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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 Strominger's equations ： ウィキペディア英語版 Strominger's equations In heterotic string theory, the Strominger's equations are the set of equations that are necessary and sufficient conditions for spacetime supersymmetry. It is derived by requiring the 4-dimensional spacetime to be maximally symmetric, and adding a warp factor on the internal 6-dimensional manifold.〔Strominger, ''(Superstrings with Torsion )'', Nuclear Physics B274 (1986) 253-284〕 Consider a metric $\backslash omega$ on the real 6-dimensional internal manifold ''Y'' and a Hermitian metric ''h'' on a vector bundle ''V''. The equations are: # The 4-dimensional spacetime is Minkowski, i.e., $g=\backslash eta$. # The internal manifold ''Y'' must be complex, i.e., the Nijenhuis tensor must vanish $N=0$. # The Hermitian form $\backslash omega$ on the complex threefold ''Y'', and the Hermitian metric ''h'' on a vector bundle ''V'' must satisfy, ## $\backslash partial\backslash bar\backslash omega=i\backslash textF(h)\backslash wedge\; F(h)-i\backslash textR^(\backslash omega)\backslash wedge\; R^(\backslash omega),$ ## $d^\backslash omega=i(\backslash partial-\backslash bar)\backslash text||\backslash Omega\; ||,$ where $R^$ is the Hull-curvature two-form of $\backslash omega$, ''F'' is the curvature of ''h'', and $\backslash Omega$ is the holomorphic ''n''-form; ''F'' is also known in the physics literature as the Yang-Mills field strength. Li and Yau showed that the second condition is equivalent to $\backslash omega$ being conformally balanced, i.e., $d(||\backslash Omega\; ||\_\backslash omega\; \backslash omega^2)=0$.〔Li and Yau, ''(The Existence of Supersymmetric String Theory with Torsion )'', J. Differential Geom. Volume 70, Number 1 (2005), 143-181〕 # The Yang-Mills field strength must satisfy, ## $\backslash omega^\}=0,$ ## $F\_=F\_\}=0.$ These equations imply the usual field equations, and thus are the only equations to be solved. However, there are topological obstructions in obtaining the solutions to the equations; # The second Chern class of the manifold, and the second Chern class of the gauge field must be equal, i.e., $c\_2(M)=c\_2(F)$ # A holomorphic ''n''-form $\backslash Omega$ must exists, i.e., $h^=1$ and $c\_1=0$. In case ''V'' is the tangent bundle $T\_Y$ and $\backslash omega$ is Kähler, we can obtain a solution of these equations by taking the Calabi-Yau metric on $Y$ and $T\_Y$. Once the solutions for the Strominger's equations are obtained, the warp factor $\backslash Delta$, dilaton $\backslash phi$ and the background flux ''H'', are determined by # $\backslash Delta(y)=\backslash phi(y)+\backslash text$, # $\backslash phi(y)=\backslash frac\; \backslash text||\backslash Omega||+\backslash text$, # $H=\backslash frac(\backslash bar-\backslash partial)\backslash omega.$ ==References== * Cardoso, Curio, Dall'Agata, Lust, Manousselis, and Zoupanos, ''(Non-Kähler String Backgrounds and their Five Torsion Classes )'', hep-th/0211118 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「Strominger's equations」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.