An algebraic quantum field theory is in particular a diagram of algebras – i.e., a functor from a small category to a category of algebras. To first order, the deformations and symmetries of a diagram of algebras are described by its asimplicial Hochschild cohomology. I have constructed an operad, Quilt, that describes natural operations on the Hochschild bicomplex of a diagram of algebras, and used this to describe the Gerstenhaber algebra structure of the cohomology. I have further used Quilt to construct an L-infinity algebra structure on the bicomplex itself. This determines a (generalized) Maurer-Cartan equation, the solutions of which are precisely the finite deformations of the diagram of algebras.
These structures may form the basis for a general theory of deformations of algebraic quantum field theories, along the lines of Kontsevich’s work on formality and deformation quantization.