Homotopical Quantum Field Theory

The following sections are included:

• Dedication
• Preface
• Contents

Algebraic quantum field theory and prefactorization algebra are two mathematical approaches to quantum field theory. One of the main aims of this book is to provide robust definitions of homotopy algebraic quantum field theories and homotopy prefactorization algebras using a new definition of the Boardman-Vogt construction of a colored operad. To compare the two mathematical approaches to quantum field theory as well as their homotopy coherent analogues, we work within the framework of operads. This approach allows us to employ the powerful machinery from operad theory to quantum field theory. In the rest of this introduction, we briefly introduce each of these topics without going into too much details.

In this chapter, we recall some basic concepts of category theory and some relevant examples. The reader who is familiar with basic category theory can just read the examples in Section 2.2 and skip the rest of this chapter. Our references for category theory are [Borceux (1994,b);Mac Lane (1998)]. Categories were originally introduced by Eilenberg and Mac Lane [Eilenberg and Mac Lane (1945)]…

One of the important descriptions of operads uses the language of trees, which we discuss in this chapter. The definitions of the Boardman-Vogt construction of a colored operad, homotopy algebraic quantum field theories, and homotopy prefactorization algebras also use trees. The following material on graphs and trees are adapted from [Yau and Johnson (2015)] Part 1, where much more details and many more examples can be found. In practice, since we mostly work with isomorphism classes of trees, it is sufficient to work pictorially as in the examples below.

In this chapter, we define colored operads and their algebras. A colored operad is a generalization of a category in which the domain of each morphism is a finite sequence of objects. Colored operads provide an efficient way to encode operations with multiple inputs and one output. This efficient bookkeeping aspect of operad theory is especially important when we discuss homotopy algebraic quantum field theories and homotopy prefactorization algebras, in which the desired structures are too complicated to encode without colored operads…

In this chapter, we discuss several important constructions and properties of colored operads. In Section 5.1 to Section 5.3, we discuss the category of algebras over a colored operad under a change of operads and a change of base categories. In Section 5.4 and Section 5.5 we study localizations of colored operads, analogous to localizations of categories, and algebras over localized operads. The material in the last two sections about localizations of operads is new. Localizations of operads are needed later when we discuss the time-slice axiom in prefactorization algebras…

In this chapter we define the Boardman-Vogt construction of a colored operad in a symmetric monoidal category as an entrywise coend and study its naturality properties.

This chapter is about the structure of algebras over the Boardman-Vogt construction of a colored operad and some key examples. The categorical setting is the same as before, so (M,⊗,𝟙) is a cocomplete symmetric monoidal closed category with an initial object ∅. Unless otherwise specified, ℭ is an arbitrary non-empty set.

This chapter is about algebraic quantum field theories in the operadic framework of [Benini et. al. (2017)]. In Section 8.1 we provide a brief description of the traditional Haag-Kastler approach to algebraic quantum field theories and how it may be generalized to an operadic framework. In Section 8.2 we discuss orthogonal categories and algebraic quantum field theories defined on them. In Section 8.3 we discuss the colored operads in [Benini et. al. (2017)] whose algebras are algebraic quantum field theories. Many examples are discussed in Section 8.4, including diagrams of (commutative) monoids, chiral conformal, Euclidean, and locally covariant quantum field theories, various flavors of quantum gauge theories, and quantum field theories on spacetimes with timelike boundary. In Section 8.5 we study homotopical properties of the category of algebraic quantum field theories…

In this chapter, we define homotopy algebraic quantum field theories on an orthogonal category $\overline{\text{C}}$. We observe that each of them has a homotopy coherent ∁-diagram structure and a compatible objectwise A_∞-algebra structure, and satisfies a homotopy coherent version of the causality axiom. If a set of morphisms in ∁ is chosen, then each homotopy algebraic quantum field theory also satisfies a homotopy coherent version of the time-slice axiom.

In this chapter, we define prefactorization algebras on a configured category as algebras over a suitable colored operad and observe their basic structure. In Section 10.1 we briefly review prefactorization algebras on a topological space in the original sense of Costello-Gwilliam [Costello and Gwilliam (2017)]. Configured categories are abstractions of the category Open(X) for a topological space X and are defined in Section 10.2. The colored operad and prefactorization algebras associated to a configured category are defined in Section 10.3. The coherence theorems for prefactorization algebras, with or without the time-slice axiom, are also recorded in that section…

In this chapter, we define homotopy prefactorization algebras on a configured category and study their structure.

In this chapter, we compare (homotopy) prefactorization algebras and (homotopy) algebraic quantum field theories…

The following sections are included:

• List of Notations
• Bibliography
• Index