Green hyperbolic complexes on Lorentzian manifolds and quantizations of gauge field theories

Gauge field theories have a central role in modern mathematical physics. The space of solutions of a gauge field theory is described by the moduli problem associated with the derived critical locus of its action functional. As such it is expected to carry a (-1)-shifted Poisson structure. On the other hand, its quantization on Lorentzian manifolds, especially in the algebraic approach, requires an unshifted Poisson structure. In this thesis we develop a framework, covering many examples of derived critical loci of gauge-theoretic quadratic action functionals, including Abelian Chern-Simons theory and Maxwell p-forms, in which these (-1)-shifted Poisson structures can be unshifted. The key ingredient of our approach is a homological generalization of Green hyperbolic operators, called Green hyperbolic complexes. We define the latter through a generalization of retarded and advanced Green's operators, called retarded and advanced Green's homotopies. We show that these generalized notions admit homological variants of the main features of their ordinary counterparts. Namely, retarded and advanced Green's homotopies are unique up to contractible spaces of choices, they induce a retarded-minus-advanced quasi-isomorphism, replacing the causal propagator, and, together with a differential pairing generalizing fiber metrics, they lead to unshifted covariant and fixed-time Poisson structures which are compatible (up to homotopy) with the retarded-minus-advanced quasi-isomorphism. Furthermore, we exploit these constructions to quantize Green hyperbolic complexes in two alternative approaches: as time-orderable prefactorization algebras via BV quantization and as algebraic quantum field theories via canonical quantization. Finally, we compare the two approaches by constructing an explicit isomorphism of time-orderable prefactorization algebras from the BV quantization to the time-orderable prefactorization algebra canonically associated to the algebraic quantum field theory.