Posted in
Organiser(s):
Christian Blohmann
Date:
Wed, 23/04/2014 - 08:00 - Wed, 25/06/2014 - 22:00 Plan of the seminar:
- Organizational meeting.
Motivation, brief introduction to the subject, discussion of the seminar plan, distribution of talks.
Literature: Introduction in: [CG]
- Prefactorization algebras.
Definition of a prequantization algebra; example: associative algebras as prefactorization algebras; example: symmetric algebra of a precosheaf; recovering an associative algebra from its prefactorization algebra; the universal enveloping prefactorization algebra of a Lie algebra (Prop. 2.3.0.5).
Literature: Sec. 2.1.1, 2.2, 2.3 in [CG]
- The factorization envelope.
Definition of dg-Lie algebras; constructions on sections of vector bundles (Sec. 2.4); prefactorization envelope of a sheaf of Lie algebras; definition of a local dg Lie algebra; twisted factorization envelope; example: Kac-Moody factorization algebra.
Literature: 2.4, 2.5 in [CG]
- Free 1-dimensional field theory and standard quantum mechanics.
Classical observables of a free scalar field theory; general free field theory; example: the 1-dimensional case; Poisson bracket; recovering the Weyl algebra.
Literature: 3.1, 3.2 in [CG]
- Factorization algebras: definitions, motivation, and easy examples.
Weiss topology; definition of factorization algebras; locally constant factorization algebras; relation to $E_n$-algebras; factorization algebras from cosheaves; factorization algebras from local Lie algebras; example: free 1-dimensional scalar field theory.
Literature: Sec. 5 in [CG] - Further structures on factorization algebras.
Hopf operads; strucutred factorization algebras; commutative factorization algebras; $P_0$ Poisson structure; BD quantization; the statement of the quantization theorem
Literature: Sec. 7 in [CG]; more literature will be provided - Elliptic moduli problems.
Formal moduli problems and Lie algebras; relation to $L_\infty$-algebras; elliptic moduli problems; example: free scalar field theory; example: $\phi^4$-theory; example: gauge theory
Literature: 9.1, 9.2, 9.3 in [CG]
- The classical BV formalism:
BV formalism in finite dimension; infinite dimension; derived critical locus of action functional; symplectic structure; reformulation of classical field theory; example: free field theories; cotangent field theories; more examples.
Literature: Sec 10 in [CG]
- Outlook: The factorization Noether theorem.
Literature: [CG]
Main reference:
[CG] Kevin Costello and Owen Gwilliam: Factorization algebras in quantum field theory, in preparation. Latest draft available on request from the seminar organizer.
Additional literature:
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