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I'm sure most of you have lost interest in my "tale of n-categories",
because it takes a fair amount of work to keep up with all the abstract
concepts involved. However, we are now at a point where we can have
some fun with what we've got, even if you haven't really followed all
the previous stuff. So what follows is a rambling tour through monads,
adjunctions, the 4-color theorem and the large-N limit of SU(N) gauge
theory....
Okay, so in "week89" we defined a gadget called a "monad". Using
the string diagrams we talked about, you can think of a monad as
involving a process like this:
I can hear you wonder, what exactly is this thing s? What
is this
process M? Well, I gave the technical answer in "week89" - but the
point is that n-category theory is deliberately designed to be so
general that it covers pretty much anything you could want! For example,
s could be the set of real numbers and M could be multiplication of real
numbers, which is a function from s × s to s. Or we could be doing
topology in the plane, in which case the picture above stands for exactly
what it looks like: two lines merging to form one line! These and many
other situations are analogous, and the formalism allows us to treat them
all at once. Here I will not review all the rules of the game. If you
just play along and trust me everything will be all right. If you
don't trust me, go back and check the definitions.
Let me turn to the axioms for a monad. In addition to the multiplication
M we want to have a "multiplicative identity", I, looking like this:
\ /
\ /
s\ s/
\ /
\ /
\ /
\ /
\ /
|M
|
|
|
|
|
s|
|
which we read downwards as describing the "fusion" of two copies of something
called s into one copy of the same thing s. The fusion process itself is
called M.
I
|
|
|
|
|s
Here nothing is coming in, and a copy of s is going out. Because ordinary
multiplication has 1x = x and x1 = x for all x, we want the following axioms
to hold:
/ |
/ |
s/ |s
I / |
\ / |
\ / |
\ / |
\ / |
|M = |
| |
| |
| |
| |
| |
s| |
| |
and
\ |
\ |
\ |s
s\ I |
\ / |
\ / |
\ / |
\ / |
|M = |
| |
| |
| |
| |
| |
s| |
| |
Also, since ordinary multiplication has (xy)z = x(yz), we want
the following associativity law to hold, too:
\ / / \ \ /
\ / / \ \ /
s\ /s s/ s\ s\ /s
\/ / \ \/
M\ / \ /M
\ / \ /
s\ / \ /s
\ / \ /
|M |M
| |
| = |
| |
| |
| |
s| s|
| |
These rules are a translation of the rules given in "week89" into
string diagram form.
If you are a physicist, you can think of these diagrams as being funny Feynman diagrams where you've got some kind of particle s and two processes M and I. M is a bit like what you'd call a "cubic self-interaction", where two particles combine to form a third. These interactions show up in simple textbook theories like the "φ3 theory" and, more importantly, in nonabelian gauge field theories like quantum chromodynamics, where the gauge bosons have cubic self-interactions. On the other hand, I is a bit like what you'd usually call a "source" or an "external potential", some sort of field imposed from outside that can create particles of type s. You shouldn't take the analogy with Feynman diagrams too seriously yet, because the context we're working in is so general, and the most interesting physics theories don't correspond to monads but to more elaborate setups. However, we could flesh out the analogy to make it very precise and accurate if we wanted, and this is especially important in topological quantum field theory. More later about that.
Now in "week83" I discussed a different sort of gadget, called an "adjunction". Here you have two guys x and x*, and two processes U and C called the "unit" and "counit", which look like this:
U
/ \
/ \
x/ \x*
/ \
and
\ /
x*\ /x
\ /
\ /
C
They satisfy the following axioms:
| |
U x| x|
/ \ | |
/ \ | |
/ \ | |
| x*\ / = |
| \ / |
| \ / |
x| C |
| |
| |
x*| U x*|
| / \ |
| / \ |
| / \ |
\ x/ | = |
\ / | |
\ / | |
C x*| |
| |
Physically, we can think of x* as the antiparticle of x, and then U is
the process of creation of a particle-antiparticle pair, while C is the
process of annihilation. The axioms just say that for a particle or
antiparticle to "double back in time" by means of these processes isn't
really different than for it to march obediently along forwards.
Mathematically, one nice example of an adjunction involves a vector
space x and its dual vector space x*. This is really the same example,
since if the behavior of a particle under symmetry transformations is
described by some group representation, its antiparticle is described by
the dual representation. For more details on the math, see "week83".
Now, let's see how to get a monad from an adjunction! We need to get s, M, and I from x, x*, U, and C. To do this, we first define s to be xx*. Then define M to be
\ \ / /
x\ \x* x/ /x*
\ \ / /
\ \ / /
\ C /
\ /
| |
| |
x| |x*
| |
Again, to really understand the rules of the game you need to learn
a bit about string diagrams and 2-categories, but the basic idea is
supposed to be simple: we can get two xx*'s to turn into one
xx* by letting an x* and x annihilate each other!
Finally, we define I to be
U
/ \
| |
| |
| |
x| |x*
| |
In other words, an xx* can be created out of nothing since it's
a "particle/antiparticle pair".
Now one can check that all the axioms for a monad hold. You really need to know a bit about 2-categories to do it carefully, but basically you just let yourself deform the pictures, in part with the help of the axioms for an adjunction, which let you straighten out curves that "double back in time." So for example, we can prove the identity law
/ / | |
/ / | |
U x/ /x* x| |x*
/\ / / | |
x\ \x* / / | |
\ \ / / | |
\ \ / / | |
\ \/ / | |
|C | = | |
| | | |
| | | |
| | | |
| | | |
| | | |
x| |x* | |
| | | |
by canceling the U and the C on the left using one of the
axioms for an adjunction. Similarly, associativity holds
because the following two pictures are topologically the same:
x\ \x* x/ /x* / / \ \ x\ \x* x/ /x*
\ \ / / / / \ \ \ \ / /
\ \/ / / / \ \ \ \/ /
\ C/ x/ /x* x\ \x* \ C/
\ \ / / \ \ / /
\ \ / / \ \ / /
\ C / \ C /
| | | |
| | | |
| | = | |
| | | |
| | | |
| | | |
x| |x* x| |x*
| | | |
Whew! Drawing these is tough work.
Now, as I said, an example of an adjunction is a vector space x and its dual x*. What monad do we get in this case? Well, the vector space x tensored with x* is just the vector space of linear transformations of x, so that's our monad in this case. In less high-brow terms, we've proven that matrices form an algebra when you define matrix multiplication in the usual way! In particular, the above picture serves as a diagrammatic proof that matrix multiplication is associative.
Of course, people didn't invent all this fancy-looking (but actually very basic) stuff just to deal with matrix multiplication! Or did they? Well, actually, Penrose did invent a diagrammatic notation for tensors which is just a slightly souped-up version of the above stuff. You can find it in:
1) Applications of negative dimensional tensors, by Roger Penrose, in Combinatorial Mathematics and its Applications, ed. D. J. A. Welsh, Academic Press, 1971.
But most of the work on this sort of thing has been aimed at applications of other sorts.
Now let me drift over to a related subject, the large-N limit of SU(N) gauge theory. Quantum chromodynamics, or QCD, is an SU(N) gauge theory with N = 3, but it turns out that things simplify a lot in the limit as N → ∞, and one gets some nice qualitative insight into the strong force by considering this simplified theory. One can even treat the number 3 as a small perturbation around the number ∞ and get some decent answers! A good introduction to this appears in Coleman's delightful book, essential reading for anyone learning particle physics:
2) Sidney Coleman, Aspects of Symmetry, Cambridge University Press, Cambrdige, 1989.
Check out section 8.3.1, entitled "the double line representation and the dominance of planar graphs". Coleman considers Yang-Mills theories, like QCD, but many of the same ideas apply to other gauge theories.
The idea is that if we start out studying the Feynman diagrams for a gauge field theory with gauge group SU(N), and see how much various diagrams contribute to any process for large N, the diagrams that contribute the most are those that can be drawn on a plane without any lines crossing. Technically, the reason is that diagrams that can only be drawn on a surface of genus g grow like N2 - 2g as N increases. This number 2 - 2g is called the Euler characteristic and it's biggest when your surface has no handles.
Even better, in the N → ∞ limit we can think of the Feynman diagrams using diagrams like the ones above. For example, we can think of the cubic self-interaction in Yang-Mills theory as simply matrix multiplication:
\ \ / /
x\ \x* x/ /x*
\ \ / /
\ \ / /
\ C /
\ /
| |
| |
x| |x*
| |
and the quartic self-interaction as something a wee bit fancier:
\ \ / /
x\ \x* x/ /x*
\ \ / /
\ \ / /
\ C /
\ /
/ \
/ U \
/ / \ \
/ / \ \
x/ /x* x\ \x*
/ / \ \
Apparently these ideas have spawned a whole field of physics called
"matrix models".
These ideas work not only for Yang-Mills theory but also for Chern-Simons theory, which is a topological quantum field theory: a theory that doesn't require any metric on spacetime to make sense. Here they have been exploited by Dror Bar-Natan to come up with a new formulation of the famous 4-color theorem:
3) Dror Bar-Natan, Lie algebras and the four color theorem, preprint available as q-alg/9606016.
As I explained in "week8" and "week22", there is a way to formulate about the 4-color theorem as a statement about trivalent graphs. In particular, Penrose invented a little recipe that lets us calculate an invariant of trivalent graphs, which is zero for some planar graph only if some corresponding map can't be 4-colored. This recipe involves the vector cross product, or equivalently, the Lie algebra of the group SU(2). You can generalize it to work for SU(N). And if you then consider the N → ∞ limit, you get the above stuff! (The point is that the above stuff also gives a rule for computing a number from any trivalent graph.)
Now as I said, in the N → ∞ limit all the nonplanar Feynman diagrams give negligible results compared to the planar ones. So another way to state the 4-color theorem is this: if the SU(2) invariant of a trivalent graph is zero, the SU(N) invariant is negligible in the N → ∞ limit.
This doesn't yet give a new proof of the 4-color theorem. But it makes it into sort of a physics problem: a problem about the relation of SU(2) Chern-Simons theory and the N → ∞ limit of Chern-Simons theory.
Now, the 4-color theorem is one of the two deep mysteries of 2-dimensional topology - a subject too often considered trivial. The other mystery is the Andrews-Curtis conjecture, discussed in "week23". Often a problem is hard or unsolvable until you get the right tools. Topological quantum field theory is a new tool in topology, so one could hope it'll shed some light on these problems. Bar-Natan's paper is a tantalizing piece of evidence that maybe, just maybe, it will.
One can't really tell yet.
Anyway, I don't really care much about the 4-color theorem per se. If I ever need to color a map I'll hire a cartographer. It's the connections between seemingly disparate subjects that I find interesting. 2-categories are a very abstract formalism developed to describe 2-dimensional ways of glomming things together. Starting from the study of 2-categories, we very naturally get the notions of "monad" and "adjunction". And before we know it, this leads us to some interesting questions about 2-dimensional quantum field theory: for really, the dominance of planar diagrams in the N → ∞ limit of gauge theory is saying that in this limit the theory becomes essentially a 2-dimensional field theory, in some funny sense. And then, lo and behold, this turns out to be related to the 4-color theorem!
By the way, I guess you all know that the 4-color theorem was proved using a computer, by breaking things down into lots of separate cases. (See "week22" for references.) Well, there's a new proof out, which also uses a computer, but is supposed to be simpler:
4) Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas, A new proof of the four-colour theorem, Electronic Research Announcements of the American Mathematical Society 2 (1996), 17-25. Available at http://www.ams.org/journals/era/1996-02-01/
I'm still hoping for the 2-page "physicist's proof" using
path integrals. 
To continue reading the "Tale of n-Categories", click here.
For more on adjunctions and monoid objects, try "week173" and especially "week174".
© 1996 John Baez
baez@math.removethis.ucr.andthis.edu
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