Also available at http://math.ucr.edu/home/baez/week50.html
March 12, 1995
This Week's Finds in Mathematical Physics (Week 50)
John Baez
Your roving mathematical physics reporter is now in Milan,
where (when not busy eating various kinds of cheese) he
is discussing BF theory with his hosts, Paolo Cotta-Ramusino
and Maurizio Martellini. This is rather a long way to go to
stumble on the October 1994 issue of the Journal of Mathematical
Physics, but such is life. This issue is a special issue on
"Topology and Physics", a nexus dear to my heart, so let me
say a bit about some of the papers in it.
1) Supersymmetric Yang-Mills theory on a four-manifold, by
Edward Witten, Jour. Math. Phys. 35 (1994), 5101-5135.
This paper concerns the relation between supersymmetric Yang-Mills
theory and Donaldson theory, discovered by Seiberg and Witten,
which not too long ago hit the front page of various newspapers.
(See "week44' and "week45" for my own yellow journalism on the
subject.) I don't have anything new to say about this stuff, of
which I am quite ignorant. If you are an expert on N = 2 supersymmetric
Yang-Mills theory, hyper-Kaehler manifolds, cosmic strings and the
renormalization group, the paper should be a piece of cake. Seriously,
he does seem to be making a serious effort to communicate the
ideas in simple terms to us mere mortals, so it's worth looking at.
2) Four-dimensional topological quantum field theory, Hopf
categories, and the canonical bases, by Louis Crane and Igor
Frenkel, Jour. Math. Phys. 35 (1994), 5136-5154.
I discussed this paper a wee bit in "week38". Now you can actually
see the pictures. As we begin to understand n-categories (see
"week49") our concept of symmetry gets deeper and deeper. This
isn't surprising. When all we knew about was 0-categories --- that
is, sets! --- our concept of symmetry revolved around the notion of
a "group". This is a set G where you can multiply elements in an
associative way, with an identity element 1 such that 1g = g1 = g
for all elements g of G, and where every element g has an inverse
g^{-1} with gg^{-1} = g^{-1}g = 1. For example, the group of rotations
in n-dimensional Euclidean space. When we started understanding
1-categories --- that is, categories! --- the real idea behind groups
and symmetry could be more clearly expressed. Sets have elements,
and they are either equal or not --- no two ways about it. Categories
have "objects", and even though two objects aren't equal, they can
still be "isomorphic". An object can be isomorphic to itself in
lots of different wasy: these are its symmetries, and the symmetries
form a group. But this is really just the tip of a still mysterious
iceberg. For example, in a 2-category, even though two objects aren't
equal, or even isomorphic, they can be "equivalent", or maybe I should
say "2-equivalent". This is a still more general notion of "sameness".
I won't try to define it just now, but I'll just note that it arises
from the fact that in a 2-category one can ask whether two morphisms
are isomorphic! (For people who followed "week49" and know some category
theory, let me note that the standard notion of equivalence of categories
is a good example of this "equivalence" business.) As we climb up
the n-categorical ladder, this keeps going. We get ever more subtle
refinements on the notion of "sameness", hence ever subtler notions
of symmetry. It's all rather mind-boggling at first, but not really
very hard once you get the hang of it, and since there's lots of
evidence that n-dimensional topological quantum field theories are
related to n-categories, I think these subtler notions of symmetry
are going to be quite interesting for physics.
And now, to wax technical for a bit (skip this paragraph if
the last one made you dizzy), it's beginning to seem that the
symmetry groups physicists know and love have glorious reincarnations,
or avatars if one prefers, at these higher n-categorical levels.
Take your favorite group --- mine is SU(2), which describes
3d rotational symmetry hence angular momentum in quantum mechanics.
It's category of representations isn't just any old category,
its a symmetric monoidal category! See the chart in "week49"
if you forget what this is. Now, there are more general things
than groups whose categories of representations are symmetric
monoidal categories -- for example, cocommutative Hopf algebras.
And there are other kinds of Hopf algebras --- "quasitriangular" ones,
often known as "quantum groups" --- whose categories of representations
are *braided* monoidal categories. The cool thing is that SU(2)
has an avatar called "quantum SU(2)" which is one of these quantum
groups. Again, eyeball the chart in "week49". Symmetric monoidal categories
are a special kind of 4-categories, which is why they show up so
much in 4d physics, while braided monoidal categories are a
special kind of 3-categories, which is why they (and quantum
groups) show up in 3d physics. For example, quantum SU(2) shows up
in the study of 3d quantum gravity (see "week16"). Now the even
cooler thing is that while a quaistriangular Hopf algebra is a set with a
bunch of operations, there is a souped-up gadget, a "quasitriangular Hopf
category", which is a CATEGORY with an analogous bunch of operations,
and these have a 2-CATEGORY of representations, but not just any
old 2-category, but in fact a BRAIDED MONOIDAL 2-CATEGORY. If
you again eyeball the chart, you'll see this is a special kind of
4-category, so it should be related to 4d topology and --- this is
the big hope --- 4d physics. Now the *really* cool thing, which
is what Crane and Frenkel show here, is that SU(2) has yet another
avatar which is one of these quasitriangular Hopf categories.
Regardless of whether it has anything to do with physics, this business
about how symmetry groups have avatars living on all sorts of rungs
of the n-categorical ladder is such beautiful math that I'm sure
it's trying to tell us something. Right now I'm trying to figure
out just what.
3) On the self-linking of knots, by Raoul Bott and Clifford Taubes,
Jour. Math. Phys. 35 (1994), 5247-5287.
I'd need to look at this a few more times before I could say
anything intelligent about it, but it looks to be a very exciting
way of understanding what the heck is really going on as far
as Vassiliev invariants, Feynman diagrams in Chern-Simons theory,
and so on are concerned --- especially if you wanted to generalize
it all to higher dimensions.
Alas, I'm getting worn out, so let me simply *list* a few more
papers, which are every bit as fun as the previous ones... no
disrespect intended... I just have to call it quits soon. As
always, it should be clear that what I write about and what I
don't is purely a matter of whim, caprice, chance, and my own
ignorance.
4) An explicit description of the symplectic struture of moduli
spaces of flat connections, by Christopher King and Ambar Sengupta, Jour.
Math. Phys. 35 (1994), 5338-5353.
The semiclassical limit of the two-dimensional quantum Yang-Mills
model, same authors, Jour. Math. Phys. 35 (1994), 5354-5363.
5) Topological interpretations of quantum Hall conductance, by
D. J. Thouless, Jour. Math. Phys. 35 (1994), 5362-5372.
6) The noncommutative geometry of the quantum Hall effect, by
J. Bellisard, A. van Elst, and H. Schulz-Baldes, Jour. Math. Phys. 35 (1994),
5373-5451.
7) Topology change in (2+1)-dimensional gravity, by Steve Carlip and
R. Cosgrove, Jour. Math. Phys. 35 (1994), 5477-5493.
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