Also available as http://math.ucr.edu/home/baez/week92.html
October 17, 1996
This Week's Finds in Mathematical Physics  Week 92
John Baez
I'm sure most of you have lost interest in my "tale of ncategories",
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 4color theorem and the largeN 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:
\ /
\ /
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 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 ncategory 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 times 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:
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
selfinteraction", where two particles combine to form a third. These
interactions show up in simple textbook theories like the "phi^3 theory"
and, more importantly, in nonabelian gauge field theories like quantum
chromodynamics, where the gauge bosons have cubic selfinteractions. 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 particleantiparticle 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 2categories, 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 2categories 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 highbrow 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 fancylooking (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 soupedup 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 largeN 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 > infinity, 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
infinity 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,
Cambridge, 1989.
Check out section 8.3.1, entitled "the double line representation
and the dominance of planar graphs". Coleman considers YangMills
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 N^{2  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 > infinity limit we can think of the Feynman
diagrams using diagrams like the ones above. For example, we can
think of the cubic selfinteraction in YangMills theory as simply
matrix multiplication:
\ \ / /
x\ \x* x/ /x*
\ \ / /
\ \ / /
\ C /
\ /
 
 
x x*
 
and the quartic selfinteraction 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 YangMills theory but also for ChernSimons
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 BarNatan to come up with a new formulation of the famous 4color
theorem:
3) Dror BarNatan, Lie algebras and the four color theorem, preprint
available as qalg/9606016.
As I explained in "week8" and "week22", there is a way to formulate
about the 4color 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 4colored. 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 > infinity 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 > infinity limit all the nonplanar Feynman diagrams
give negligible results compared to the planar ones. So another way to state
the 4color theorem is this: if the SU(2) invariant of a trivalent graph
is zero, the SU(N) invariant is negligible in the N > infinity limit.
This doesn't yet give a new proof of the 4color theorem. But it makes
it into sort of a *physics* problem: a problem about the relation of
SU(2) ChernSimons theory and the N > infinity limit of ChernSimons
theory.
Now, the 4color theorem is one of the two deep mysteries of 2dimensional
topology  a subject too often considered trivial. The other mystery is
the AndrewsCurtis 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. BarNatan's paper is a tantalizing piece
of evidence that maybe, just maybe, it will.
One can't really tell yet.
I don't really care much about the 4color 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. 2categories
are a very abstract formalism developed to describe 2dimensional ways
of glomming things together. Starting from the study of 2categories,
we very naturally get the notions of "monad" and "adjunction". And
before we know it, this leads us to some interesting questions about
2dimensional quantum field theory: for really, the dominance of planar
diagrams in the N > infinity limit of gauge theory is saying that in
this limit the theory becomes essentially a 2dimensional field theory,
in some funny sense. And then, lo and behold, this turns out to be related to
the 4color theorem!
By the way, I guess you all know that the 4color 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 fourcolour theorem, Electronic Research Announcements
of the American Mathematical Society 2 (1996), 1725. Available at
http://www.ams.org/journals/era/19960201/
I'm still hoping for the 2page "physicist's proof" using path integrals.
:)

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