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May 8, 2007

Quantization and Cohomology (Week 22)

Posted by John Baez

This week our class on Quantization and Cohomology was a bit slow-paced, in part because I was jet-lagged, but also because Derek Wise and Jeffrey Morton were gone, attending the Ottawa conference on traces. So, nobody taking the class knew much about principal bundles. Review time!

  • Week 22 (Apr. 30) - Smooth functors and beyond. Review of what we’ve done so far. Principal U(1)\mathrm{U}(1) bundles and phases in quantum mechanics. The need for nontrivial principal bundles in geometric quantization. Torsors.

Last week’s notes are here; next week’s notes are here.

Aficionados of this course will sense that today’s notes were not taken by Derek Wise. Instead, they were taken by Alex Hoffnung.

I think the UCR math PhD program should give one scholarship a year based on penmanship.

Posted at May 8, 2007 1:27 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1267

7 Comments & 1 Trackback

Re: Quantization and Cohomology (Week 22)

The link is to the cohomology and computation notes!

:D

Posted by: David Roberts on May 8, 2007 6:32 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 22)

Whoops! Weird.

Fixed. Thanks!

Posted by: John Baez on May 8, 2007 6:53 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 22)

Could either or both of Derek and Jeffrey be persuaded to give a brief summary of the Traces conference as a guest post?

Posted by: David Corfield on May 8, 2007 11:30 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 22)

David wrote:

Could either or both of Derek and Jeffrey be persuaded to give a brief summary of the Traces conference as a guest post?

No. They are both having thesis defenses at the end of this month, and they’re desperately writing stuff up. No blogging for them!

Jeff’s thesis is on extended TQFTs — rigorously constructing the Dijkgraaf-Witten model as a 2-functor

Z:nCob 22VectZ: nCob_2 \to 2Vect

for all nn. Lots of nice spans of groupoids show up here!

Derek’s thesis is on topological gauge theory, Cartan geometry and gravity. Right now he’s busily classifying conjugacy classes in SO(4,1)SO(4,1) and showing how they match up to various kinds of particles.

What I should really do is get them to write guest blogs about their theses after they’re done!

The conference on traces sounded fun, but I have not been able to extract any specific new insights from their descriptions of what went on.

Posted by: John Baez on May 8, 2007 7:59 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 22)

There has been an ominous silence surrounding my attempt to view 2-geometry as being about bundles (perhaps also orbifolds). But looking at your U(1)U(1)-principal bundles, isn’t it natural to think of figures such as particle trajectories being preserved?

E.g., on page 9 of your notes, a 1-dimensional subspace of the total space of the bundle would have xx follow a loop on the sphere, while yy travelled a number of times around the perpendicular circle.

I suppose there’s the issue of whether you can have the geometry of the fibres more flexible than the geometry of the base, e.g., up to rigid transformation in the base, but diffeomorphism in the fibres.

By the way, does ordinary Klein 1-geometry work for looser ‘geometries’? E.g., can you quotient the set of homeomorphisms of the plane by the set of homeomorphisms preserving a region of the plane to get the collection of such regions?

Posted by: David Corfield on May 9, 2007 9:53 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 22)

There has been an ominous silence surrounding my attempt to view 2-geometry as being about bundles (perhaps also orbifolds).

I thought I said something concenring the path groupoid, regarded as a “finite” version of the tangent bundle.

Notice that every smooth category for which both source and target maps are surjective submersions may be regarded as a bundle (of morphisms) over the space of objects in two ways.

Regarding a vector bundle as a smooth category where composition of morphisms is addition of vectors in a given fibre is just one special degenrate case of this:

the category is the skeletal Lie groupoid whose vertex groups are the abelian groups given by addition in the fibers.

I am just saying this to emphasize that – indeed – (certain kinds of) bundles naturally fit into the realm of 2-geometry, if the latter is the study of categories internal to suitable categories of “spaces” (smooth spaces in the above examples).

Posted by: urs on May 9, 2007 11:58 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 22)

Yes, you did say something concerning the path groupoid along those lines.

I suppose I’m wondering why we didn’t look straight away at this kind of situation. Should it be so hard to line up the class of a certain type of smooth subcategory of a smooth category with a quotient of the symmetry 2-group by a 2-group which fixes a specific example of that type of subcategory?

Posted by: David Corfield on May 9, 2007 1:00 PM | Permalink | Reply to this
Read the post That Shift in Dimension
Weblog: The n-Category Café
Excerpt: What makes the Kontsevich-Cattaneo-Felder theorem tick? How can it be that an n-dimensional quantum field theory is encoded in an (n+1)-dimensional one?
Tracked: August 25, 2007 2:28 AM

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