PG minicourse: BRST cohomology

This is the website for the PG minicourse on BRST cohomology and symplectic reduction taught by José Figueroa-O'Farrill in the academic year 2006-7. The first part of the course will consist of 5 lectures, covering the following topics at an introductory level: Lie algebra cohomology, symplectic reduction, and the construction of the BRST complex. The second part of the course, given after Nils Scheithauer's introductory lectures on vertex algebras will consist of 2-3 lectures on the applications of BRST cohomology to string theory. Lectures will last approximately 90 minutes.


The first thing one notices in the hamiltonian treatment of gauge theories is that the dynamics are constrained. Dirac studied this problem and set up a formalism of constrained hamiltonian dynamics on a reduced phase space. The case relevant to gauge theories, and the one we will study in this course, is that of first-class constraints or, in more modern terminology, coisotropic reduction, a special case of which was studied by Marsden and Weinstein in a celebrated paper. The aim of the course is the construction of a complex of Poisson superalgebras built naturally out of the functions on the unconstrained phase space, whose cohomology (in degree zero) are the functions on the reduced phase space. Being a complex of Poisson superalgebras, it can be readily quantized; although we will not discuss the quantization in this course. Neither will we discuss Dirac's theory of constraints except as motivation.


The first lecture will take place on Tuesday 19 September at 4:10pm in JCMB 5326.

The time and place of further lectures will be discussed at the end of the first lecture and announced in this web page.

  1. Tuesday, 19 September 2006
    4:10pm, JCMB 5326
    Introduction; Lie algebra cohomology
  2. Friday, 22 September 2006
    2:00pm, JCMB 5326
    Symplectic reduction (I)
  3. Tuesday, 26 September 2006
    4:10pm, JCMB 3218
    Symplectic reduction (II)
  4. Thursday, 28 September 2006
    2pm, JCMB 5215
    The BRST complex (I)
  5. Wednesday, 4 October 2006
    4.10pm, JCMB 6324
    The BRST complex (II)
  6. Tuesday, 24 October 2006
    4.10pm, JCMB 6203
  7. Monday, 8 January 2007
    4.10pm, JCMB 4312
    BRST quantisation of the bosonic string
  8. Thursday, 11 January 2007
    4.10pm, JCMB 4312
    Operator product technology (I): Basics
  9. Tuesday, 16 January 2007
    4.10pm, JCMB Lecture Theatre C
    Operator product technology (II): Applications
  10. Friday, 19 January 2007
    3.30pm, JCMB Lecture Theatre C
    BRST cohomology = semi-infinite cohomology

Lecture notes

The notes are organised into logical lectures. The number of physical lectures per logical lecture will depend on the topic. Some time between writing the second and third (logical) lectures, I decided to incorporate the tutorial problems into the lecture notes: this explains the discontinuity in the page numbers. (Just in case you were wondering.)

  1. Lie algebra cohomology
  2. Symplectic reduction
  3. The BRST complex for a group action

There is now also a full set of notes incorporating the problems. They differ somewhat from the ones above, in that I have reorganised the material slightly.

There are no lecture notes (thus far) for the second half of the course. This may change in due course.

Problem sheets

The following tutorial sheets contain problems (some of them hard) which complete, complement or extend what is covered in lectures. You should at least look at them and attempt those you find more interesting.

  1. Lie algebra cohomology
  2. Symplectic reduction
  3. The BRST complex
  4. Operator product technology
  5. BRST differential in conformal field theory
  6. BRST cohomology of critical bosonic string

This cheat sheet on operator product technology might come in handy when doing calculations, but not as handy as its computer implementation (Zip archive of Mathematica files), courtesy of Kris Thielemans.

Online references

Other references

Last modified: 6 May 2009 at 10:27 JST