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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Gromov–Witten invariant ： ウィキペディア英語版 Gromov–Witten invariant 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 $\backslash overline$ be the Deligne–Mumford moduli space of curves of genus ''g'' with ''n'' marked points and $\backslash overline(X,\; A)$ 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 $\backslash overline(X,\; A)$ are of the form: :::$(C,\; x\_1,\; \backslash ldots,\; x\_n,\; f)$, 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 :::$d\; :=\; 2\; c\_1^X\; (A)\; +\; (2k\; -\; 6)\; (1\; -\; g)\; +\; 2\; n.$ Let :::$\backslash mathrm(C,\; x\_1,\; \backslash ldots,\; x\_n)\; \backslash in\; \backslash overline\; (X,\; A)$ denote the stabilization of the curve. Let :::$Y\; :=\; \backslash overline\; \backslash times\; X^n,$ which has real dimension 6''g'' - 6 + 2''kn''. There is an evaluation map :::$\backslash begin$ \mathrm: \overline(X, A) \to Y \\ \mathrm(C, x_1, \ldots, x_n, f) = \left(\mathrm(C, x_1, \ldots, x_n), f(x_1), \ldots, f(x_n) \right). \end The evaluation map sends the fundamental class of ''M'' to a ''d''-dimensional rational homology class in ''Y'', denoted :::$GW\_^\; \backslash in\; H\_d(Y,\; \backslash mathbf).$ 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 $\backslash overline$ 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 :$GW\_^(\backslash beta,\; \backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_n)\; :=\; GW\_^\; \backslash cdot\; \backslash beta\; \backslash cdot\; \backslash alpha\_1\; \backslash cdots\; \backslash alpha\_n\; \backslash in\; H\_0(Y,\; \backslash mathbf),$ where $\backslash cdot$ 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 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.