Operads for algebraic quantum field theory Marco Benini1,a, Alexander Schenkel2,b and Lukas Woike1,c 1 Fachbereich Mathematik, Universit¨at Hamburg, Bundesstr. 55, 20146 Hamburg, Germany. 2 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom. Email: a
[email protected] b
[email protected] c
[email protected] September 2017 Abstract We construct a colored operad whose category of algebras is canonically isomorphic to the cat- egory of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped with an additional structure that we call an orthogonality relation. This allows us to describe different types of quantum field theories, including theories on a fixed Lorentzian manifold, locally covariant theories and also chiral conformal and Euclidean theories. Moreover, because the colored operad depends functorially on the orthogonal category, we obtain adjunctions between cat- egories of different types of quantum field theories. These include novel and physically very interesting constructions, such as time-slicification and local-to-global extensions of quantum field theories. We compare the latter to Fredenhagen’s universal algebra. Keywords: algebraic quantum field theory, locally covariant quantum field theory, colored operads, change of color adjunctions, Fredenhagen’s universal algebra MSC 2010: 81Txx, 18D50 arXiv:1709.08657v1 [math-ph] 25 Sep 2017 1 Contents 1 Introduction and summary 2 2 Categorical preliminaries 5 2.1 Closed symmetric monoidal categories . .......... 5 2.2 Endsandcoends ................................... 7 2.3 Dayconvolution .................................. ... 9 2.4 Monoids,monadsandalgebras .