Gromov-Witten invariant

Gromov-Witten invariants are a special form of topological invariants that establish a connection between topology and algebra .

More precisely, in symplectic topology and algebraic geometry , they denote rational numbers that count pseudoholomorphic curves (with certain additional conditions) on a symplectic manifold and serve to distinguish symplectic manifolds. They can be understood as a homology or cohomology class of an associated space or as a deformed cup product of a quantum cohomology . The Gromov-Witten invariants are named after Michail Gromow and Edward Witten . They also play an important role in topological string theory .

The exact mathematical construction is dealt with in a separate article " Stable mapping ".

definition

Let be a closed symplectic manifold of dimension , a 2-dimensional homology class in and , arbitrary natural numbers including zero. Be further ${\displaystyle X}$ ${\displaystyle 2k}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle g}$ ${\displaystyle n}$

${\displaystyle {\bar {M}}_{g,n}}$

the Deligne-Mumford modular space of curves of sex with marked (distinguished) points, and ${\displaystyle g}$ ${\displaystyle n}$

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

the module space of stable mappings according to the class , that of the real dimension ${\displaystyle X}$ ${\displaystyle A}$

${\displaystyle d:=2c_{1}^{X}(A)+(2k-6)(1-g)+2n.}$

Has. Finally be

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

with the real dimension . The implementation mapping maps the fundamental class from to a -dimensional rational homology class in : ${\displaystyle 6g-6+2kn}$ ${\displaystyle M}$ ${\displaystyle d}$ ${\displaystyle Y}$

${\displaystyle GW_{g,n}^{X,A}\in H_{d}(Y,\mathbb {Q} ).}$

This homology class is in a sense the Gromov-Witten invariant of to the values , and . It is an invariant of the symplectic isotopy of the symplectic manifold . ${\displaystyle X}$ ${\displaystyle g}$ ${\displaystyle n}$ ${\displaystyle A}$ ${\displaystyle X}$

To interpret the Gromov-Witten invariant geometrically, let a homology class in and homology classes in , so that the sum of the codimensions of is equal . This includes homology classes in via the Künneth formula . Be ${\displaystyle \gamma }$ ${\displaystyle {\bar {M}}_{g,n}}$ ${\displaystyle \alpha _{1},\ldots ,\alpha _{n}}$ ${\displaystyle X}$ ${\displaystyle \gamma ,\alpha _{1},\ldots ,\alpha _{n}}$ ${\displaystyle d}$ ${\displaystyle Y}$

${\displaystyle GW_{g,n}^{X,A}(\gamma ,\alpha _{1},\ldots ,\alpha _{n}):=GW_{g,n}^{X,A}\cdot \gamma \cdot \alpha _{1}\cdot \cdots \cdot \alpha _{n}\in H_{0}(Y,\mathbb {Q} ),}$

wherein the average product ( intersection product ) in the rational homology referred to. This is a rational number, the Gromov-Witten invariant for these classes. It counts the pseudo-alcoholomorphic curves (in the class with gender , with definition area in the “ part” of the Deligne-Mumford space) “virtually”, with the marked points being mapped to the cycles represented. ${\displaystyle \cdot }$ ${\displaystyle Y}$ ${\displaystyle A}$ ${\displaystyle g}$ ${\displaystyle \gamma }$ ${\displaystyle n}$ ${\displaystyle \alpha _{i}}$

In simple terms, the Gromov-Witten invariant counts how many curves intersect selected submanifolds of . Because of the nature of this enumeration, indicated by the term “virtual”, these do not have to be natural numbers, since the space of the stable mappings is an orbifold , at whose isotropy points incomplete numbers can contribute to the invariants. ${\displaystyle n}$ ${\displaystyle X}$

There are many variations of this design in which e.g. B. instead of homology cohomology is used or instead of cuts an integration. Sometimes the " pull-back " (from the Deligne-Mumford room) Chern classes are also integrated.

Calculation method

Gromov-Witten invariants are generally difficult to compute. While they are defined for every generic almost complex structure for which the linearization of the operator is surjective , in practice a specific one must be chosen. Usually one with special properties is chosen, such as special symmetries or integrability. In fact, the calculations are often performed on Kahler manifolds using algebraic geometry techniques. ${\displaystyle J}$ ${\displaystyle D}$ ${\displaystyle {\bar {\partial }}_{j,J}}$ ${\displaystyle J}$ ${\displaystyle J}$

However, a special one can lead to a non-surjective one and thus to a module space of pseudoholomorphic curves that is larger than expected. Roughly speaking, one corrects this effect by forming a vector bundle out of the coke of, called an obstruction bundle , and then defining the Gromov-Witten invariant as an integral on the Euler class of this bundle. Technically, the theory of polyfolds is used. ${\displaystyle J}$ ${\displaystyle D}$ ${\displaystyle D}$

The main calculation method is localization . It is applicable if there is a torus-manifold, that is, if the action of a complex torus is present on it or if it is at least locally a torus. Then you can reduce the Atiyah-Bott fixed point theorem (by Michael Atiyah and Raoul Bott ) to the calculation of the invariants to an integration over the location of the fixed points of the effect (“localize”). ${\displaystyle X}$

Another approach uses symplectic " surgery " ( surgery ) to break down into manifolds on which the calculation of the Gromov-Witten invariants is easier. Of course, one must first understand the behavior of the manifolds under surgery. For these applications, the more elaborately defined “relative Gromov-Witten invariants” are often used, which count curves with prescribed tangential properties along symplectic submanifolds with a real codimension of 2. ${\displaystyle X}$ ${\displaystyle X}$

Related invariants and constructions

The Gromov-Witten invariants are closely related to other geometric concepts such as the Donaldson invariants and the Seiberg-Witten invariants . For compact symplectic 4-manifolds, Clifford Taubes showed that a variant of the Gromov-Witten invariants ( Taubes' Gromov invariant ) is equivalent to the Seiberg-Witten invariants. It is believed that they contain the same information as the Donaldson-Thomas invariant and the Gopakumar-Vafa invariants , which are both integers.

Gromov-Witten invariants can also be formulated in the language of algebraic geometry. In some cases they agree with the classical counting invariants, but are generally also characterized by a law of composition for the “sticking together” of curves. The invariants can be summarized in the quantum cohomology ring of the manifold , a deformation of the ordinary cohomology. The law of composition of the invariants then makes the deformed cup product associative. ${\displaystyle X}$

The quantum cohomology ring is isomorphic to the symplectic Floer homology with its “ pair of pants ” product.

Applications in physics

Gromov-Witten invariants are of interest in string theory , in which the elementary particles are represented as excitations of 1 + 1-dimensional strings. “1 + 1” refers to the space-time dimension of the string “World Sheets”, which spreads out in a 10-dimensional space-time background. Since the module space of such surfaces (the number of its degrees of freedom) is infinitely dimensional and no mathematical measure is known for it, the path integral description of this theory lacks a mathematically strict basis.

In the case of mathematical models called topological string theories that have 6 space-time dimensions that make up a symplectic manifold, the situation is better. The world surfaces are parameterized by pseudo-alcoholomorphic curves, the module spaces of which are finite-dimensional. Gromov-Witten invariants are integrals over these module spaces and correspond to the path integrals in these theories. In particular, the partition function of the topological string theory for gender is equal to the generating function of the Gromov-Witten invariant for gender . ${\displaystyle g}$ ${\displaystyle g}$

literature

• Dusa McDuff , Dietmar Salamon : J-Holomorphic Curves and Symplectic Topology (= American Mathematical Society. Colloquium Publications 52). American Mathematical Society, Providence RI 2004, ISBN 0-8218-3485-1 .
• Sergei Piunikhin, Dietmar Salamon, Matthias Schwarz: Symplectic Floer-Donaldson theory and quantum cohomology. In CB Thomas (Ed.): Contact and Symplectic Geometry (= Publications of the Newton Institute 8). Cambridge University Press, Cambridge et al. 1996, ISBN 0-521-57086-7 , pp. 171-200.