Geometry and Gauge Theory

Announcements (new ones first)

  • Here you can find the second version of my preliminary lecture notes (until April 2, 2012).
  • No lectures on Thursday, April 5 and on Monday, April 9 because of Easter! So next lecture on Thursday, April 12. Happy Easter!
  • Here you can find preliminary lecture notes of the course so far (until March 5, 2012). They were mainly intended as notes for myself, so they are quite informal and probably contain many typos and are not always self-explanatory. But they might still be useful.
  • On Monday, February 20 there will be no lecture because of Carneval
  • Ok, first meeting is done, courses will be
    Mo 14:00-16:00 : P8 (math-building)
    Thu 14:00-16:00 : V1.07 Pavilhão de Civil, 1st floor
  • First meeting on Monday February 13, 2012, 14pm to 15pm in room 4.35 (math department, fourth floor):
    In this first meeting I will sketch my idea of the course and we can informally discuss about the topics and the schedule.

Summary

The course will start with a geometric discussion of Hamiltonian systems with so-called second and first class constraints. The latter are related to local symmetries (gauge symmetries). Explicitly solving the constraints usually breaks covariance under the symmetry groups of the system. This can be avoided by setting the constraints to zero only cohomologically. There will be two differentials with a clear geometrical meaning, the Koszul Tate differential and the longitudinal exterior derivative , whose relative cohomology will provide the physical subspace. The same can be obtained in a single step, by combining the two to the so-called BRST-differential.
Also the corresponding machinery in the Lagrangian formalism will be discussed. It is known under the names "antifield-formalism" or "BV formalism" and is based on an odd bracket, the "antibracket" (which is a Gerstenhaber bracket).
The discussions will be first at classical level. However, an important motivation for the cohomological description of gauge symmetries is the task of covariantly quantizing a theory. We will thus continue to discuss quantum corrections as well as anomalies.
The formalism will be applied in particular to Yang Mills theory whose fibre bundle geometry is also one of the main topics.
It is rather improbable that we will make it through the whole list of topics sketched at the fenix-announcement of this course. We will see, how much time will be left and then choose among the remaining subjects depending on your, but in particular also on my preference ;-)

Please note that at present I have no possibility to edit the content on the fenix-site myself, so all new announcements will be done on this website here. However, rough summaries of the previous lectures are announced by now on the fenix site.

Bibliography

  • Among the bibliography listed on fenix, I will in the first lectures mainly use "Quantization of Gauge Systems" (Marc Henneaux and Claudio Teitelboim, 1992, Princeton University Press).
  • For the discussion of fiber bundle geometry I will mainly use "Geometry, Topology and Physics" (M. Nakahara, Bristol, UK: Hilger, 1990).
  • The discussion of anomalies will probably be based on "Anomalies in Quantum Field Theory" (Reinhold A. Bertlmann, Oxford University Press 1995/2000)


For further information please drop me an email (don't forget to delete the capital letters in the email-address) or step by (office 2.09 in the math building).