Linear Yang-Mills theory as a homotopy AQFT

Abstract

It is observed that the shifted Poisson structure (antibracket) on the solution complex of Klein-Gordon and linear Yang-Mills theory on globally hyperbolic Lorentzian manifolds admits retarded/advanced trivializations (analogs of retarded/advanced Green's operators). Quantization of the associated unshifted Poisson structure determines a unique (up to equivalence) homotopy algebraic quantum field theory (AQFT), i.e. a functor that assigns differential graded $\ast$-algebras of observables and fulfills homotopical analogs of the AQFT axioms. For Klein-Gordon theory the construction is equivalent to the standard one, while for linear Yang-Mills it is richer and reproduces the BRST/BV field content (gauge fields, ghosts and antifields).

31 pages - v2: Final version to appear in Communications in Mathematical Physics