# Green hyperbolic complexes on Lorentzian manifolds

July 08, 2022

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.

Keywords:

homological methods in gauge theory, globally hyperbolic Lorentzian manifolds, Green hyperbolic operators, dg-categories