re: quantum groups in physics

[John Baez <john.c.baez@gmail.com>]

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


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