystyle n}$ , and ${\displaystyle A}$ . It is an invariant of the symplectic isotopy class of the symplectic manifold ${\displaystyle X}$ .

We can interpret it geometrically as follows. Let

${\displaystyle m:=6g-6+n\dim _{\mathbb {R} }X,}$ 

be the real dimension of ${\displaystyle Y}$ , and let ${\displaystyle \gamma }$  be a homology class in the Deligne-Mumford space and ${\displaystyle \alpha _{1},\ldots ,\alpha _{n}}$  homology classes in ${\displaystyle X}$ , such that the total dimension of these classes equals ${\displaystyle m-d}$ . They induce classes in the homology of ${\displaystyle Y}$  by the equation above. The intersection of these classes with the class ${\displaystyle GW_{g,n}^{X,A}}$  is zero-dimensional, so it corresponds to a rational number, the Gromov-Witten invariant

${\displaystyle GW_{g,n}^{X,A}(\gamma ,\alpha _{1},\ldots ,\alpha _{n})}$ 

for the given data. This number gives a "virtual" count of the number of pseudoholomorphic curves (in the class ${\displaystyle A}$ , of genus ${\displaystyle g}$ , with ___domain in the ${\displaystyle \gamma }$  part of the Deligne-Mumford space) whose ${\displaystyle n}$  marked points are mapped to cycles representing the ${\displaystyle \alpha _{i}}$ .

Put simply, the GW invariant counts how many curves there are that intersect ${\displaystyle n}$  chosen submanifolds of ${\displaystyle X}$ . 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. Also, the evaluation map may intersect the constraints ${\displaystyle \gamma ,\alpha _{i}}$  negatively. In principle, the virtual count may be any rational number.

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.

Computation of Gromov-Witten invariants

GW invariants are generally difficult to compute. While they are defined for any generic almost complex structure ${\displaystyle J}$ , for which the linearization ${\displaystyle D}$  of the ${\displaystyle {\bar {\partial }}_{j,J}}$ operator is surjective, they must actually be computed with respect to a specific chosen ${\displaystyle J}$ . It is most convenient to choose ${\displaystyle J}$  with special properties, such as nongeneric symmetries or integrability. Computations are often carried out on Kähler manifolds using the techniques of algebraic geometry.

However, a special ${\displaystyle J}$  may induce a nonsurjective ${\displaystyle D}$  and thus a moduli space of pseudoholomorphic curves that is larger than expected. To correct for this, one forms from the cokernel of ${\displaystyle D}$  a vector bundle, called the obstruction bundle. A GW invariant can then be realized as the integral of the Euler class of the obstruction bundle.

The main computational technique is localization. This applies when ${\displaystyle X}$  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 Atiyah and 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 ${\displaystyle X}$  to one or more other spaces whose GW invariants are more easily computed. Of course, one must first understand how the invariants behave under the surgeries. For such applications one often uses the more elaborate relative GW invariants, which count curves with prescribed tangency conditions along a symplectic submanifold of ${\displaystyle X}$  of real codimension two.

Related invariants and other constructions

The Gromov-Witten invariants are closely related to a number of other concepts in geometry, including the Donaldson invariants and Seiberg-Witten invariants. For compact symplectic four-manifolds, Cliff Taubes showed that a variant of the Gromov-Witten invariants (see Taubes's Gromov invariant) are equivalent to the Seiberg-Witten invariants. They are conjectured to contain the same information as Donaldson-Thomas invariants and Gopakumar-Vafa invariants, both of which are integer-valued.

GW invariants can also be defined using the language of algebraic geometry. In some cases, GW invariants agree with classical enumerative invariants of algebraic geometry. However, in general GW invariants enjoy one important advantage over the enumerative invariants, namely the existence of a composition law which describes how curves glue. The GW invariants can be bundled up into the quantum cohomology ring of the manifold ${\displaystyle X}$ , which is a deformation of the ordinary cohomology. The composition law of GW invariants is what makes the deformed cup product associative.

The quantum cohomology is known to be isomorphic to the pair of pants product in symplectic Floer homology.

Applications in physics

Gromov-Witten invariants are of interest in type IIA string theory. In the type IIA theory, the path of a closed string is a pseudoholomorphic curve. Gromov-Witten invariants, as integrals over spaces of such curves, are the path integrals of the theory, used to compute certain important probability amplitudes.

References

• McDuff, Dusa & Salamon, Dietmar (2004). J-Holomorphic Curves and Symplectic Topology, American Mathematical Society colloquium publications. ISBN 0-8218-3485-1.

• Piunikhin, Sergey; Salamon, Dietmar & Schwarz, Matthias (1996). Symplectic Floer-Donaldson theory and quantum cohomology. In C. B. Thomas (Ed.), Contact and Symplectic Geometry, pp. 171–200. Cambridge University Press. ISBN 0-5215-7086-7