From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5455 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: re: quantum groups in physics Date: Wed, 30 Dec 2009 17:46:45 -0800 Message-ID: References: <4B3B5E1E.8000103@math.upenn.edu> Reply-To: John Baez NNTP-Posting-Host: lo.gmane.org Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1262526890 704 80.91.229.12 (3 Jan 2010 13:54:50 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 3 Jan 2010 13:54:50 +0000 (UTC) To: categories Original-X-From: categories@mta.ca Sun Jan 03 14:54:43 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NRQum-0001vs-Af for gsmc-categories@m.gmane.org; Sun, 03 Jan 2010 14:54:40 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NRQYz-0004f8-Ax for categories-list@mta.ca; Sun, 03 Jan 2010 09:32:09 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5455 Archived-At: Happy New Year! I wrote: > On the other hand, it's been known in theoretical physics ever since the >> work of Witten, Reshetihkin and Turaev that Chern-Simons theory can be >> described in a purely algebraic way using quantum groups! >> > Jim Stasheff wrote: > Can you give me a clue as to how quantum groups enter? - > since > > Chern-Simons theory can be > described in a purely algebraic way > > for ordinary Lie groups. > In my remark, I was speaking of Chern-Simons theory as a 3d field theory with the Lagrangian tr(A dA + (2/3) A^3) Here A is a connection on some bundle with an ordinary Lie group as structure group. As a classical field theory, the solutions of Chern-Simons theory are just flat connections. But when you quantize it, quantum groups come in! The moduli space of flat connections has a symplectic structure, and when you geometrically quantize it, the resulting Hilbert space has a nice description in terms of the category of representations of the quantum group associated to your original Lie group. This is what Witten initiated with his famous paper "Quantum field theory and the Jones polynomial". For a really nice account, try this book: Bojko Bakalov and Alexander Kirillov, Jr., Lectures on Tensor Categories and Modular Functors, American Mathematical Society, Providence, Rhode Island, 2001. Preliminary version available http://www.math.sunysb.edu/~kirillov/tensor/tensor.html For many mathematicians physicists, this connection to field theory is a big part of why quantum groups are interesting! Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]