{"id":292,"date":"2005-11-10T22:09:17","date_gmt":"2005-11-11T03:09:17","guid":{"rendered":"http:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=292"},"modified":"2005-11-28T16:57:47","modified_gmt":"2005-11-28T21:57:47","slug":"latest-freed-hopkins-teleman","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=292","title":{"rendered":"Latest Freed-Hopkins-Teleman"},"content":{"rendered":"

A wonderful long-promised paper by Dan Freed, Mike Hopkins and Constantin Teleman entitled Loop Groups and Twisted K-theory II<\/a> has just appeared. They have advertised it in the past under various names such as “K-theory, Loop Groups and Dirac Families”, but their latest way of organizing their work seems to be to relabel the two-year old Twisted K-theory and Loop Group Representations<\/a> (which recently has been updated, improved and expanded with new material) as “Loop Groups and Twisted K-theory III”. Working backwards it seems, they now advertise a “Loop Groups and Twisted K-theory I” as still to appear, hopefully in less than two years. <\/p>\n

I don’t mean to give them a hard time about this. They are doing wonderful work, continually refining and improving on their results, and the paper is worth the wait. At the moment I don’t have time to do them justice by explaining much about their results or the conjectural relations that I see to quantum field theory, but I wrote a little bit about this a while back in another context<\/a>. In the future I’ll try and find time to write some more entries about this material.<\/p>\n

Also related to this is a new paper of Michael Atiyah and Graeme Segal called Twisted K-theory and cohomology<\/a> which discusses the relation of twisted K-theory to twisted and untwisted cohomology.<\/p>\n

Teleman has also recently made available on his web-site<\/a> a preliminary version of notes from his fascinating talk at the algebraic geometry conference in Seattle<\/a> this past summer, entitled Loop Groups, G-bundles on curves<\/a>. He starts off with some philosophy he claims comes from lessons learned in working with moduli of bundles:<\/p>\n

(i) K-theory is better than cohomology
\n(ii) Stacks are better than spaces
\n(iii) Symmetry<\/i><\/p>\n

The first and third points I’m well aware of, and he has convinced me to spend some more time learning about stacks by his next point, which I hope may clarify some issues that confused me when I was writing my notes on Quantum Field Theory and Representation Theory<\/a>. According to Teleman, the fundamental K-homology class of a classifying stack BG gives a notion of “integration over BG” in K-theory that corresponds precisely to that of taking the G-invariants of a representation. This idea has been a fundamental motivation for me for quite a while. It seems to me that one fundamental question about the path integral formulation of the standard model is “why are we looking at the space of connections and trying to integrate over it?” The K-theory philosophy gives a potential answer to this: we’re looking at the space of connections because it is the classifying space of the gauge group, and we’re integrating over it because we want to be able to pick out the invariant piece of a gauge group representation. I’ll try and write up more about this later, especially if learning some more about stacks ends up really clarifying things for me as I hope.<\/p>\n

On a somewhat different topic, Teleman recently gave a very interesting talk at Santa Barbara entitled The Structure of 2D Semi-simple Field Theories<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

A wonderful long-promised paper by Dan Freed, Mike Hopkins and Constantin Teleman entitled Loop Groups and Twisted K-theory II has just appeared. They have advertised it in the past under various names such as “K-theory, Loop Groups and Dirac Families”, … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-292","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/292","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=292"}],"version-history":[{"count":0,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/292\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=292"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=292"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=292"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}