Advanced Geometry I: Geometry of Classical Field Theory (SoSe 2019)

Instructor

Dr. Christian Blohmann [1]

Assistant

Lory Aintablian-Kadiyan

Time and Place

Lecture
Tue: 14:00-16:00, Wegelerstr. 10 - Kleiner Hörsaal.
Wed: 14:00-16:00, Wegelerstr. 10 - Kleiner Hörsaal.

Exercise session
Mon: 12:00-14:00, Endenicher Allee 60 - SemR 1.008.

Course description

Classical field theory is a conceptual and mathematical framework to derive and study differential equations that describe a large class of physical phenomena and geometric structures, such as electromagnetism, gauge theories, general relativity, Poisson-Sigma models, Chern-Simons theory, factorization algebras, etc. While its origins go back to the 18th century, the mathematical formulation of classical field theories has been continously modernized to keep up with the increasing requirements of rigor and generality. The goal of the course is an exposition of the mathematical framework and structure lagrangian field theory (LFT) as it is used as a concept and tool in current research in geometry, topology, and mathematical physics.

Contents: Pinciples of LFT: fields, action principle, lagrangian; examples of field theories: classical mechanics, gauge theory, general relativity; locality, jet bundles, Peetre's theorem; catgegorical digression: pro-objects and ind-objects, pro-finite vector spaces vs. Fréchet spaces; the infinite jet bundle: as pro-manifold, horizontal and vertical splitting, vector fields on the infinite jet bundle; the variational bicomplex: definition, jet coordinates, Cartan distribution, acyclicity theorems of Takens and Bauderon-Anderson; cohomological action principle: Euler-Lagrange form, universal current, presymplectic form on the variety of solutions, Helmholtz problem; symmetries of an LFT: local vector fields, symmetries of the action, Noether currents and charges; Noether’s first and second theorem; initial value problem of LFT: the initial data map; formal solutions and formal well-posedness, constraints and gauge fixing, local symmetries, hamiltonian formulation, symplectic reduction of gauge theories, LFTs with extrinsic symmetries.

Bibliography

• Deligne P., Freed D.S.: Classical field theory. In: Quantum fields and strings: a course for mathematicians, vol. 1, Amer. Math. Soc., Providence, RI (1999), 137–225.

• Zuckerman G.J.: Action principles and global geometry. In: Mathematical aspects of string theory, Adv. Ser. Math. Phys., vol. 1, World Sci. Publishing, Singapore (1987), 259–284.

• Anderson, I. M.: The Variational Bicomplex, preprint, available online [2].

Exams

There will be oral exams at the end of the term.

Exercise sheets

• Exercise sheet 1 [3] (03.04.2019)

• Exercise sheet 2 [4] (17.04.2019)

• Exercise sheet 3  [5](28.04.2019)

• Exercise sheet 4 [6] (02.05.2019)

• Exercise sheet 5 [7] (10.05.2019)

• Exercise sheet 6 [8]  (28.05.2019)

• Exercise sheet 7 [9] (06.06.2019)

• Review questions [10]