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 the first volume of this set, the authors develop the theory of factorization algebras in depth, 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 ChernSimons theory. In the second volume, they 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.
Ideal for researchers and graduate students at the interface between mathematics and physics, this two-volume set discusses factorization algebras. The first volume highlights examples exhibiting their use in field theory, while the second develops quantum field theory from the ground up using a rich mix of modern mathematics.
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Over two volumes, the authors develop factorization algebras, creating an essential reference for graduates and researchers.
Volume 1:
1. Introduction; Part I. Prefactorization Algebras:
2. From
Gaussian Measures to Factorization Algebras;
3. Prefactorization Algebras and
Basic Examples; Part II. First Examples of Field Theories:
4. Free Field
Theories;
5. Holomorphic Field Theories and Vertex Algebras; Part III.
Factorization Algebras:
6. Factorization Algebras Definitions and
Constructions;
7. Formal Aspects of Factorization Algebras;
8. Factorization
Algebras Examples; Appendix A. Background; Appendix B. Functional Analysis;
Appendix C. Homological Algebra in Differentiable Vector Spaces; Appendix D.
The Atiyah-Bott Lemma; References; Index; Volume 2:
1. Introduction and
Overview; Part I. Classical Field Theory:
2. Introduction to Classical Field
Theory;
3. Elliptic Moduli Problems;
4. The Classical Batalin-Vilkovisky
Formalism;
5. The Observables of a Classical Field Theory; Part II. Quantum
Field Theory:
6. Introduction to Quantum Field Theory;
7. Effective Field
Theories and Batalin-Vilkovisky Quantization;
8. The Observables of a Quantum
Field Theory;
9. Further Aspects of Quantum Observables;
10. Operator Product
Expansions, with Examples; Part III. A Factorization Enhancement of Noether's
Theorem:
11. Introduction to Noether's Theorems;
12. Noether's Theorem in
Classical Field Theory;
13. Noether's Theorem in Quantum Field Theory;
14.
Examples of the Noether Theorems; Appendix A. Background; Appendix B.
Functions on Spaces of Sections; Appendix C. A Formal Darboux Lemma;
References; Index.
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.
