From: bryson@nas.nasa.gov (Steve T. Bryson)
Newsgroups: sci.math
Subject: Re: jones polynomials in stat mech
Date: 5 Mar 91 21:59:42 GMT

In article <1991Mar2.234319.23665@agate.berkeley.edu> greg@garnet.berkeley.edu (Greg Kuperberg) writes:

   In article <2604@bnlux0.bnl.gov> kyee@bnlux0.bnl.gov (kenton yee) writes:
   >	hi, i'm looking for a pedagogical (but substantial)
   >	article describing examples of uses of the jones
   >	polynomial in statistical mechanics and field 
   >	theory.   can anyone recommend something?
   >	thanks in advance,  -ken

   It won't look like the Jones polynomial when you read it, but the book
   *Exactly Solved Models in Statistical Mechanics*, by Baxter, and the
   paper "Quantum groups", by Drinfel'd, in the Proceedings of the 1986
   ICM at Berkeley are two good sources if you are really interested in
   the beef of this connection.

   Historically, the advances in statistical mechanics and quantum
   scattering did not follow from the discovery of the Jones polynomial.
   Rather, all three followed from a set of amazing algebraic coincidences
   which still don't have a good name.  Baxter, Drinfel'd, and Jones
   independently discovered a part of these coincidences without knowledge
   of the work of the others.  It is high time for a unifying survey
   article, but I don't know of any satisfactory ones.

   Quantum field theory is a horse of a different color.  Witten's paper
   "Quantum field theory and the Jones polynomial", in Communications of
   Mathematical Physics, 1989?, is at once one of the most edifying and
   one of the most frustrating treatises on the subject, in addition to
   being the first.  At a superficial level, it is pedagogical.  At a
   substantive level, it is still very important, but pedagogical it is
   not.
   ----
   Greg Kuperberg                 Reply only to postings you like.
   greg@math.berkeley.edu         Ignore postings you dislike.

I like this posting!  It harbingers truly substantive awareness of active
issues in mathematics!

Anyway, on to the point.  Greg's suggestions are (I am assured) good approaches
for those who have a VERY solid grounding in modern statistical mechanics. 
For the same material from a knot theory point of view, the best thing I've 
read is a short paper called "Statistical Mechanics and the Jones Polynomial"
by Louis Kauffman, Proceedings of the 1986 Santa Cruz conference on Artin's 
braid group, AMS contemporary math series, 1989, vol 79.  This also appears
in the book "Problems and Methods and Techniques in Quantum Field Theory
and Statistical Mechanics", edited by Rasetti, World Scientific, 1990.  In this
paper, a polynomial invariant of knots (the bracket or Kauffman polynomial) 
is shown to be, in different special cases, the Jones polynomial for knots, and
the partition function for the Potts model in statistical mechanics.  The 
discussion is very accessible for the sophisticated non-specialist.  

Lou Kauffman has just finished a book called "Knots and Physics" which should
appear in the next couple of months and covers all of this stuff (in >500 
pages!).  It is being published by World Scientific.  I have seen the preprint 
and it starts from a very basic level and develops much of the state of the
art on this subject.

So what is this subject?  (I'm all hyped up because I just saw Lou Kauffman
give a talk on it at Berkeley)  It seems that the (smooth) topology of low
dimensional manifolds, knot theory, quantum field theory, and statistical
mechanics are all coming together in many ways from many directions.  What is
exciting about all this is that these four fields are (at first sight) very
different fields that each have fundamental unsolved problems (i.e. we do not
know how to classify smooth 4 and 3 dimensional manifolds, we do not know why
quantum field theory works, etc.).  Now it is being shown that each of these 
four fields is tied to all of the other four fields in some useful way.  This
started in the early 80's when Donaldson used quantum field theory techniques
(functional integration of gauge theories) to prove theorems about the topology
of 4-dimensional manifolds.  Statistical mechanics has used techniques from 
quantum field theory, and now stat mech has been giving great insight into 
quantum field theory through the various lattice models.  Now along comes the
Jones polynomial and its relations to the Yang-Baxter equations in stat mech,
which Kauffman has generalized to other invariants of knot theory.   
Witten has shown (well, described) that one can use knot theory in the context
of quantum field theory to produce invariants of 3 dimensional manifolds.  
Finally, Michael Atiyah is using the Jones-Witten theory to try to make sense
out of functional integration in gauge theories and quantization.

This is a very exciting time, with some very heady stuff.  I am told that
no one can see where this is all going to lead, but it looks like there may
be some revolutions in the making.

Steve Bryson
bryson@nas.nasa.gov