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E-raamat: Factorization Algebras in Quantum Field Theory: Volume 2

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, (University of Massachusetts, Amherst)
  • Formaat: PDF+DRM
  • Sari: New Mathematical Monographs
  • Ilmumisaeg: 23-Sep-2021
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316730188
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Factorization Algebras in Quantum Field Theory: Volume 2Suurem pilt
  • Formaat: PDF+DRM
  • Sari: New Mathematical Monographs
  • Ilmumisaeg: 23-Sep-2021
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316730188
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Factorization algebras are local-to-global objects that play a role in classical and quantum field theory that is similar to the role of sheaves in geometry: they conveniently organize complicated information. Their local structure encompasses examples like associative and vertex algebras; in these examples, their global structure encompasses Hochschild homology and conformal blocks. In this second volume, the authors show how factorization algebras arise from interacting field theories, both classical and quantum, and how they encode essential information such as operator product expansions, Noether currents, and anomalies. Along with a systematic reworking of the Batalin–Vilkovisky formalism via derived geometry and factorization algebras, this book offers concrete examples from physics, ranging from angular momentum and Virasoro symmetries to a five-dimensional gauge theory.

Ideal for researchers and graduate students at the interface between mathematics and physics, this text develops quantum field theory from the ground up using a rich mix of modern mathematics. It provides a unified approach to deformation quantization, Hochschild homology, vertex algebras, conformal field theory, quantum groups, and gauge theory.

Arvustused

'The central achievement of the book is in its development of a formalism that leads to classical and quantum versions of Noether's theorem, itself a familiar topic in physics, using the language of factorization algebras Institutions employing mathematicians and theoretical physicists actively working in this area should acquire the book Recommended.' M. C. Ogilvie, Choice Connect ' perfectly suitable for self-study by an interested scholar with little to almost no previous exposure to factorization algebras, or for use as a reference text for a lecture series on the subject.' Domenico Fiorenza, MathSciNet

Muu info

This second volume shows how factorization algebras arise from interacting field theories, both classical and quantum.
Contents of Volume I xi
1 Introduction and Overview
1(14)
1.1 The Factorization Algebra of Classical Observablcs
1(1)
1.2 The Factorization Algebra of Quantum Observables
2(1)
1.3 The Physical Importance of Factorization Algebras
3(3)
1.4 Poisson Structures and Deformation Quantization
6(2)
1.5 The Noether Theorem
8(3)
1.6 Brief Orienting Remarks toward the Literature
11(1)
1.7 Acknowledgments
12(3)
PART I CLASSICAL FIELD THEORY
15(58)
2 Introduction to Classical Field Theory
17(3)
2.1 The Euler-Lagrange Equations
17(1)
2.2 Observables
18(1)
2.3 The Symplectic Structure
19(1)
2.4 The P0 Structure
19(1)
3 Elliptic Moduli Problems
20(23)
3.1 Formal Moduli Problems and Lie Algebras
21(4)
3.2 Examples of Elliptic Moduli Problems Related to Scalar Field Theories
25(3)
3.3 Examples of Elliptic Moduli Problems Related to Gauge Theories
28(4)
3.4 Cochains of a Local Algebra
32(2)
3.5 D-modules and Local Lqc Algebras
34(9)
4 The Classical Batalin-Vilkovisky Formalism
43(20)
4.1 The Classical B V Formalism in Finite Dimensions
43(2)
4.2 The Classical BV Formalism in Infinite Dimensions
45(3)
4.3 The Derived Critical Locus of an Action Functional
48(6)
4.4 A Succinct Definition of a Classical Field Theory
54(3)
4.5 Examples of Scalar Field Theories from Action Functionals
57(1)
4.6 Cotangent Field Theories
58(5)
5 The Observables of a Classical Field Theory
63(10)
5.1 The Factorization Algebra of Classical Observables
63(1)
5.2 The Graded Poisson Structure on Classical Observables
64(2)
5.3 The Poisson Structure for Free Field Theories
66(2)
5.4 The Poisson Structure for a General Classical Field Theory
68(5)
PART II QUANTUM FIELD THEORY
73(152)
6 Introduction to Quantum Field Theory
75(11)
6.1 The Quantum BV Formalism in Finite Dimensions
76(3)
6.2 The "Free Scalar Field" in Finite Dimensions
79(2)
6.3 An Operadic Description
81(1)
6.4 Equivariant BD Quantization and Volume Forms
82(1)
6.5 How Renormalization Group Flow Interlocks with the BV Formalism
83(1)
6.6 Overview of the Rest of This Part
84(2)
7 Effective Field Theories and Batalin-Vilkovisky Quantization
86(25)
7.1 Local Action Functionals
87(1)
7.2 The Definition of a Quantum Field Theory
88(11)
7.3 Families of Theories over Nilpotent dg Manifolds
99(6)
7.4 The Simplicial Set of Theories
105(4)
7.5 The Theorem on Quantization
109(2)
8 The Observables of a Quantum Field Theory
111(33)
8.1 Free Fields
111(4)
8.2 The BD Algebra of Global Observables
115(9)
8.3 Global Observables
124(2)
8.4 Local Observables
126(2)
8.5 Local Observables Form a Prefactorization Algebra
128(4)
8.6 Local Observables Form a Factorization Algebra
132(6)
8.7 The Map from Theories to Factorization Algebras Is a Map of Presheaves
138(6)
9 Further Aspects of Quantum Observables
144(35)
9.1 Translation Invariance for Field Theories and Observables
144(4)
9.2 Holomorphically Translation-Invariant Theories and Observables
148(6)
9.3 Renormalizability and Factorization Algebras
154(14)
9.4 Cotangent Theories and Volume Forms
168(9)
9.5 Correlation Functions
177(2)
10 Operator Product Expansions, with Examples
179(46)
10.1 Point Observables
179(6)
10.2 The Operator Product Expansion
185(3)
10.3 The OPE to First Order in h
188(9)
10.4 The OPE in the ø4 Theory
197(4)
10.5 The Operator Product for Holomorphic Theories
201(9)
10.6 Quantum Groups and Higher-Dimensional Gauge Theories
210(15)
PART III A FACTORIZATION ENHANCEMENT OF THE NOETHER THEOREM
225(135)
11 Introduction to the Noether Theorems
227(18)
11.1 Symmetries in the Classical BV Formalism
228(5)
11.2 Koszul Duality and Symmetries via the Classical Master Equation
233(6)
11.3 Symmetries in the Quantum BV Formalism
239(6)
12 The Noether Theorem in Classical Field Theory
245(44)
12.1 An Overview of the Main Theorem
245(1)
12.2 Symmetries of a Classical Field Theory
246(13)
12.3 The Factorization Algebra of Equivariant Classical Observables
259(4)
12.4 The Classical Noether Theorem
263(5)
12.5 Conserved Currents
268(2)
12.6 Examples of the Classical Noether Theorem
270(9)
12.7 The Noether Theorem and the Operator Product Expansion
279(10)
13 The Noether Theorem in Quantum Field Theory
289(36)
13.1 The Quantum Noether Theorem
289(5)
13.2 Actions of a Local Lx Algebra on a Quantum Field Theory
294(5)
13.3 Obstruction Theory for Quantizing Equivariant Theories
299(4)
13.4 The Factorization Algebra of an Equivariant Quantum Field Theory
303(1)
13.5 The Quantum Noether Theorem Redux
304(9)
13.6 Trivializing the Action on Factorization Homology
313(1)
13.7 The Noether Theorem and the Local Index Theorem
314(9)
13.8 The Partition Function and the Quantum Noether Theorem
323(2)
14 Examples of the Noether Theorems
325(35)
14.1 Examples from Mechanics
325(19)
14.2 Examples from Chiral Conformal Field Theory
344(10)
14.3 An Example from Topological Field Theory
354(6)
Appendix A Background 360(15)
Appendix B Functions on Spaces of Sections 375(10)
Appendix C A Formal Darboux Lemma 385(8)
References 393(6)
Index 399
Kevin Costello is Krembil William Rowan Hamilton Chair in Theoretical Physics at the Perimeter Institute for Theoretical Physics, Waterloo, Canada. He is an honorary member of the Royal Irish Academy and a Fellow of the Royal Society. He has won several awards, including the Berwick Prize of the London Mathematical Society (2017) and the Eisenbud Prize of the American Mathematical Society (2020). Owen Gwilliam is Assistant Professor in the Department of Mathematics and Statistics at the University of Massachusetts, Amherst.
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