Also available at http://math.ucr.edu/home/baez/week182.html
June 19, 2002
This Week's Finds in Mathematical Physics (Week 182)
John Baez
It's been a long time, but in the last Week's Finds I was telling you
about my adventures this spring in northern California, and I hadn't
quite gotten around to telling you about that cool conference on
"Nonabelian Hodge Theory" at the MSRI in Berkeley. I'll continue my
story about that now...
...but first, a little detour through the Nile valley!
Egyptians liked to write fractions as the sum of reciprocals of
integers. For example, instead of writing
5/6
those folks would write something like
1/2 + 1/3
Nobody is sure why, but one possibility is that they started with a
neat notation for 1/n, and then wanted to extend this to handle other
fractions, and couldn't think of anything better.
Of course they *could* have written m/n as
1/n + ........... + 1/n
m terms
but they preferred to use as few terms as possible. This leads to some
tricky questions. For example: clearly every fraction of the form 4/n
can be written using 4 terms  but can you always make do with just 3?
Nobody knows! Alan Swett claims to have shown you only need 3 terms if
n is less than or equal to 10^14. For example:
4/8689 = 1/2175 + 1/1718250 + 1/14929874250.
For much more on this, see:
1) David Eppstein, Egyptian fractions,
http://www.ics.uci.edu/~eppstein/numth/egypt/
2) Alan Swett, The ErdosStrauss conjecture,
http://math.uindy.edu/swett/esc.htm
Egyptian fraction problems have a spooky way of showing up in
different unrelated mathematical contexts... which have a spooky way
of turning out not to be unrelated after all!
For example, suppose we are trying to classify all the Platonic
solids. We're looking for ways to tile the surface of a sphere
with regular ngons, with m meeting at each vertex. Suppose there
is a total of V vertices, E edges, and F faces. Since the Euler
characteristic of the sphere is 2, we have
V  E + F = 2.
Since each face has n edges but 2 faces meet along each edge,
we have
nF = 2E.
Since each vertex has m edges meeting it but each edge
meets 2 vertices, we also have
mV = 2E.
Putting these equations together we get
2E(1/n + 1/m  1/2) = 2
or
1/n + 1/m = 1/2 + 1/E.
An Egyptian fractions problem! It's obvious that this can only
have solutions if 1/n + 1/m > 1/2. And interestingly, all the
solutions of this inequality do indeed correspond to Platonic solids...
at least if n,m > 2. Here they are:
(n,m) = (3,3) tetrahedron
(n,m) = (3,4) octahedron
(n,m) = (4,3) cube
(n,m) = (3,5) icosahedron
(n,m) = (5,3) dodecahedron
The cases n = 1,2 don't give Platonic solids in the usual sense:
after all, most people don't like polygons to have just 1 or 2 edges.
Neither do the cases m = 1,2, since most people don't like polyhedra
to have just 1 or 2 faces meeting at a vertex!
One can argue about whether these are irrational prejudices. But it's
actually good to study *all* unordered pairs of natural numbers with
1/n + 1/m > 1/2
since they correspond to *all* the isomorphism classes of finite subgroups
of the rotation group! The Platonic solids have their symmetry groups,
which don't change when we switch n and m. The solution (n,1)
corresponds to the cyclic group Z_n: the symmetries of a regular ngon,
where you're not allowed to flip it over. The solution (n,2)
corresponds to the dihedral group D_n: the symmetries of a regular ngon
where you *are* allowed to flip it over.
In some weird sense, maybe we should think of Z_n and D_n as the
symmetry groups of Platonic solids with only 1 or 2 faces. I'll
leave you to ponder the Platonic solids with only 1 or 2 vertices.
If you get stuck, look up the word "hosohedron"!
The story gets better if we also consider solutions of
1/n + 1/m = 1/2
which formally correspond to Platonic solids where the number
E of edges is infinite. In fact, these correspond to tilings
of the plane by regular polygons:
(n,m) = (3,6): tiling by triangles
(n,m) = (6,3): tiling by hexagons
(n,m) = (4,4): tiling by squares
Similarly, solutions of
1/n + 1/m < 1/2
give tilings of the hyperbolic plane: for example, Escher used
(n,m) = (6,4) in some of his prints.
Let me try to arrange this information in a table, using lines
to separate the spherical, planar and hyperbolic regions:
n=1 n=2 n=3 n=4 n=5 n=6 n=7
m=1 Z_1 Z_2 Z_3 Z_4 Z_5 Z_6 Z_7
m=2 Z_2 D_2 D_3 D_4 D_5 D_6 D_7
=======
m=3 Z_3 D_3 tetrahedron cube dodecahedron  hexagonal 
 tiling 
=============
m=4 Z_4 D_4 octahedron  square 
 tiling 
 
m=5 Z_5 D_5 isosahedron 

  hyperbolic tilings
m=6 Z_6 D_6  triangular 
 tiling 
 
m=7 Z_7 D_7 

It's not very pretty in ASCII, but hopefully you get the idea!
Now, the same Egyptian fraction problem comes up when studying other
problems, too. For example, suppose you are trying to find a basis of
R^n consisting of unit vectors that are all at 90degree or 120degree
angles from each other. We can describe a problem like this by drawing
a bunch of dots, one for each vector, and connecting two dots with an
edge when they're supposed to be at a 120degree angle from each other.
If two dots are not connected, they should be at right angles to one
another.
So, for example, this diagram tells us to find a basis for R^3
consisting of unit vectors all at 120 degree angles from each other:
o
/ \
/ \
oo
It's easy to see this is impossible, since three vectors all
at 120 degrees from each must lie in a plane  so they can't be
linearly independent. On the other hand, this diagram gives a
solvable problem:
ooo
You just pick two unit vectors at right angles to each other and
wiggle the third one around until it's at a 120degree angle to both.
It's not hard.
So, the question is: which diagrams give solvable problems?
This is actually a very fun puzzle: it's very famous, but most books
manage to make it seem really boring and "technical", so you should
really spend some time thinking about it for yourself. I'll give away
the answer, but I won't say how you prove it's true.
First, it's easy to see that if a diagram consists of a bunch of separate
pieces, and you can solve the problem for each piece, you can solve
the problem for the whole diagram. So, it's sufficient to consider
the case of connected diagrams.
Second, a connected diagram can only give a solvable problem if it's
Yshaped, like this:
o

o

oooooooooo
Third, a diagram like this gives a solvable problem only if
1/k + 1/n + 1/m > 1
where (k,n,m) are the numbers labelling the tips of the Y when we
number it like this:
3

2

4321234567
So for example, this particular problem is not solvable because
1/4 + 1/3 + 1/7 < 1.
Now, it's easy to see what we can only get 1/k + 1/n + 1/m > 1 if
one of the numbers is 1 or 2. If one of the numbers is 1, our
"Yshaped" diagram is actually just a straight line of dots!
We can also describe this straight line by taking one of the
numbers to be 2, like this:
2123456
except for the boring case where we have just a single dot.
So, let's assume one of the numbers is 2. By symmetry we can
assume this number is k. We are thus looking for pairs (n,m)
with
1/2 + 1/n + 1/m > 1
or in other words
1/n + 1/m > 1/2.
This is the same problem as before! So the problem we're dealing
with now is very much like classifying Platonic solids!
Even better, these diagrams I've been drawing are called "Dynkin
diagrams", and we can use them to get certain incredibly important
finitedimensional Lie algebras called "simplylaced simple Lie algebras".
For a taste of how this works, reread "week65" and some previous Weeks.
Similarly, we get certain *infinitedimensional* Lie algebras
called "simplylaced affine Lie algebras" when
1/n + 1/m = 1/2,
and "simplylaced hyperbolic KacMoody algebras" when
1/n + 1/m < 1/2.
So, our whole big table above translates into a table of Lie algebras!
Let me draw it with the standard names of these Lie algebras below their
diagrams. Unfortunately, I'll have to make it very small to fit
everything in. So, for example, I'll draw the socalled E8 Dynkin
diagram:
o

ooooooo
as this puny miserable thing:
o
oooooo
This is what we get:
n=1 n=2 n=3 n=4 n=5 n=6
o o o o o o
m=1 o oo ooo oooo ooooo ooooooo
A2 A3 A4 A5 A6 A7
o o o o o o
m=2 oo ooo oooo ooooo oooooo oooooooo
A3 D4 D5 D6 D7 D8
==
 
o o o o o  o 
m=3 ooo oooo ooooo oooooo ooooooo  ooooooooo 
 
A4 D5 E6 E7 E8  E8^1 
 
===========
 
o o o  o  o o
m=4 oooo ooooo oooooo  ooooooo  oooooooo oooooooooo
 
A5 D6 E7  E7^1 
 
 

o o o  o o o
m=5 ooooo oooooo ooooooo  oooooooo ooooooooo ooooooooooo

A6 D7 E8 
 hyperbolic KacMoody algebras
 
 
o o  o  o o o
m=6 oooooo ooooooo  oooooooo  ooooooooo oooooooooo oooooooooooo
A7 D8  
 E8^1 
 
 


This mysterious way that the same Egyptian fraction problem shows up
in classifying Platonic solids and simplylaced simple Lie algebras is
actually the tip an iceberg sometimes called the "McKay correspondence" 
though important aspects of it go back to the theory of Kleinian
singularities. I talked about the McKay correspondence in "week65",
so that's a good place to dig deeper, but you should really
look at some of the references in there, and also these two 
both of which explain the mysterious word "hosohedron":
3) H. S. M. Coxeter, Generators and relations for discrete groups,
Springer, Berlin, 1984.
4) Joris van Hoboken, Platonic solids, binary polyhedral groups,
Kleinian singularities and Lie algebras of type A,D,E, Master's
Thesis, University of Amsterdam, 2002, available at
http://home.student.uva.nl/joris.vanhoboken/scriptiejoris.ps
or http://math.ucr.edu/home/baez/joris_van_hoboken_platonic.pdf
Okay. Now  back to that conference at the Mathematical Sciences
Research Institute! You can look at transparencies and watch videos
of the talks here:
5) MSRI streaming video archive, Spring 2002,
http://www.msri.org/publications/video/index04.html
If you like watching math talks, there's a lot to see here  not just
this one conference, but all the MSRI conferences! For example, right
after the nonabelian Hodge theory conference there was one on conformal
field theory and supersymmetry, featuring talks by bigshots like Richard
Borcherds, Dan Freed, Igor Frenkel, Victor Kac, and JeanBernard Zuber 
just to name a few. You can see talks by all these folks.
But anyway, let me start by telling you what nonabelian Hodge theory is....
Hmm. I guess I should *start* by telling you what *abelian* Hodge
theory is!
In its simplest form, Hodge theory talks about how differential forms on
a smooth manifold get extra interesting structure when the manifold has
extra interesting structure. To warm up, let me remind you about what
we can do when our manifold has *no* extra interesting structure.
Whenever we have a smooth manifold M there's an "exterior derivative"
operator d going from pforms on M to (p+1)forms on M. This is just a
generalization of grad, curl, div and all that. In particular it
satisfies
d^2 = 0,
so the space of "closed" pforms:
{w: dw = 0}
contains the space of "exact" pforms:
{w: w = du for some u}.
This makes it fun to look at the vector space of closed pforms modulo
exact pforms. This is called the "pth de Rham cohomology group of M",
or
H^p(M)
for short. It only depends on the topology of M; its size keeps track
of the number of pdimensional holes in M. When M is compact, it
agrees with the cohomology computed in a bunch of other ways that
topologists like.
Fine. But now, suppose M has a Riemannian metric on it! Then we can
write down a version of the Laplacian for differential forms. A
function is a 0form, so we're just generalizing the Laplacian you
already know and love. Differential forms whose Laplacian is zero are
called "harmonic". Every harmonic pform is closed, but if M is compact
life is even better: the vector space of harmonic pforms is isomorphic
to the pth de Rham cohomology of M.
This is great: it means the de Rham cohomology, which only depends on
the *topology* of M, can also be thought of as the space of solutions
of a *differential equation* on M! This gets topologists and analysts
talking to each other, and has all sorts of marvelous spinoffs and
generalizations.
Some people call this stuff "Hodge theory". But Hodge theory
goes further when M has more structure  most notably, when it's
a Kaehler manifold!
A Kaehler manifold is to the complex plane as a Riemannian manifold is
to the real line. More precisely, it's is a manifold whose tangent
spaces have been made into *complex* vector spaces and equipped with a
*complex* inner product. Of course the real part of the inner product
makes it into a Riemannian manifold. That lets us parallel transport
vectors, so we demand a compatibility condition: parallel transporting
a vector and then multiplying it by i is the same as multiplying it by i
and then parallel transporting it! This makes complex analysis work
well on Kaehler manifolds.
Now, if you've taken complex analysis, you may remember how people use
it to find solutions of Laplace's equation... like when they're studying
electrostatics, or the flow of fluids with no viscosity or vorticity 
an idealization that von Neumann mockingly called "dry water". On
the complex plane we can talk about "holomorphic" functions, which
satisfy the CauchyRiemann equation:
_ _
df/dz = 0 (note: df/dz = df/dx + i df/dy)
and also the complex conjugates of these, called "antiholomorphic"
functions, which satisfy
df/dz = 0 (note: df/dz = df/dx  i df/dy)
Both holomorphic and antiholomorphic functions are automatically harmonic,
so we can find solutions of Laplace's equation this way. But even better,
every harmonic function is a linear combination of a holomorphic and an
antiholomorphic one!
All this stuff works much more generally for pforms on Kaehler
manifolds. To get going, let's think a bit more about the complex
plane. If we have any 1form on the complex plane we can write it as a
linear combination of dx and dy, where x and y are the usual coordinates
on the plane. But things get nicer if we work with *complexvalued*
differential forms. Then we can form linear combinations like
dz = dx + idy
and
_
dz = dx  idy
and express any 1form as a linear combination of *these* in a unique way.
We call these the (1,0) and (0,1) parts of our 1form.
This means that if we have a function f, we can take its exterior
derivative of f and chop it into its (1,0) part and (0,1) part:
_
df = Df + Df
_
These guys D and D are called "Dolbeault operators". People usually
write them using nice curly lowercase d's like you see in a partial
derivative, but I can't do that here: I'm a prisoner of low technology!
Anyway, it turns out that
_
Df = 0
is just a slick way of writing CauchyRiemann equation, which says that
f is holomorphic. You should check this for yourself! Similarly,
Df = 0
says that f is antiholomorphic.
Now let me say how all this stuff generalizes to arbitrary Kaehler
manifolds. We can decompose any pform on a Kaehler manifold into
its (i,j) parts where i+j = p. For example, a (1,2)form
in 4 dimensions might look something like this in complex coordinates:
_ _ _ _
f dz_1 ^ dz_3 ^ dz_2 + g dz_2 ^ dz_3 ^ dz_4.
We have
_
d = D + D
_
where D maps (i,j)forms to (i+1,j)forms, while D maps (i,j)forms
to (i,j+1)forms. This allows us to take the de Rham cohomology
groups of our manifold M and write them as a direct sum of smaller
vector spaces, which I'll call
H^{i,j}(M)
for short.
So far I don't think I've used anything about the metric on M, so all
this would work whenever M is a socalled "complex manifold". But if we
really have a Kaehler manifold, and it's compact, we can say more: a
pform is harmonic if and only if all its (i,j) parts are. This
means H^{i,j}(M) is isomorphic to the space of harmonic (i,j)forms.
_
Alternatively, you can describe H^{i,j}(M) just in terms of D:
you just take the (i,j)forms in here:
_
{w: Dw = 0}
modulo those in here:
_
{w: w = Du for some u}
This is called the "(i,j)th Dolbeault cohomology group of M".
That's Hodge theory in a nutshell. There's even *more* you can
do when M is a Kaehler manifold, but I'm getting a little tired,
so I'll just let you read about that here:
6) R. O. Wells, Differential analysis on complex manifolds,
Springer, Berlin, 1980.
This is a really *great* book for learning about all sorts of good
geometry stuff, starting with differential forms and working on up
through Hodge theory, pseudodifferential operators, sheaves and so on.
But anyway, I've given you a little taste of Hodge theory.
The main thing to remember is that when your manifold is complex,
the cohomology becomes "bigraded": instead of just
H^p(M)
you get
H^{i,j}(M).
So now, what's nonabelian Hodge theory?
The basic idea is simple: instead of askng what extra structure the
*homology groups* get when M is a complex manifold, we ask what
extra structure the *homotopy type* of M gets when M is a complex
manifold. The homotopy type includes invariants like the homotopy
groups, but also more. How are these constrained by the fact that
M is complex?
Unfortunately, to describe the answer  even a little teeny part of the
answer  I need to turn up the math level a notch.
For starters we can consider the fundamental group pi_1(M). But this is
hard to relate to differential geometry, so we will immediately water it
down by picking an algebraic group G and looking at homomorphisms of
pi_1(M) into G. These are basically the same thing as flat Gbundles
over M, so it's easier to see how M being a complex manifold affects
things. We can even be sneaky and study this for all G at once by forming
a group PI_1(M) called the "proalgebraic completion" of pi_1(M).
This is a proalgebraic group (an inverse limit of algebraic groups)
which a contains pi_1(M) and has the property that any homomorphism from
pi_1(M) into an algebraic group G extends uniquely to a proalgebraic
group homomorphism from PI_1(M) to G.
It's nice to ask what extra structure PI_1(M) gets when M is
a complex manifold, because this question has a nice answer.
To get ready for how nice the answer is, first go back to plain old
abelian Hodge theory. Note that making the cohomology of M bigraded
gives an obvious way for the algebraic group C*, the nonzero complex
numbers, to act on the cohomology. The reason is that for each integer
there's a representation of C* where the number z acts as multiplication
by z^n, so gradings are just another way of talking about C* actions.
Since the cohomology of M is automatically graded, putting *another*
grading on it amounts to letting C* act on it.
So in plain old Hodge theory, the answer to "What extra structure
does the cohomology of M get when M is complex?" is:
"It gets an action of C*!"
And it turns out that in nonabelian Hodge theory, the answer to
"What extra structure does Pi_1(M) get when M is complex?" is:
"It gets an action of C*!"
This is incredibly cool, but the story goes a lot further. The
fundamental group is just the beginning; you can do something similar
for the higher homotopy groups  but it's a lot more subtle. In fact,
you can do something similar directly to the homotopy type of M! When M
is a compact complex manifold, there's a homotopy type called the
"schematization of M" whose fundamental group is PI_1(M)  and there's
an action of C* on this homotopy type!
By the way, when M is a compact Kaehler manifold the action of C* on its
cohomology extends to a natural action of SL(2,C), as explained in
Wells' book. I wonder if SL(2,C) acts on the schematization of M?
I learned about most of this fancy stuff from an incredibly lucid
talk by Bertrand Toen. Unfortunately there seems to be no video of his
talk, since he gave it down the hill at U. C. Berkeley instead of at
the MSRI  and the handwritten notes at the MSRI website are rather
illegible. So you want to learn more about this, you should probably
start with this quick summary of abelian Hodge theory:
7) Tony Pantev, Review of abelian Hodge theory,
http://www.msri.org/publications/ln/msri/2002/introstacks/pantev/1/index.html
and then take the deep plunge into this paper:
8) Ludmil Katzarkov, Tony Pantev and Bertrand Toen, Schematic homotopy
types and nonabelian Hodge theory I: The Hodge decomposition,
available at math.AG/0107129.
There are a lot of model categories and ncategories lurking in
the background of this subject, as well as ideas that originated
in physics, like "Higgs bundles". For the brave reader I recommend
these papers:
9) Bertrand Toen, Toward a Galoisian interpretation of homotopy theory,
available as math.AT/0007157.
This answers the question: "the fundamental group is to covering
spaces as the whole homotopy type is to... what?" The fact that
it's in French probably makes it easier to understand.
10) Bertrand Toen and Gabriele Vezzosi, Algebraic geometry over model
categories (a general approach to derived algebraic geometry),
available as math.AG/0110109.
This is only for badass mathematicians who find algebraic geometry
and homotopy theory insufficiently mindblowing when taken separately.
Ever wondered what an affine scheme would be like if you replaced the
ground field by an E_infinity ring spectrum? Then this is for you.
(I thank David Eppstein for pointing out the work of Alan Swett.)

Geometry may sometimes appear
to take the lead over analysis,
but in fact precedes it only
as a servant goes before his master
to clear the path and light him
on his way.  James Sylvester

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
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http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html