Green hyperbolic complexes on Lorentzian manifolds

TL;DR

This paper introduces Green hyperbolic complexes, a homological extension of Green hyperbolic operators, applicable to gauge theories in Lorentzian manifolds, and explores their properties and structures.

Contribution

It develops the theory of Green hyperbolic complexes, including their retarded and advanced homotopies, and proves key homological properties and compatibility with Poisson structures.

Findings

Retarded-minus-advanced cochain map is a quasi-isomorphism

Differential pairing induces covariant Poisson structures

Compatibility of cochain map with Poisson structures up to homotopy

Abstract

We develop a homological generalization of Green hyperbolic operators, called Green hyperbolic complexes, which cover many examples of derived critical loci for gauge-theoretic quadratic action functionals in Lorentzian signature. We define Green hyperbolic complexes through a generalization of retarded and advanced Green's operators, called retarded and advanced Green's homotopies, which are shown to be unique up to a contractible space of choices. We prove homological generalizations of the most relevant features of Green hyperbolic operators, namely that (1) the retarded-minus-advanced cochain map is a quasi-isomorphism, (2) a differential pairing (generalizing the usual fiber-wise metric) on a Green hyperbolic complex leads to covariant and fixed-time Poisson structures and (3) the retarded-minus-advanced cochain map is compatible with these Poisson structures up to homotopy.