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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 Supermetric ： ウィキペディア英語版 Supermetric Supersymmetry gauge theory including supergravity is mainly developed as a Yang - Mills type theory with spontaneous breakdown of supersymmetries. There are various superextensions of pseudo-orthogonal Lie algebras and the Poincaré Lie algebra. The nonlinear realization of some Lie superalgebras have been studied. However, supergravity introduced in SUSY gauge theory has no geometric feature as a supermetric. In gauge theory on a principal bundle $P\backslash to\; M$ with a structure group $K$, spontaneous symmetry breaking is characterized as a reduction of $K$ to some closed subgroup $H$. By the well-known theorem, such a reduction takes place if and only if there exists a global section $h$ of the quotient bundle $P/H\backslash to\; M$. This section is treated as a classical Higgs field. In particular, this is the case of gauge gravitation theory where $P=FM$ is a principal frame bundle of linear frames in the tangent bundle $TM$ of a world manifold $M$. In accordance with the geometric equivalence principle, its structure group $GL(n,\backslash mathbb\; R)$ is reduced to the Lorentz group $O(1,3)$, and the associated global section of the quotient bundle $FM/O(1,3)$ is a pseudo-Riemannian metric on $M$, i.e., a gravitational field in General Relativity. Similarly, a supermetric can be defined as a global section of a certain quotient superbundle. It should be emphasized that there are different notions of a supermanifold. Lie supergroups and principal superbundles are considered in the category of $G$-supermanifolds. Let $\backslash widehat\; P\backslash to\; \backslash widehat\; M$ be a principal superbundle with a structure Lie supergroup $\backslash widehat\; K$, and let $\backslash widehat\; H$ be a closed Lie supersubgroup of $\backslash widehat\; K$ such that $\backslash widehat\; K\backslash to\; \backslash widehat\; K/\backslash widehat\; H$ is a principal superbundle. There is one-to-one correspondence between the principal supersubbundles of $\backslash widehat\; P$ with the structure Lie supergroup $\backslash widehat\; H$ and the global sections of the quotient superbundle $\backslash widehat\; P/\backslash widehat\; H\backslash to\; \backslash widehat\; M$ with a typical fiber $\backslash widehat\; K/\backslash widehat\; H$. A key point is that underlying spaces of $G$-supermanifolds are smooth real manifolds, but possessing very particular transition functions. Therefore, the condition of local triviality of the quotient $\backslash widehat\; K\backslash to\; \backslash widehat\; K/\backslash widehat\; H$ is rather restrictive. It is satisfied in the most interesting case for applications when $\backslash widehat\; K$ is a supermatrix group and $\backslash widehat\; H$ is its Cartan supersubgroup. For instance, let $\backslash widehat\; P=F\backslash widehat\; M$ be a principal superbundle of graded frames in the tangent superspaces over a supermanifold $\backslash widehat\; M$ of even-odd dimensione $(n,2m)$. If its structure general linear supergroup $\backslash widehat\; K=\backslash widehat(n|2m;\; \backslash Lambda)$ is reduced to the orthogonal-symplectic supersubgroup $\backslash widehat\; H=\backslash widehatp(n|m;\backslash Lambda)$, one can think of the corresponding global section of the quotient superbundle $F\backslash widehat\; M/\backslash widehat\; H\backslash to\; \backslash widehat\; M$ as being a supermetric on a supermanifold $\backslash widehat\; M$. In particular, this is the case of a super-Euclidean metric on a superspace $B^$. == References == * Deligne, P. and Morgan, J. (1999) Notes on supersymmetry (following Joseph Bernstein). In: ''Quantum Field Theory and Strings: A Course for Mathematicians, Vol. 1'' (Providence, RI: Amer. Math. Soc.) pp. 41-97 ISBN 978-0-8218-1198-6. * Sardanashvily, G. (2008) Supermetrics on supermanifolds, ''Int. J. Geom. Methods Mod. Phys.'' 5, 271. 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「Supermetric」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.