Batalin-Vilkovisky Quantization

Annotation

Gauge invariance is a key principle of constructing theoretical models of fundamental interactions, including the standard model, gravity and (super)strings theories. The course gives a systematic introduction to the theory of gauge systems and Batalin-Vilkovisky (BV) formalism which is considered the most powerful method of quantizing gauge systems. The course covers classical dynamics of gauge systems in Lagrangian and Hamiltonian approaches, systematically introduces ghost variables and Becchi-Rouet-Stora-Tyutin (BRST)  transformations, (quantum) master equation, and gauge fixing procedure.   Particular attention is paid to the relevant mathematical methods including graded geometry, elements of homological algebra, and the jet-bundle description of local gauge theories. Applications of the BV approach to the study of consistent interactions, anomalies, renormalization, and generalized symmetries are also considered.

Course plan

1. Gauge systems in Lagrangian and Hamiltonian formalisms. Physical interpretation and examples of gauge theories.
2. Gauge systems in the Hamiltonian approach. Constraints, evolution and observables. Dirac quantization of constrained systems. Geometry of the constrained surface.
3. Ghost variables and BRST invariance. De Rham and Chevalley–Eilenberg complexes as BRST complexes. Q-manifolds.
4. Hamiltonian BRST quantization. Physical states, evolution, observables.
5. From Hamiltonian to Lagrangian description. BV formulation of the extended Hamiltonian action.
6. Ghost fields and antifields in the Lagrangian formalism. The physical and geometric meaning of ghost fields and antifields. Koszul resolution of the stationary surface.
7. Global symmetries in BV approach. Generalized Noether theorem.  Local BRST cohomology.
8. Main structures of the BV formalism: antibracket, master equation, anticanonical transformations, gauge fixing fermion.
9. Quantum master equation and Delta operator. BV path integral. Gauge invariance of physical quantities.
10. Interactions as consistent deformations. Elements of the deformation theory. Renormalization and deformations.
11. Gauge theories in the first quantized BRST formalism. L_\infty-algebras. Spinning particles. Algebraic structures of string  field  theory.