Plebanski complex

TL;DR

This paper introduces the Plebański complex, an elliptic differential complex derived from linearising equations for hyper-Kähler and Einstein 4-manifolds, revealing new geometric and analytical structures.

Contribution

It defines the Plebański complex, studies its properties, and shows its relation to known complexes, including its connection to Dirac operators and elliptic theory in 4-manifold geometry.

Findings

The Plebański complex is elliptic and encodes hyper-Kähler structures.

It extends to Einstein equations, fitting into the elliptic complex framework.

Operators in the complex assemble into Dirac operators, linking to spectral theory.

Abstract

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.