Lagrangian Field Theory

Lecture course, University of Bonn, Summer 2017

Tuesday 10:15-12:00, Lecture Hall MPIM

Lagrangian field theory (LFT) 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 LFT 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 of LFT as it is used as a concept and tool in current research in geometry, topology, and mathematical physics.

Syllabus

Principles of LFT: fields, action principle, lagrangian, locality; examples of field theories: classical mechanics, gauge theory, general relativity; mathematical framework of LFT: jet bundles, Peetre’s theorem, pro-objects and ind-objects, pro-spaces vs. Fréchet spaces; the variational bicomplex: definition, jet coordinates, Cartan distribution, acyclicity theorems of Takens and Bauderon-Anderson; cohomological action principle: Euler-Lagrange form, universal current, symplectic 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.

Prerequisites

Basic knowledge of differential geometry (incl. prinicpal bundles, connections, riemannian metrics, curvature), symplectic geometry (incl. hamiltonian actions, symplectic reduction), cohomological algebra (incl. spectral sequences), category theory, and functional analysis is required or should be acquired in parallel. A background on PDE theory and a physics background on classical field theory is helpful but not necessary.

Course credit

The course work required for credit will consist of an individual literature research assignment in the form of a term paper and an oral presentation. More details will be given in the first lecture.

Lecture notes

While there is a number of math books on classical field theory, there is no text or monograph that would be suitable for this course. Therefore, I plan to write lecture notes during the semester that will be distributed to the participants. Comments on the notes are very welcome.

  1. 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.

  2. 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.

  3. Anderson, I. M.: The Variational Bicomplex, preprint, available online