Related papers: Coloring Operads for Algebraic Field Theory
We generalize the operadic approach to algebraic quantum field theory [arXiv:1709.08657] to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is…
We construct a colored operad whose category of algebras is the category 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…
This text, based on the author's Bachelor's thesis, introduces the theory of Algebraic Operads, a mathematical formalism that provides a unifying framework for modern algebra. We demonstrate how the fundamental theories of associative,…
This survey provides an elementary introduction to operads and to their applications in homotopical algebra. The aim is to explain how the notion of an operad was prompted by the necessity to have an algebraic object which encodes higher…
In this series of lectures directed towards a mainly mathematically oriented audience I try to motivate the use of operator algebra methods in quantum field theory. Therefore a title as ``why mathematicians are/should be interested in…
Operads may be represented as symmetric monoidal functors on a small symmetric monoidal category. We discuss the axioms which must be imposed on a symmetric monoidal functor in order that it give rise to a theory similar to the theory of…
Motivated by gauge theory, we develop a general framework for chain complex valued algebraic quantum field theories. Building upon our recent operadic approach to this subject, we show that the category of such theories carries a canonical…
Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -- the theory of…
Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form bigger and more complex ones. Coming historically from algebraic…
Process theories provide a powerful framework for describing compositional structures across diverse fields, from quantum mechanics to computational linguistics. Traditionally, they have been formalized using symmetric monoidal categories…
We review recent progress in operator algebraic approach to conformal quantum field theory. Our emphasis is on use of representation theory in classification theory. This is based on a series of joint works with R. Longo.
Involutive category theory provides a flexible framework to describe involutive structures on algebraic objects, such as anti-linear involutions on complex vector spaces. Motivated by the prominent role of involutions in quantum (field)…
In various subjects including mathematics, one can hope to use mathematical thinking well when the right kinds of algebraic structure to consider can be discovered or spotted. Therefore, it would help to understand kinds of algebraic…
In this paper, we describe a general theory of modules over an algebra over an operad. We also study functors between categories of modules. Specializing to the operad E_d of little d-dimensional disks, we show that each (d-1)-manifold…
Algebraic operads provide a powerful tool to understand the homotopy theory of the types of (co)algebras they encode. So far, the principal results and methods that this theory provides were only available in characteristic zero. The reason…
Algebraic quantum field theory and prefactorization algebra are two mathematical approaches to quantum field theory. In this monograph, using a new coend definition of the Boardman-Vogt construction of a colored operad, we define homotopy…
This chapter provides a non-technical overview and motivation for the recent interactions between algebraic quantum field theory (AQFT) and rather abstract mathematical disciplines such as operads, model categories and higher categories.
A brief overview of the recent developments of operadic and higher categorical techniques in algebraic quantum field theory is given. The relevance of such mathematical structures for the description of gauge theories is discussed.
The essential parts of the operad algebra are concisely presented, which should be useful when confronting with the operadic physics. It is also clarified how the Gerstenhaber algebras can be associated with the linear pre-operads (comp…
This is a set of lecture notes on the operator algebraic approach to 2-dimensional conformal field theory. Representation theoretic aspects and connections to vertex operator algebras are emphasized. No knowledge on operator algebras or…