{"id":804,"date":"2008-08-20T09:08:30","date_gmt":"2008-08-20T14:08:30","guid":{"rendered":"http:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=804"},"modified":"2017-09-25T11:11:50","modified_gmt":"2017-09-25T15:11:50","slug":"freed-on-chern-simons","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=804","title":{"rendered":"Freed on Chern-Simons"},"content":{"rendered":"

Dan Freed has a wonderful preprint out on the arXiv this evening, based on a talk he gave at the celebration of MSRI’s 25th anniversary, entitled Remarks on Chern-Simons Theory<\/a>. It’s mainly about the current state of attempts to better understand the mathematical significance of the Chern-Simons-Witten quantum field theory.<\/p>\n

This is a truly remarkable and very simple 3d quantum gauge theory, the significance of which Witten first came to understand back in 1988. He quickly showed that the theory brought together in an unexpected way several quite different but important areas of mathematics and physics (3d topology, moduli spaces of vector bundles, loop group representations, quantum groups, 2d conformal field theory among others). This work was the main reason he was awarded a Fields medal in mathematics in 1990. While Witten and others worked out many important aspects of this story back then, many important puzzles still remain, and it is these that Freed concentrates on.<\/p>\n

Perhaps the biggest puzzle is that of how to actually define the theory in a local manner. The standard definition thrown around is that this is just the QFT with Lagrangian given by the Chern-Simons number CS[A] of a connection A, so all one has to do is evaluate the path integral
\n$$\\int [dA] e^{i2\\pi k CS(A)}$$
\nWhile this is a good starting point for a perturbative expansion at large k, it doesn’t appear to make much sense non-perturbatively. Freed points out that it is known that the theory must depend on additional topological structure on the 3-manifold (e.g. a 2-framing), whereas the path integral looks like it only depends on the orientation. If you try and think about how you would actually calculate such an integral numerically, by discretizing it and taking a limit, it looks like you will end up with something hopelessly dependent on the details of the discretization and the limit. For a much simpler toy example with some of the same problems, consider the path integral on closed curves on a sphere, taking as Lagrangian the enclosed area.<\/p>\n

Freed describes in detail the state of attempts to rigorously define the theory without dealing with the path integral, but instead exploiting the fact that it is supposed to be a topological qft, and thus may have an abstract definition in terms of generators and relations. He describes the current situation as follows:<\/p>\n

\n
  • There is a generators-and-relations construction of the 1-2-3 theory via modular tensor categories for many classes of compact Lie groups G. This includes finite groups, tori, and simply connected groups, the latter via quantum groups or operator algebras.<\/li>\n
  • There are new generators-and-relations constructions – at this stage still conjectural – of the 0-1-2-3 theory for certain groups, including finite groups and tori.<\/li>\n
  • There is an a priori<\/em> construction of the 0-1-2-3 theory for a finite group.<\/li>\n
  • There is an a priori<\/em> construction of the dimensionally reduced 1-2 theory for all<\/em> compact Lie groups G<\/li>\n<\/blockquote>\n

    The bottom line is that we only have a local construction of the theory for the case of finite groups, where one can make perfectly good sense of the path integral. For the case of a 3-manifold that is a product of a circle and a Riemann surface, one can define things in terms of a 2d theory, and Freed explains the connections to the Freed-Hopkins-Teleman theorem.<\/p>\n

    To convince mathematicians that there is something to the path integral, Freed writes down the asymptotic expansion for large k that it leads to, and shows that this gives a highly non-trivial conjecture relating quite different mathematical objects associated with a 3-manifold. He shows strong numerical evidence for this conjecture.<\/p>\n

    Finally, he ends with some extensive and interesting comments about the relationship between quantum field theory and mathematics, as it has been pursued by both physicists and mathematicians over the past quarter-century, with some speculation about what direction this might take in the future.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Dan Freed has a wonderful preprint out on the arXiv this evening, based on a talk he gave at the celebration of MSRI’s 25th anniversary, entitled Remarks on Chern-Simons Theory. It’s mainly about the current state of attempts to better … 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-804","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\/804","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=804"}],"version-history":[{"count":15,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/804\/revisions"}],"predecessor-version":[{"id":9571,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/804\/revisions\/9571"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=804"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=804"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=804"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}