Seminar on Factorization Algebras and QFTs

Plan of the seminar:

1. Organizational meeting.
   Motivation, brief introduction to the subject, discussion of the seminar plan, distribution of talks.
   Literature: Introduction in: [CG]
    
2. 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]
    
3. 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]
    
4. 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]
    
5. 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]
   
   
6. 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
   
   
7. 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]
    
8. 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]
    
9. 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:

[G] Owen Gwilliam: Factorization algebras and free field theories, PhD thesis, Northwestern University, 2012 [4]