Beijing Institute of Mathematical Sciences and Applications

1, Quantization
This part is devoted to the rigorous quantization of symplectic vector spaces and Lagrangian subspaces of these spaces. To quantize, we should fix Darboux coordinates; the result does not depend on this choice (up to a constant factor). In the formalism of L-functionals, it is not necessary to fix Darboux coordinates. These ideas can be used to quantize quadratic action functionals, in particular, the Dirichlet functional, and to develop a simplified version of operator formalism in CFT.

2. Introduction to homological algebra and derived geometry. Applications to physics
Differential graded algebras. Homology. Euler characteristic and Lefschetz Trace formula. Quasi-isomorphism. BRST quantization. Projective modules and projective resolution. Koszul-Tate resolution. Cohomology of Lie algebras.

Supermanifolds. Q-manifolds and QP-manifolds. Derived geometry.
BV formalism.

3. Local field theories.
Sewing of two domains. Axiomatics of conformal theory. BV formalism and BFV formalism. Cattaneo-Mnev-Reshetikhin theory.

4. New approach to string theory and superstring theory [4],[5]
A classical theory with first-class constraints can be quantized in terms of the BRST formalism. One can describe a broad class of physical quantities for quantum theory and show that multiloop amplitudes of string theory are among these quantities.