标题： 佩莱班斯基复数 显示英文标题

标题： Plebanski complex

摘要： 正如众所周知的，4-流形上的瞬子方程的线性化会产生一个椭圆复形，即截断的（扭曲）Hodge复形$\Lambda^0(\mathfrak{g}) \to \Lambda^1(\mathfrak{g})\to \Lambda^2_+(\mathfrak{g})$。 此外，完整的YM方程的线性化也符合这个框架，因为它由第二个映射及其伴随映射组成。 我们定义并研究了我们称之为Plebański复形的性质。 这是一个微分复形，它通过导致黎曼4-流形为超凯勒的方程的线性化而出现。 我们回忆一下，这些条件最自然地表述为存在一个完美的$\Sigma^i\wedge \Sigma^j\sim\delta^{ij}$三元组$\Sigma^i, i=1,2,3$的2-形式，这些2-形式是闭合的$d\Sigma^i=0$。 黎曼度量由2-形式$\Sigma^i$编码。 我们证明结果是一个椭圆微分复形$TM \to S\to E\times \Lambda^1 \to E$，其中$S$是完美三元组空间的切空间，且$E=\mathbb{R}^3$。 我们还证明，如同瞬子方程的情况一样，完整的爱因斯坦方程$Ric=0$也符合这个框架，其线性化由第二个映射及其伴随映射给出。 我们的第二个结果涉及普莱巴内斯基复形定义的椭圆算子。 在瞬子复形的情况下，复形中出现的算子与其伴随算子一起组合成狄拉克算子。 我们展示了对于普莱巴内斯基复形同样成立。 通过适当的伴随算子，算子组合成一个平方等于拉普拉斯算子且由两个狄拉克算子的直和给出的椭圆算子。 显示英文摘要

摘要： As is very well-known, linearisation of the instanton equations on a 4-manifold gives rise to an elliptic complex of differential operators, the truncated (twisted) Hodge complex $\Lambda^0(\mathfrak{g}) \to \Lambda^1(\mathfrak{g})\to \Lambda^2_+(\mathfrak{g})$. Moreover, the linearisation of the full YM equations also fits into this framework, as it is given by the second map followed by its adjoint. We define and study properties of what we call the Pleba\'nski complex. This is a differential complex that arises by linearisation of the equations implying that a Riemannian 4-manifold is hyper-K\"ahler. We recall that these are most naturally stated as the condition that there exists a perfect $\Sigma^i\wedge \Sigma^j\sim\delta^{ij}$ triple $\Sigma^i, i=1,2,3$ of 2-forms that are closed $d\Sigma^i=0$. The Riemannian metric is encoded by the 2-forms $\Sigma^i$. We show that what results is an elliptic differential complex $TM \to S\to E\times \Lambda^1 \to E$, where $S$ is the tangent space to the space of perfect triples, and $E=\mathbb{R}^3$. We also show that, as in the case with instanton equations, the full Einstein equations $Ric=0$ also fit into this framework, their linearisation being given by the second map followed by its adjoint. Our second result concerns the elliptic operator that the Pleba\'nski complex defines. In the case of the instanton complex, operators appearing in the complex supplemented with their adjoints assemble to give the Dirac operator. We show how the same holds true for the Pleba\'nski complex. Supplemented by suitable adjoints, operators assemble into an elliptic operator that squares to the Laplacian and is given by the direct sum of two Dirac operators.