Factorization Algebras in Quantum Field Theory

Description

Factorization algebras are local-to-global objects that play a role in classical and quantum field theory which 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 first volume, the authors develop the theory of factorization algebras in depth, but with a focus upon examples exhibiting their use in field theory, such as the recovery of a vertex algebra from a chiral conformal field theory and a quantum group from Abelian Chern-Simons theory. Expositions of the relevant background in homological algebra, sheaves and functional analysis are also included, thus making this book ideal for researchers and graduates working at the interface between mathematics and physics. Contents......Page 8 1 Introduction......Page 12 1.1 The Motivating Example of Quantum Mechanics......Page 14 1.3 Prefactorization Algebras in Quantum Field Theory......Page 19 1.4 Comparisons with Other Formalizations of Quantum Field Theory......Page 22 1.5 Overview of This Volume......Page 27 1.6 Acknowledgments......Page 29 PART I PREFACTORIZATION ALGEBRAS......Page 32 2 From Gaussian Measures to Factorization Algebras......Page 34 2.1 Gaussian Integrals in Finite Dimensions......Page 36 2.2 Divergence in Infinite Dimensions......Page 38 2.3 The Prefactorization Structure on Observables......Page 42 2.4 From Quantum to Classical......Page 45 2.5 Correlation Functions......Page 47 2.6 Further Results on Free Field Theories......Page 50 2.7 Interacting Theories......Page 51 3.1 Prefactorization Algebras......Page 55 3.2 Associative Algebras from Prefactorization Algebras on R......Page 62 3.3 Modules as Defects......Page 63 3.4 A Construction of the Universal Enveloping Algebra......Page 70 3.5 Some Functional Analysis......Page 73 3.6 The Factorization Envelope of a Sheaf of Lie Algebras......Page 84 3.7 Equivariant Prefactorization Algebras......Page 90 PART II FIRST EXAMPLES OF FIELD THEORIES AND THEIR OBSERVABLES......Page 98 4.1 The Divergence Complex of a Measure......Page 100 4.2 The Prefactorization Algebra of a Free Field Theory......Page 104 4.3 Quantum Mechanics and the Weyl Algebra......Page 117 4.4 Pushforward and Canonical Quantization......Page 123 4.5 Abelian Chern–Simons Theory......Page 126 4.6 Another Take on Quantizing Classical Observables......Page 135 4.7 Correlation Functions......Page 140 4.8 Translation-Invariant Prefactorization Algebras......Page 142 4.9 States and Vacua for Translation Invariant Theories......Page 150 5.1 Vertex Algebras and Holomorphic Prefactorization Algebras on C......Page 156 5.2 Holomorphically Translation-Invariant Prefactorization Algebras......Page 160 5.3 A General Method for Constructing Vertex Algebras......Page 168 5.4 The βγ System and Vertex Algebras......Page 182 5.5 Kac–Moody Algebras and Factorization Envelopes......Page 199 PART III FACTORIZATION ALGEBRAS......Page 216 6.1 Factorization Algebras......Page 218 6.2 Factorization Algebras in Quantum Field Theory......Page 226 6.3 Variant Definitions of Factorization Algebras......Page 227 6.4 Locally Constant Factorization Algebras......Page 231 6.5 Factorization Algebras from Cosheaves......Page 236 6.6 Factorization Algebras from Local Lie Algebras......Page 241 7.1 Pushing Forward Factorization Algebras......Page 243 7.3 Pulling Back Along an Open Immersion......Page 251 7.4 Descent Along a Torsor......Page 252 8.1 Some Examples of Computations......Page 254 8.2 Abelian Chern–Simons Theory and Quantum Groups......Page 260 Appendix A Background......Page 284 Appendix B Functional Analysis......Page 321 Appendix C Homological Algebra in Differentiable Vector Spaces......Page 362 Appendix D The Atiyah–Bott Lemma......Page 385 References......Page 388 Index......Page 394