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	The ~~name~~ of ~~the writer is Numbers~~. ~~Years ago we moved~~ to ~~North Dakota~~. The ~~factor she adores most~~ is ~~physique developing~~ and ~~now she~~ is ~~attempting~~ to ~~make money~~ with it. ~~I am~~ a ~~meter reader~~.<br><br>~~Look at my weblog~~ ... [~~http~~://~~www~~.~~hotporn123~~.~~com~~/~~blog~~/~~154316 http~~://~~www~~.~~hotporn123~~.~~com~~/~~blog~~/~~154316~~]		{{string theory}}

			In [[mathematics]], specifically in [[symplectic topology]] and [[algebraic geometry]], '''Gromov–Witten (GW) invariants''' are [[rational number]]s that, in certain situations, count [[pseudoholomorphic curve]]s meeting prescribed conditions in a given [[symplectic manifold]]. The GW invariants may be packaged as a [[Homology (mathematics)\|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\|type IIA]] [[string theory]]. They are named for [[Mikhail Leonidovich Gromov\|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 manifold\|closed]] [[symplectic manifold]] of dimension 2''k'',
			*''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 are associated to the 4-tuple: (''X'', ''A'', ''g'', ''n''). Let <math>\overline{\mathcal{M}}_{g, n}</math>: the [[Deligne–Mumford moduli space of curves]] of genus ''g'' with ''n'' marked points and <math>\overline{\mathcal{M}}_{g, n}(X, A)</math> denote the moduli space of [[stable map]]s into ''X'' of class ''A'', for some chosen [[almost complex manifold\|almost complex structure]] ''J'' on ''X'' compatible with its symplectic form. The elements of <math>\overline{\mathcal{M}}_{g, n}(X, A)</math> are of the form:
			:::<math>(C, x_1, \cdots, x_n, f)</math>,
			where ''C'' is a (not necessarily stable) curve with ''n'' marked points ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> and ''f'' : ''C'' → ''X'' is pseudoholomorphic. The moduli space has real dimension
			:::<math>d := 2 c_1^X (A) + (2k - 6) (1 - g) + 2 n.</math>

			Let
			:::<math>\mathrm{st}(C, x_1, \cdots, x_n) \in \overline{\mathcal{M}}_{g, n} (X, A)</math>
			denote the [[stable map\|stabilization]] of the curve. Let
			:::<math>Y := \overline{\mathcal{M}}_{g, n} \times X^n,</math>
			which has real dimension 6''g'' - 6 + 2''kn''. There is an evaluation map
			:::<math> \begin{cases}
			\mathrm{ev}: \overline{\mathcal{M}}_{g, n}(X, A) \to Y \\
			\mathrm{ev}(C, x_1, \cdots, x_n, f) = \left(\mathrm{st}(C, x_1, \cdots, x_n), f(x_1), \cdots, f(x_n) \right).
			\end{cases}</math>
			The evaluation map sends the [[fundamental class]] of ''M'' to a ''d''-dimensional rational homology class in ''Y'', denoted
			:::<math>GW_{g, n}^{X, A} \in H_d(Y, \mathbf{Q}).</math>
			In a sense, this homology class is the '''Gromov–Witten invariant''' of ''X'' for the data ''g'', ''n'', and ''A''. It is an [[invariant (mathematics)\|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 <math>\overline{\mathcal{M}}_{g, n}</math> and α<sub>1</sub>, ..., α<sub>''n''</sub> homology classes in ''X'', such that the sum of the codimensions of β, α<sub>1</sub>, ..., α<sub>''n''</sub> equals ''d''. These induce homology classes in ''Y'' by the [[Künneth formula]]. Let
			:<math>GW_{g, n}^{X, A}(\beta, \alpha_1, \ldots, \alpha_n) := GW_{g, n}^{X, A} \cdot \beta \cdot \alpha_1 \cdot \cdots \cdot \alpha_n \in H_0(Y, \mathbf{Q}),</math>
			where <math>\cdot</math> denotes the [[intersection theory\|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 α<sub>''i''</sub>.

			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 class]]es 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 manifold\|almost complex structure]] ''J'', for which the [[linearization]] ''D'' of the <math>\bar \partial_{j, J}</math>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 manifold]]s 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 argument using Kuranishi structures.

			The main computational technique is '''localization'''. This applies when ''X'' is [[toric geometry\|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\|Atiyah]] and [[Raoul Bott\|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. 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 ''X'' of real codimension two.

			==Related invariants and other constructions==
			The GW invariants are closely related to a number of other concepts in geometry, including the [[Donaldson invariant]]s and [[Seiberg–Witten invariant]]s in the symplectic category, and [[Donaldson–Thomas theory]] in the algebraic category. For compact symplectic four-manifolds, [[Clifford Taubes]] showed that a variant of the GW invariants (see [[Taubes's Gromov invariant]]) are equivalent to the Seiberg–Witten invariants. They are conjectured to contain the same information as [[Donaldson–Thomas invariant]]s and [[BPS state\|Gopakumar–Vafa invariant]]s, 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 ''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 ring is known to be isomorphic to the symplectic [[Floer homology]] with its pair-of-pants product.

			==Application in physics==
			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 particle]]s, is made of tiny [[string (physics)\|string]]s. As a string travels through spacetime it traces out a surface, called the worldsheet of the string. Unfortunately, the moduli space of such parametrized surfaces, at least ''a priori'', is infinite-dimensional; no appropriate [[measure (mathematics)\|measure]] on this space is known, and thus the [[path integral formulation\|path integrals]] of the theory lack a rigorous definition.

			The situation improves in the variation known as [[topological string theory\|closed A-model]]. 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. GW invariants, as integrals over these moduli spaces, are then path integrals of the theory. In particular, the [[Helmholtz free energy\|free energy]] of the A-model at [[genus (mathematics)\|genus]] ''g'' is the [[generating function]] of the genus ''g'' GW invariants.

			==References==
			*{{Cite book \|last=McDuff \|first=Dusa \|lastauthoramp=yes \|last2=Salamon \|first2=Dietmar \|year=2004 \|title=J-Holomorphic Curves and Symplectic Topology \|publisher=American Mathematical Society colloquium publications \|isbn=0-8218-3485-1 }} An analytically flavoured overview of Gromov–Witten invariants and Quantum cohomology for symplectic manifolds, very technically complete
			*{{Cite book \|last=Piunikhin \|first=Sergey \|last2=Salamon \|first2=Dietmar \|lastauthoramp=yes \|last3=Schwarz \|first3=Matthias \|year=1996 \|chapter=Symplectic Floer–Donaldson theory and quantum cohomology \|editor-first=C. B. \|editor-last=Thomas \|title=Contact and Symplectic Geometry \|pages=171–200 \|publisher=Cambridge University Press \|location= \|isbn=0-521-57086-7 }}
			*Joachim Kock, Israel Vainsencher, ''An invitation to quantum cohomology: Kontsevich's formula for rational plane curves'' A nice introduction with history and exercises to the formal notion of [[moduli space]], treats extensively the case of projective spaces using the basics in the language of [[Scheme (mathematics)\|schemes]].

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			[[Category:Symplectic topology]]
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			[[Category:String theory]]

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