Homotopical quantum field theory

311 Pages·2020·14.72 MB·English

by  Yau, Donald Ying

Homotopical
Quantum Field
Theory
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Homotopical
Quantum Field
Theory
Donald Yau
The Ohio State University at Newark, USA
World Scientific
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Library of Congress Control Number: 2019049784
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HOMOTOPICAL QUANTUM FIELD THEORY
Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd.
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ISBN 978-981-121-285-7
For any available supplementary material, please visit
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To Eun Soo and Jacqueline
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Preface
Algebraic quantum field theory and prefactorization algebra are two mathemati-
cal approaches to quantum field theory. In this monograph, using a new coend
definition of the Boardman-Vogt construction of a colored operad, we define ho-
motopy algebraic quantum field theories and homotopy prefactorization algebras,
and investigate their homotopy coherent structures. Homotopy coherent diagrams,
homotopy inverses, A -algebras, E -algebras, and E -modules arise naturally in
∞ ∞ ∞
this context. Each homotopy algebraic quantum field theory has the structure of
a homotopy coherent diagram of A -algebras and satisfies a homotopy coherent
∞
version of the causality axiom. When the time-slice axiom is defined for algebraic
quantum field theory, a homotopy coherent versionof the time-slice axiom is satis-
fiedbyeachhomotopyalgebraicquantumfieldtheory. Overeachtopologicalspace,
every homotopy prefactorization algebra has the structure of a homotopy coherent
diagram of E -modules over an E -algebra. To compare the two approaches, we
∞ ∞
construct a comparison morphism from the colored operad for (homotopy) prefac-
torization algebras to the colored operad for (homotopy) algebraic quantum field
theories, and study the induced adjunctions on algebras.
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Contents
Preface vii
1. Introduction 1
1.1 Algebraic Quantum Field Theory . . . . . . . . . . . . . . . . . . . . 1
1.2 Homotopy Algebraic Quantum Field Theory . . . . . . . . . . . . . 3
1.3 Homotopy Prefactorization Algebra . . . . . . . . . . . . . . . . . . . 5
1.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2. Category Theory 11
2.1 Basics of Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Examples of Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Limits and Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Adjoint Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Symmetric Monoidal Categories . . . . . . . . . . . . . . . . . . . . . 24
2.6 Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.7 Monads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.8 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3. Trees 39
3.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Tree Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Grafting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4. Colored Operads 53
4.1 Operads as Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Operads in Terms of Generating Operations. . . . . . . . . . . . . . 56
4.3 Operads in Terms of Partial Compositions. . . . . . . . . . . . . . . 59
4.4 Operads in Terms of Trees . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 Algebras over Operads. . . . . . . . . . . . . . . . . . . . . . . . . . . 67
ix