The Calabi complex and Killing sheaf cohomology

TL;DR

This paper explores the Calabi complex and Killing sheaf cohomology, providing explicit formulas, local exactness proofs, and cohomology calculations relevant to linearized gravity and gauge theories on various backgrounds.

Contribution

It offers a novel review and explicit calculations of the Calabi complex's structure, connecting it to sheaf cohomology and extending tools for analyzing linearized gravity on different manifolds.

Findings

Explicit formulas for differential operators in the Calabi complex

Proof of local exactness of the Calabi complex

Cohomology calculations for black hole and cosmological manifolds

Abstract

It has recently been noticed that the degeneracies of the Poisson bracket of linearized gravity on constant curvature Lorentzian manifold can be described in terms of the cohomologies of a certain complex of differential operators. This complex was first introduced by Calabi and its cohomology is known to be isomorphic to that of the (locally constant) sheaf of Killing vectors. We review the structure of the Calabi complex in a novel way, with explicit calculations based on representation theory of GL(n), and also some tools for studying its cohomology in terms of of locally constant sheaves. We also conjecture how these tools would adapt to linearized gravity on other backgrounds and to other gauge theories. The presentation includes explicit formulas for the differential operators in the Calabi complex, arguments for its local exactness, discussion of generalized Poincar\'e duality, methods of computing the cohomology of locally constant sheaves, and example calculations of Killing sheaf cohomologies of some black hole and cosmological Lorentzian manifolds.