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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 Metaplectic structure ： ウィキペディア英語版 Metaplectic structure In differential geometry, a metaplectic structure is the symplectic analog of spin structure on orientable Riemannian manifolds. A metaplectic structure on a symplectic manifold allows one to define the symplectic spinor bundle, which is the Hilbert space bundle associated to the metaplectic structure via the metaplectic representation, giving rise to the notion of a symplectic spinor field in differential geometry. Symplectic spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in establishing the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for symplectic spin geometry. == Formal definition == A metaplectic structure 〔 page 35 〕 on a symplectic manifold $(M,\; \backslash omega)$ is an equivariant lift of the symplectic frame bundle $\backslash pi\_\backslash colon\backslash to\; M\backslash ,$ with respect to the double covering $\backslash rho\backslash colon\; )\backslash to\; ).\backslash ,$ In other words, a pair $(,F\_)$ is a metaplectic structure on the principal bundle $\backslash pi\_\backslash colon\backslash to\; M\backslash ,$ when :a) $\backslash pi\_\backslash colon\backslash to\; M\backslash ,$ is a principal $)$-bundle over $M$, :b) $F\_\backslash colon\backslash to\; \backslash ,$ is an equivariant $2$-fold covering map such that :$\backslash pi\_\backslash circ\; F\_=\backslash pi\_$ and $F\_(q)=F\_()\backslash rho(q)$ for all $\backslash in$ and $q\backslash in\; ).$ The principal bundle $\backslash pi\_\backslash colon\backslash to\; M\backslash ,$ is also called the bundle of metaplectic frames over $M$. Two metaplectic structures $(,F\_)$ and $(,F\_)$ on the same symplectic manifold $(M,\; \backslash omega)$ are called equivalent if there exists a $)$-equivariant map $f\backslash colon\; \backslash to$ such that :$F\_\backslash circ\; f=F\_$ and $f(q)=f()q$ for all $\backslash in$ and $q\backslash in\; ).$ Of course, in this case $F\_$ and $F\_$ are two equivalent double coverings of the symplectic frame $)$-bundle $\backslash pi\_\backslash colon\backslash to\; M\backslash ,$ of the given symplectic manifold $(M,\; \backslash omega)$. 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「Metaplectic structure」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.