Green hyperbolic complexes and their quantization

Abstract: Green hyperbolic operators are linear differential operators on globally hyperbolic Lorentzian manifolds which admit retarded and advanced Green’s operators. It is well known that Green hyperbolicity allows to endow the space of observables of a field theory with a Poisson structure and thus to quantize it. Unfortunately, (linear) gauge field theories cannot be treated along these lines, unless a particular gauge is chosen. To overcome this issue we will introduce the notion of Green hyperbolic complexes, which instead cover many examples of linear gauge field theories. They are defined through a homological generalization of retarded and advanced Green’s operators, called retarded and advanced Green’s homotopies, and present homotopy coherent variants of the main properties of Green hyperbolic operators. In particular we will show that they can be endowed with a covariant Poisson structure, which turns out to be crucial for constructing an algebraic QFT that quantizes them. Furthermore, we will show that Green hyperbolic complexes can also be quantized as prefactorization algebras and a comparison of the two different approaches will be presented.