标题： 四维中的SU(2)结构和广义相对论的Plebanski形式主义 显示英文标题

标题： SU(2) structures in four dimensions and Plebanski formalism for GR

摘要： 一个四维的SU(2)结构可以描述为三个2-形式Sigma^i属于Lambda^2(M)，i=1,2,3，满足Sigma^i wedge Sigma^j ~ delta^{ij}。这样的三元组定义了M上的一个黎曼度量。如果这个黎曼度量的全纯性包含在SU(2)中，则称该SU(2)结构是可积的。 众所周知，当且仅当2-形式是闭合的dSigma^i=0时，这种情况成立。 本文的主要目的是分析对于SU(2)结构可以构造的二阶导数微分同胚不变的作用泛函。 主要结果是，如果施加一个额外的要求，即作用泛函也具有SU(2)不变性，其中SU(2)以它的矢量表示作用于三元组Sigma^i，则存在唯一的此类作用泛函。 该作用泛函在SU(2)结构的内在扭率方面具有非常简单的表达式。 我们证明其临界点是关联度量为爱因斯坦的SU(2)结构。 我们描述的作用泛函也有一个一阶导数版本，我们展示了它如何与物理学文献中已知的GR的Plebanski形式有关。 显示英文摘要

摘要： An SU(2) structure in four dimensions can be described as a triple of 2-forms Sigma^i in Lambda^2(M), i=1,2,3 satisfying Sigma^i wedge Sigma^j ~ delta^{ij}. Such a triple defines a Riemannian signature metric on M. An SU(2) structure is said to be integrable if the holonomy of this Riemannian metric is contained in SU(2). It is well-known that this is the case if and only if the 2-forms are closed dSigma^i=0. The main purpose of the paper is to analyse the second order in derivatives diffeomorphism invariant action functionals that can be constructed for an SU(2) structure. The main result is that there is a unique such action functional if one imposes an additional requirement that the action is also SU(2) invariant, with SU(2) acting on the triple Sigma^i as in its vector representation. This action functional has a very simple expression in terms of the intrinsic torsion of the SU(2) structure. We show that its critical points are SU(2) structures whose associated metric is Einstein. The action we describe has also a first order in derivatives version, and we show how this is related to what in the physics literature is known as Plebanski formalism for GR.