Plebanski complex

TL;DR

This work develops a Plebański complex to encode the Einstein equations in a first-order, elliptic-differential framework, mirroring the YM instanton complex. By exploiting an ${\rm SU}(2)$-structure through a perfect triple $\Sigma^i$, the authors construct a 3-term complex with maps $d_1:d\!M\to S$, $d_2:S\to E\otimes \Lambda^1$, and $d_3:E\otimes \Lambda^1\to E$, and prove its ellipticity via a detailed symbol analysis. They identify inner products on the representation-theoretic components and show that the linearised Einstein equations arise from $d_2^* d_2$, constructing a gauge-fixed elliptic operator $\tilde{D}$ that squares to the Laplacian and decomposes into two Dirac-type blocks $D_4$ and $D_{12}$ after a suitable change of variables. A corresponding Lagrangian is derived, revealing a well-posed, gauge-fixed second-order action and a clear path to applications in numerical relativity, hyper-Kähler/K3 constructions, and potential extensions to Spin(7) holonomy contexts. The framework provides a robust bridge between Plebański's first-order formulation and elliptic operator theory, enabling precise control over gauge, adjoints, and spectral properties of the linearised gravitational system.

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 $Λ^0(\mathfrak{g}) \to Λ^1(\mathfrak{g})\to Λ^2_+(\mathfrak{g})$. 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 Plebański complex. This is a differential complex that arises by linearisation of the equations implying that a Riemannian 4-manifold is hyper-Kähler. We recall that these are most naturally stated as the condition that there exists a perfect $Σ^i\wedge Σ^j\simδ^{ij}$ triple $Σ^i, i=1,2,3$ of 2-forms that are closed $dΣ^i=0$. The Riemannian metric is encoded by the 2-forms $Σ^i$. We show that what results is an elliptic differential complex $TM \to S\to E\times Λ^1 \to E$, 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 Plebański 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 Plebański 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.

Theorems & Definitions (3)

• Theorem A
• Theorem B
• Theorem C