General Covariance from the Viewpoint of Stacks

TL;DR

This paper explores the mathematical structure of general covariance in field theories using stacks and groupoids, deriving a new expression for the stress-energy tensor and broadening the understanding of covariance in physics.

Contribution

It introduces a stack-theoretic approach to general covariance, connecting the tangent complex of the metric stack to a differential graded Lie algebra and deriving a novel stress-energy tensor expression.

Findings

Derived a new expression for the stress-energy tensor.

Connected the tangent complex to a differential graded Lie algebra.

Provided a broader formalism for understanding covariance in field theories.

Abstract

General covariance is a crucial notion in the study of field theories in curved spacetime. A field theory defined with respect to a semi-Riemannian metric is generally covariant if two metrics which are related by a diffeomorphism produce equivalent physics. From a purely mathematical perspective, this suggest that we try to understand the quotient stack of metrics modulo diffeomorphism. We'll use the language of groupoids to do this concretely. Then we'll inspect the tangent complex of this stack at a fixed metric, which when shifted up by one defines a differential graded Lie algebra. By considering the action of this Lie algebra on the observables for the Batalin-Vilkovisky free scalar field theory, we recover a novel expression of the stress-energy tensor for that example. We'll describe how this construction nicely encapsulates but also broadens the usual presentation in the physics literature and discuss applications of the formalism.