Coloured operads for quantum field theory

• Applicant: Dr. Marco Benini
• Subject Area: Mathematics
• Term: from 2017 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 367130593

Combining the theory of coloured operads with homotopical algebra, we propose a new operadic approach to locally covariant quantum field theory (LCQFT). This perspective leads to novel techniques for constructing quantum field theories (QFTs) on Lorentzian manifolds and it allows us to develop a natural homotopical generalisation to investigate the structure of quantum gauge theories.LCQFTs are described by functors assigning algebras to spacetimes, subject to a set of axioms reflecting the causal structure of spacetime. Constructing examples of LCQFTs is a difficult task: in addition to identifying a suitable functor, one has to verify a posteriori that it satisfies the desired axioms, which generically is very hard. As a consequence, there are currently only few successful constructions of LCQFTs.The primary goal of the proposed research is to circumvent these problems by developing an operadic approach to LCQFT that treats functoriality and the LCQFT axioms on the same footing. We shall encode all algebraic operations of LCQFT (and their compatibility conditions) in a single structure, namely a coloured operad, whose algebras will be genuine LCQFTs. This perspective will lead us to new techniques for constructing examples of LCQFTs. As these techniques rely on categorical properties of operads and their algebras, they have no counterpart in the traditional approach to LCQFT. In particular, as the LCQFT axioms are encoded in our coloured operad, all operadic constructions will automatically respect these axioms, making it unnecessary to confirm them a posteriori.The proposed research also provides an interesting non-trivial generalisation of the recent factorisation algebra approach of Costello and Gwilliam. While the latter is used to axiomatise QFTs on Riemannian manifolds in terms of algebras over a suitable coloured operad, we shall design our coloured operad to reproduce Lorentzian QFTs encoding Einstein causality, so that algebras localised in causally disjoint regions will commute. In order to achieve this result, we shall use the causal relations between spacetime regions to parametrise the "degree of commutativity" of the operations in our coloured operad, thus interpolating between the associative and the commutative operads.Our approach does not only reformulate LCQFT more effectively, but it also suggests its natural homotopical extension, which is particularly suitable to describe quantum gauge theories. As A-infinity algebras, which are associative algebras "up to homotopy", arise as algebras over a resolution of the associative operad, a resolution of the envisaged coloured operad is a natural candidate for the formulation of the LCQFT axioms "up to homotopy". Our final goal is to confirm that the proposed homotopical extension of LCQFT is capable to describe examples of quantum gauge theories.

• DFG Programme: Research Grants