For many years I’ve been fascinated by the topic of “Dirac cohomology” and its possible relations to various questions about quantization and quantum field theory. At first I was mainly trying to understand the relation to BRST, and wrote some things here on the blog about that. As time has gone on, my perspective on the subject has kept changing, and for a long time I’ve been wanting to write something here about these newer ideas. Last year I gave a talk at Dartmouth, explaining some of my point of view at the time. Over the last few months I’ve unfortunately yet again changed direction on where this is going. I’ll write about this new direction here in some detail next week, but in the meantime, have decided to make available the slides from the Dartmouth talk, and a version of the document I was writing on Quantization and Dirac Cohomology.
- Best to ignore the comments at the end of the slides about applications to Poincaré group representations and BRST. Both of these applications require getting the Dirac cohomology machinery to work in cases of non-reductive Lie algebras. As far as Poincaré goes, I’ve recently come to the conclusion that doing things with the conformal group (which is reductive) is both more interesting and works better. I’ll write more about this next week. For BRST, there is a lot one can say, but I likely won’t get back to writing more about that for a while.
- The Quantization and Dirac Cohomology document is kind of a mess. It’s an amalgam of various pieces written from different perspectives, and some lecture notes from a course on representation theory. Some day I hope to find the time for a massive rewrite from a new perspective, but maybe some people will find interesting what’s there now.