Homotopical Quantum Field Theory

✍ Scribed by Donald Yau

• Publisher: WSPC
• Year: 2020
• Tongue: English
• Leaves: 312
• Category: Library

✦ Synopsis

This book provides a general and powerful definition of homotopy algebraic quantum field theory and homotopy prefactorization algebra using a new coend definition of the Boardman-Vogt construction for a colored operad. All of their homotopy coherent structures are explained in details, along with a comparison between the two approaches at the operad level. With chapters on basic category theory, trees, and operads, this book is self-contained and is accessible to graduate students.

✦ Table of Contents

Preface
Contents
1. Introduction
1.1 Algebraic Quantum Field Theory
1.2 Homotopy Algebraic Quantum Field Theory
1.3 Homotopy Prefactorization Algebra
1.4 Comparison
1.5 Organization
2. Category Theory
2.1 Basics of Categories Definition
2.2 Examples of Categories
2.3 Limits and Colimits Definition 2.3.1.
2.4 Adjoint Functors
2.5 Symmetric Monoidal Categories
2.6 Monoids
2.7 Monads Definition
2.8 Localization
Theorem 2.8.3.
3. Trees
3.1 Graphs
3.2 Tree Substitution
3.3 Grafting
4. Colored Operads
4.1 Operads as Monoids
4.2 Operads in Terms of Generating Operations
4.3 Operads in Terms of Partial Compositions
4.4 Operads in Terms of Trees
4.5 Algebras over Operads
5. Constructions on Operads
5.1 Change-of-Operad Adjunctions
5.2 Model Category Structures
5.3 Changing the Base Categories
5.4 Localizations of Operads
5.5 Algebras over Localized Operads
6. Boardman-Vogt Construction of Operads
6.1 Overview
6.2 Commutative Segments
6.3 Coend Definition of the BV Construction
6.4 Augmentation
6.5 Homotopy Morita Equivalence
6.6 Filtration
7. Algebras over the Boardman-Vogt Construction
7.1 Overview
7.2 Coherence Theorem
7.3 Homotopy Coherent Diagrams
7.4 Homotopy Inverses
7.5 Aª-Algebras
7.6 Eª-Algebras
7.7 Homotopy Coherent Diagrams of Aª-Algebras
7.8 Homotopy Coherent Diagrams of Eª-Algebras
8. Algebraic Quantum Field Theories
8.1 From Haag-Kastler Axioms to Operads
8.2 AQFT as Functors
8.3 AQFT as Operad Algebras
8.4 Examples of AQFT
8.5 Homotopical Properties
9. Homotopy Algebraic Quantum Field Theories
9.1 Overview
9.2 Homotopy AQFT as Operad Algebras
9.3 Examples of Homotopy AQFT
9.4 Coherence Theorem
9.5 Homotopy Causality Axiom
9.6 Homotopy Coherent Diagrams
9.7 Homotopy Time-Slice Axiom
9.8 Objectwise
9.9 Homotopy Coherent Diagrams of
10. Prefactorization Algebras
10.1 Costello-Gwilliam Prefactorization Algebras
10.2 Configured Categories
10.3 Prefactorization Algebras as Operad Algebras
10.4 Pointed Diagram Structure
10.5 Commutative Monoid Structure
10.6 Diagrams of Modules over a Commutative Monoid
10.7 Diagrams of Commutative Monoids
10.8 Configured and Homotopy Morita Equivalences
11. Homotopy Prefactorization Algebras
11.1 Overview
11.2 Homotopy Prefactorization Algebras as Operad Algebras
11.3 Examples
11.4 Coherence Theorem
11.5 Homotopy Coherent Pointed Diagrams
11.6 Homotopy Time-Slice Axiom
11.7 Eª-Algebra Structure
11.8 Objectwise
11.9 Homotopy Coherent Diagrams of
11.10 Homotopy Coherent Diagrams of Eª-Algebras
12. Comparing Prefactorization Algebras and AQFT
12.1 Orthogonal Categories as Configured Categories
12.2 Configured Categories to Orthogonal Categories
12.3 Comparison Adjunctions
12.4 Examples of Comparison
12.5 Prefactorization Algebras from AQFT
List of Notations
Notation Page Description Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Bibliography
Index