Green Hyperbolic Complexes on Lorentzian Manifolds
Abstract
We develop a homological generalization of Green hyperbolic operators, called Green hyperbolic complexes, which cover many examples of derived critical loci for gaugetheoretic 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 retardedminusadvanced cochain map is a quasiisomorphism, (2) a differential pairing (generalizing the usual fiberwise metric) on a Green hyperbolic complex leads to covariant and fixedtime Poisson structures and (3) the retardedminusadvanced cochain map is compatible with these Poisson structures up to homotopy.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 October 2023
 DOI:
 10.1007/s00220023048075
 arXiv:
 arXiv:2207.04069
 Bibcode:
 2023CMaPh.403..699B
 Keywords:

 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 Mathematics  Algebraic Topology;
 81T70;
 81T20;
 18G35;
 58J45
 EPrint:
 41 pages  Accepted for publication in Communications in Mathematical Physics