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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 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 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 ''x''1, ..., ''x''''n'' and ''f'' : ''C'' → ''X'' is pseudoholomorphic. The moduli space has real dimension ::: Let ::: denote the stabilization of the curve. Let ::: which has real dimension 6''g'' - 6 + 2''kn''. There is an evaluation map ::: The evaluation map sends the fundamental class of ''M'' 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 α1, ..., α''n'' homology classes in ''X'', such that the sum of the codimensions of β, α1, ..., α''n'' 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 α''i''. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gromov–Witten invariant」の詳細全文を読む スポンサード リンク
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