 #1
John Baez
Also available at http://math.ucr.edu/home/baez/week230.html
May 5, 2006
This Week's Finds in Mathematical Physics (Week 230)
John Baez
As we've seen in previous weeks, Mars is a beautiful world, but a
world in a minor key, a world whose glory days  the Hesperian Epoch 
are long gone, whose once grand oceans are now reduced to windy canyons
and icy dunes. Let's say goodbye to it for now... leaving off
with this Martian sunset, photographed by the rover Spirit in
Gusev Crater on May 19th, 2005:
1) A moment frozen in time, NASA Mars Exploration Rover Mission,
http://marsrovers.nasa.gov/gallery/press/spirit/20050610a.html
This week I'll talk about Dynkin diagrams, quivers and Hall algebras.
But first, some cool identities!
My student Mike Stay did computer science before he came to UCR.
When he was applying, he mentioned a result he helped prove, which
relates Goedel's theorem to the Heisenberg uncertainty principle:
2) C. S. Calude and M. A. Stay, From Heisenberg to Goedel via Chaitin,
International Journal of Theoretical Physics, 44 (2005), 10531065.
Also available at http://math.ucr.edu/~mike/
Now, this particular combination of topics is classic crackpot fodder.
People think "Gee, uncertainty sounds like incompleteness, they're
both limitations on knowledge  they must be related!" and go off the
deep end. So I got pretty suspicious until I read his paper and
saw it was CORRECT... at which point I *definitely* wanted him around!
The connection they establish is not as precise as I'd like, but it's
solid math.
So, now Mike is here at UCR working with me on quantum logic and
quantum computation using ideas from category theory. In his spare
time, he sometimes fools around with math identities and tries to
categorify them  see "week184" and "week202" if you don't know what
that means. Anyway, maybe that's how he stumbled on this:
3) Jonathan Sondow, A faster product for pi and a new integral
for ln(pi/2), Amer. Math. Monthly 112 (2005), 729734. Also
available as math.NT/0401406.
In this paper, Sondow gives eerily similar formulas for some of
our favorite math constants. First, one for e:2 1/1 2^2 1/2 2^3 x 4 1/3 2^4 x 4^4 1/4
e = (  ) (  ) (  ) (  ) ...
1 1 x 3 1 x 3^3 1 x 3^6 x 5Then, one for pi/2:pi 2 1/2 2^2 1/4 2^3 x 4 1/8 2^4 x 4^4 1/16
 = (  ) (  ) (  ) ( ) ...
2 1 1 x 3 1 x 3^3 1 x 3^6 x 5Then one for e^{gamma}, where gamma is Euler's constant:gamma 2 1/2 2^2 1/3 2^3 x 4 1/4 2^4 x 4^4 1/5
e = (  ) (  ) (  ) (  ) ...
1 1 x 3 1 x 3^3 1 x 3^6 x 5He also points out Wallis' product for pi and Pippenger's for e:pi 2 1/1 2x4 1/1 4x6x6x8 1/1
 = (  ) (  ) (  ) ...
2 1 3x3 5x5x7x7
2 1/2 2x4 1/4 4x6x6x8 1/8
e = (  ) (  ) (  ) ...
1 3x3 5x5x7x7What does it all mean? I haven't a clue! Another mystery thrown
down to us by the math gods, like a bone from on high... we can merely
choose to chew on it or not, as we wish.
Julie Bergner gave a great talk on "derived Hall algebras" at the
Mac Lane memorial conference. I just want to explain the very
first result she mentioned, due to Ringel  a surprising trick for
constructing certain quantum groups from simplylaced Dynkin diagrams.
It's very different from the *usual* method for getting quantum groups
from Dynkin diagrams, and it's a miracle that it works.
But, I guess I should start near the beginning!
Way back in 1995, in "week62", "week63", "week64" and "week65",
I explained how "Dynkin diagrams"  little gizmos like this:
o

ooooooo
show up all over mathematics. They have a strange way of tying
together subjects that superficially seem completely unrelated.
In one sense people understand how they work, but in another sense
they're very puzzling  their power keeps growing in unexpected ways.
I love mysterious connections, so as soon I understood enough about
Dynkin diagrams to appreciate them, I became fascinated by them,
and I've been studying them ever since. I explained their relation
to geometry in "week178", "week179", "week180", "week181" and "week182",
and their relation to quantum deformation and combinatorics in
"week186" and "week187".
You might think that would be enough  but you'd be wrong, way wrong!
I haven't really talked about the most mysterious aspects of Dynkin
diagrams, like their relation to singularity theory and representations
of quivers. That's because these aspects were too mysterious!
I didn't understand them *at all*. But lately, James Dolan and Todd
Trimble and I have been making some progress understanding these aspects.
First, I should remind you how Dynkin diagrams infest so much of
mathematics. Let's start with a little puzzle mentioned in "week182".
Draw n dots and connect some of them with edges  at most one edge
between any pair of dots, please:
o
/ \
/ \
o ooooo
/
/
o oo
Now, try to find a basis of R^n consisting of one unit vector per dot,
subject to these rules: if two dots are connected by an edge, the angle
between their vectors must be 120 degrees, but otherwise their vectors
must be at right angles.
This sounds like a silly puzzle that only a mathematician could give a
hoot about. It takes a while to see its magnificent depth. But anyway,
it turns out you can solve this problem only for certain special diagrams
called "simplylaced Dynkin diagrams". The basic kinds are called A_n,
D_n, E_6, E_7, and E_8.
The A_n Dynkin diagram is a line of n dots connected by edges like this:
ooooo
The D_n diagram has n dots arranged like this:
o
/
ooooo
\
o
A line of them but then a little fishtail at the end! We should
take n to be at least 4, to make the diagram connected and different
from A_n.
The E_6, E_7, and E_8 diagrams look like this:
o o o
  
ooooo oooooo ooooooo
You're also allowed to take disjoint unions of the above diagrams.
So, a weird problem with a weird answer! Its depth is revealed
only when we see that many DIFFERENT puzzles lead us to the SAME
diagrams. For example:
A) the classification of integral lattices in R^n having a basis of
vectors whose length squared equals 2
B) the classification of simply laced semisimple Lie groups
C) the classification of finite subgroups of the 3d rotation group
D) the classification of simple singularities
E) the classification of tame quivers
Let me run through these problems and say a bit about how they're
connected:
A) An "integral lattice" in R^n is a lattice where the dot product
of any two vectors in the lattice is an integer. There are zillions
of these  but if we demand that they have a basis of vectors whose
length squared is 2, we can only get them from simplylaced Dynkin
diagrams.
It's not very hard to see that finding a lattice like this is equivalent
to the puzzle I mentioned earlier. For example, given a solution of
that puzzle, you can just multiply all your vectors by sqrt(2) and
form the lattice of their integer linear combinations.
So, this connection is not so deep. But it connects to something
a bit deeper...
B) Lie groups are fundamental throughout math and physics: they're
groups of continuous symmetries, like rotations. The nicest of the
lot are the semisimple Lie groups. Some familiar examples are the
group of rotations in ndimensional space, which is called SO(n),
and the group of unitary matrices with determinant 1, which is
called SU(n). There are more, but people know what they all are.
They're classified by Dynkin diagrams!
Why? The key point is that any semisimple Lie group has a "root lattice".
This is an integral lattice spanned by special vectors called "roots".
I won't give the details, since I explained this stuff in "week63"
and "week64", but it turns out that root lattices, and thus semisimple
Lie groups, are classified by Dynkin diagrams.
Not all these Dynkin diagrams look like the A, D and E diagrams listed
above. But, it turns out that the length squared of any root must be
either 1 or 2. If all the roots have length squared equal to 2, we
say our semisimple Lie group is "simply laced". In this case, we're
back to problem B), which we already solved! So then our Lie group
corresponds to a diagram of type A, D, or E  or a disjoint union of
such diagrams.
Here's how it goes:
* The diagram A_n gives the compact Lie group SU(n+1),
consisting of (n+1) x (n+1) unitary matrices with determinant 1.
It's the isometry group of complex projective nspace.
* The diagram D_n gives the compact Lie group SO(2n),
consisting of 2n x 2n orthogonal matrices with determinant 1.
It's the isometry group of real projective (2n)space.
* The diagram E_6 gives a 78dimensional compact Lie group
that people call E_6. It's the isometry group of the bioctonionic
projective plane.
* The diagram E_7 gives a 133dimensional compact Lie group
that people call E_7. It's the isometry group of the quateroctonionic
projective plane.
* The diagram E_8 gives a 248dimensional compact Lie group
that people call E_8. It's the isometry group of the octooctonionic
projective plane.
In short, two regular series and three exotic weirdos.
You may ask where the rotation groups SO(n) with n odd went!
Well, these correspond to fancier Dynkin diagrams that aren't
simply laced, like this:
oooo>o
The funny arrow here indicates that the last two vectors aren't
at a 120degree angle; they're at a 135degree angle, and the last
vector is shorter than the rest: it has length one instead of sqrt(2).
Why are semisimple Lie groups "better" when they're simply laced?
What's the big deal? I don't really understand this, but for one,
when all the roots have the same length, they're all alike 
a certain symmetry group called the Weyl group acts transitively on them.
Anyway, so far our A, D, E Dynkin diagrams have been classifying
things that are clearly related to lattices. But now things get
downright spooky...
C) Take a Platonic solid and look at its group of rotational symmetries.
You get a finite subgroup of the 3d rotation group SO(3). But in
general, finite subgroups of SO(3) are classified by ADE Dynkin
diagrams!
So, Platonic solids turn out to fit into the game we're playing here!
First I'll say which diagram corresponds to which subgroup of SO(3).
Then I'll explain how the correspondence works:
* The diagram A_n corresponds to the group of obvious rotational symmetries
of the regular ngon. This group is called the "cyclic group" Z/n.
* The diagram D_n corresponds to the group of rotational symmetries
of the regular ngon where you can turn it and also flip it over.
By sheer coincidence, this group is called the "dihedral group" D_n.
A cosmic stroke of good luck!
* The diagram E_6 corresponds to the group of rotational symmetries
of the tetrahedron: the "tetrahedral group". This is also the
group of even permutations of 4 elements, the "alternating group" A_4 
not to be confused with the A_n's we were just talking about.
A cosmic stroke of bad luck!
* The diagram E_7 corresponds to the group of rotational symmetries
of the octahedron or cube: the "octahedral group". This is also
the group of all permutations of 4 elements, the "symmetric group" S_4.
* The diagram E_8 corresponds to the group of rotational symmetries
of the icosahedron or dodecahedron: the "icosahedral group". This
is also the group of even permutations of 5 elements, called A_5. Darn!
So, the exceptional Lie groups E_6, E_7 and E_8 correspond to Platonic
solids in a sneaky way.
To understand what's going on here, first we need to switch from SO(3)
to SU(2). The group SU(2) is used to describe rotations in quantum
mechanics: it's the double cover of the rotation group SO(3).
It's really finite subgroups of SU(2) that are classified by ADE Dynkin
diagrams! It just so happens that these correspond, in a slightly
slippery way, to finite subgroups of SO(3).
You'll see how if I list the finite subgroups of SU(2):
* The diagram A_n corresponds to the cyclic subgroup Z/n of SU(2).
This double covers a cyclic subgroup of SO(3) when n is even.
* The diagram D_n corresponds to a subgroup of SU(2) that double
covers the dihedral group D_n.
* The diagram E_6 corresponds to a subgroup of SU(2) that double
covers the rotational symmetries of the tetrahedron. This
subgroup has 24 elements and it's called the "binary tetrahedral group".
* The diagram E_7 corresponds to a subgroup of SU(2) that double
covers the rotational symmetries of the octahedron. This
subgroup has 48 elements and it's called the "binary octahedral group".
* The diagram E_8 corresponds to a subgroup of SU(2) that double
covers the rotational symmetries of the icosahedron. This
subgroup has 120 elements and it's called the "binary icosahedral group".
Now, how does the correspondence work? For this, I'm afraid I have to
raise the sophistication level a bit  I've been trying to keep things
simple, but it's getting tough.
In his book on the icosahedron, Felix Klein noticed it was interesting
to let the icosahedral group act on the Riemann sphere, and look for
rational functions invariant under this group.
It turned out that every such function depends on a single one:
Klein's icosahedral function! The explict formula for it is pretty
disgusting, but it's a beautiful thing: you can pick it so that it
equals 0 at all the vertices of the icosahedron, 1 at the midpoints
of the edges, and infinity at the midpoints of the faces. Even
better, if you write the function like this:
w = f(z)
then Klein showed that knowing how to solve for z as a function of w
lets you solve every quintic equation! The reason is that the Galois
group of the general quintic is a close relative of the icosahedral
group: the former is S_5, the latter is A_5.
Anyway, when I said that "every such function depends on a single one",
what I really meant was this. Let C(z) be the field of rational functions
of one variable; then the icosahedral group acts on this, and the invariant
functions form a subfield C(w) where w is Klein's icosahedral function.
The Galois group of the little field in the big one is the icosahedral
group.
The same kind of thing works for the other finite subgroups of SO(3),
except of course for the connection to the quintic equation.
But, it's actually even better to think about finite subgroups of
SU(2), since SU(2) acts on C^2, and when we *projectivize* C^2 we
get SO(3) acting on the Riemann sphere. This viewpoint fits more
squarely into the worldview of algebraic geometry.
If we take the quotient of C^2 by a finite subgroup G of SU(2), we
don't get a smooth manifold: the quotient has a singularity at 0.
But we can "resolve" the singularity, finding a smooth complex manifold
with a holomorphic map
p: M > C^2/G
that has a holomorphic inverse on a dense open set. There may be
lots of ways to do this, but in the present case there's just one
"minimal" resolution, meaning a resolution that every other resolution
factors through.
Then  and here's the magic part!  the inverse image of 0 in M
turns out to be the union of a bunch of Riemann spheres. And
if we draw a dot for each sphere, and an edge between these dots
whenever their spheres intersect, we get a simply laced Dynkin
diagram on the above list!
Well, almost. We get this diagram with an extra dot thrown in,
connected by some extra edges in a specific way. This is called
the "extended" Dynkin diagram. It also shows up naturally from
the Lie group viewpoint, when we consider central extensions of
loop groups.
That's *one* way the correspondence works. Another way, discovered
by McKay, is to draw a dot for each irrep of G. There's one special
dot, since there's always a 2dimensional irrep coming from the action
of SU(2) on C^2. Let's just call this irrep C^2. Then, draw an edge
from the dot R to the dot S whenever the irrep S shows up in the rep
R tensor C^2. You get the same extended Dynkin diagram as before!
This second way is called the "McKay correspondence". The first way
is sometimes called the "geometric McKay correspondence", though I think
it was discovered earlier.
Now we're well on the road to the next item...
D) Simplylaced Dynkin diagrams also classify the simple critical points
of holomorphic functions
f: C^3 > C
A "critical point" is just a place where the gradient of f vanishes.
We can try to classify critical points up to a holomorphic change of
variables. It's better to classify their "germs", meaning we only
look at what's going on *right near* the critical point. But, even
this is hopelessly complicated unless we somehow limit our quest.
To do this, we can restrict attention to "stable" critical points,
which are those that don't change type under small perturbations.
But we can do better: we can classify "simple" critical points,
namely those that change into only finitely many other types under
small perturbations.
These correspond to simplylaced Dynkin diagrams!
First I'll say which diagram corresponds to which type of critical
point. To do this, I'll give a polynomial f(x,y,z) that has a certain
type of critical point at x = y = z = 0. Then I'll explain how
the correspondence works:
* The diagram A_n corresponds to the critical point of x^{n+1} + y^2 + z^2.
* The diagram D_n corresponds to the critical point of x^{n1} + xy^2 + z^2.
* The diagram E_6 corresponds to the critical point of x^4 + y^3 + z^2.
* The diagram E_7 corresponds to the critical point of x^3 y + y^3 + z^2.
* The diagram E_8 corresponds to the critical point of x^5 + y^3 + z^2.
Here's how the correspondence works. For each of our Dynkin diagrams
we have a finite subgroup of SU(2), thanks to item C). This subgroup
acts on the ring of polynomials on C^2, so we can form the subring of
invariant polynomials. This turns out to be generated by three polynomials
that we will arbitrarily call x, y, and z. But, they satisfy one relation,
given by the polynomial above!
Conversely, we can start with the polynomial
f: C^3 > C
The zero set
{f = 0}
has an isolated singularity at the origin. But, we can resolve
this singularity, finding a smooth complex manifold N with a
holomorphic map
q: N > {f = 0}
that has a holomorphic inverse on a dense open set. There may be
lots of ways to do this, but in the present case there's just one
"minimal" resolution, meaning one that every other resolution
factors through this one.
Then  and here's the magic part!  the inverse image of 0 in N
turns out to be the union of a bunch of Riemann spheres. And
if we draw a dot for each sphere, and an edge between these dots
whenever their spheres intersect, we get back our simply laced
Dynkin diagram!
This whole section should have given you a feeling of deja vu.
It's a lot like section D). If I were smarter, I'd probably see
how it's *exactly* the same stuff, repackaged slightly.
The last item on our list seems different...
E) A quiver is just a category freely generated by some set of
morphisms. To specify a quiver we just write down some dots
and arrows. The dots are the objects of our category; the
arrows are the generating morphisms.
A representation of a quiver Q is just a functor
F: Q > Vect
So, we get a vector space for each dot and a linear map for
each arrow, with no extra restrictions. There's an obvious
category of representations Rep(Q) of a quiver Q.
A guy named Gabriel proved a divine result about these categories
Rep(Q). We say a quiver Q is "tame" if Rep(Q) has finitely
many indecomposable objects  objects that aren't direct sums of
others. And, it turns out the tame quivers are just those coming
from simplylaced Dynkin diagrams!
Actually, for this to make sense, you need to take your Dynkin
diagram and turn it into a quiver by putting arrows along the edges.
If you have an ADE Dynkin diagram, you get a tame quiver no matter
which way you let the arrows point.
There's clearly a lot of mysterious stuff going on here. In
particular, this last item sounds completely unrelated to the
rest. But it's not! There are cool relationships between quivers
and quantum groups, which tie this item to the rest.
I'll just mention one  the one Julie Bergner started her talk with.
For this, you need to know a bit about abelian categories.
Abelian categories are categories like the category of abelian groups,
or more generally the category of modules of any ring, where you can
talk about chain complexes, exact sequences and stuff like that.
You can see the precise definition here:
4) Abelian categories, Wikipedia,
http://en.wikipedia.org/wiki/Abelian_category
and learn more here:
5) Peter Freyd, Abelian Categories, Harper and Row, New York, 1964.
Also available at http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html
It's really interesting to study the "Grothendieck group" K(A) of an
abelian category A. As a set, this consists of formal linear combinations
of isomorphism classes of objects of A, where we impose the relations
[a] + = [a + b]
where a + b on the right side is the direct sum of the objects
a and b. It becomes an abelian group in an obvious way.
For example, if A is the category of representations of some group G,
it's an abelian category and K(A) is called the "representation ring"
of A  it's a ring because we tensor representations. Or, if A is the
category of vector bundles over a space X, it's again abelian, and K(A)
is called the "Ktheory of X".
The Hall ring H(A) of an abelian category is a vaguely similar idea.
As a set, this consists of formal linear combinations of isomorphism
classes of objects of A. No extra relations! It's an abelian group
with the obvious addition. But the cool part is, with a little luck,
we can make it into a *ring* by letting the product [a] be the
sum of all isomorphism classes of objects [x] weighted by the number
of isomorphism classes of short exact sequences
0 > a > x > b > 0
This only works if the number is always finite.
So far when speaking of "formal linear combinations" I've been
implicitly using integer coefficients, but people seem to prefer
complex coefficients in the Hall case, and they get something
called the "Hall algebra" instead of the "Hall ring".
The fun starts when we take the Hall algebra of Rep(Q), where
Q is a quiver. We could look at representations in vector
spaces over any field, but let's use a finite field  necessarily
a field with q elements, where q is a prime power.
Then, Ringel proved an amazing theorem about the Hall algebra
H(Rep(Q)) when Q comes from a Dynkin diagram of type A, D, or E:
5) C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101
(1990), 583592.
He showed this Hall algebra is a quantum group! More precisely,
it's isomorphic to the qdeformed universal enveloping algebra
of a maximal nilpotent subalgebra of the Lie algebra associated
to the given Dynkin diagram.
That's a mouthful, but it's cool. For example, the Lie algebra
associated to A_n is sl(n+1), and the maximal nilpotent subalgebra
consists of strictly upper triangular matrices. We're qdeforming
the universal enveloping algebra of this. One cool thing is that
the "q" of qdeformation gets interpreted as a prime power  something
we've already seen in "week185" and subsequent weeks.
So, it seems that all the ways simplylaced Dynkin diagrams show
up in math are related. But, I don't think anyone understands
what's really going on! It's like black magic.
And, I've just described *some* of the black magic!
For example, you'll notice I portrayed the Hall algebra H(A)
as a kind of evil twin of the more familiar Grothendieck group K(A).
They have some funny relations. For example, if you take the minimal
resolution of C^2/G where G is a finite subgroup of SU(2), you get a
variety whose Ktheory (as defined above) is isomorphic to the
representation ring of G! This was shown here:
6) G. GonzalezSpringberg and J. L. Verdier, Construction geometrique
de la correspondance de McKay, Ann. ENS 16 (1983), 409449.
For further developments, try this paper, which studies the derived
category of coherent sheaves on this minimal resolution of C^2/G:
7) Mikhail Kapranov and Eric Vasserot, Kleinian singularities,
derived categories and Hall algebras, available as math.AG/9812016.
Now let me give a bunch of references for further study. For a
really quick overview of the whole ADE business, try these:
8) Andrei Gabrielov, CoxeterDynkin diagrams and singularities,
in Selected Papers of E. B. Dynkin with Commentary, eds. A. A.
Yushkevich, G. M. Seitz and A. I. Onishchik, AMS, 1999.
9) John McKay, A rapid introduction to ADE theory,
http://math.ucr.edu/home/baez/ADE.html
Here's a more detailed but still highly readable introduction:
10) 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
Here's a really nice, elementary introduction to Klein's work
on the icosahedron and the quintic:
11) Jerry Michael Shurman, Geometry of the Quintic, Wiley, New York, 1997.
This is a bit harder to get ahold of, but it's a classic:
12) M. Hazewinkel, W. Hesselink, D. Siermsa, and F. D. Veldkamp, The
ubiquity of CoxeterDynkin diagrams (an introduction to the ADE problem),
Niew. Arch. Wisk., 25 (1977), 257307.
I haven't seen this book, but I hear it's good:
13) P. Slowody, Simple Singularities and Algebraic Groups,
Lecture Notes in Mathematics 815, Springer, Berlin, 1980.
Here's a bibliography with links to online references:
14) Miles Reid, Links to papers on McKay correspondence,
http://www.maths.warwick.ac.uk/~miles/McKay/
Of those references, I especially like this:
15) Miles Reid, La Correspondence de McKay (in English),
Seminaire Bourbaki, 52eme annee, November 1999, no. 867,
to appear in Asterisque 2000. Also available as math.AG/9911165.
Here you'll also see some material about *generalizations*
of the McKay correspondence. For example, if we take a finite
subgroup G of SU(3), we get a quotient C^3/G, which has
singularities. If we take a "crepant" resolution of
p: M > C^3/G
which is the right generalization of a minimal resolution, then
M is a CalabiYau manifold. This gets string theory into the act!
Around 1985, Dixon, Harvey, Vafa and Witten used this to guess
that the Euler characteristic of M equals the number of irreps of G.
A lot of work has been done on this since then, and Reid's article
summarizes a bunch.
Apparently a "crepant" resolution is one that induces an isomorphism
of canonical bundles; when this fails to happen folks say there's a
discrepancy, so a crepant resolution is one with no discrepancy.
Get it? Since a CalabiYau manifold is one whose canonical bundle
is trivial, it shouldn't be completely shocking that crepant
resolutions yield CalabiYaus. This all works in the original
2d McKay correspondence, too  the minimal resolutions we saw
there are also crepant.
In fact, string theory also sheds light on the original McKay
correspondence. The reason is that the minimal resolution of
C^2/G is a very nice Riemannian 4manifold (when viewed as a *real*
manifold). It's an "asymptotically locally Euclidean" manifold,
or ALE manifold for short. Doing string theory on this gives a
way of seeing how the extended Dynkin diagrams sneak into the McKay
correspondence: they're the Dynkin diagrams for central extensions
of loop groups, which show up as gauge groups in string theory!
I don't really understand this, but it makes a kind of sense.
I guess this is a famous paper about this stuff:
16) Michael R. Douglas and Gregory Moore, Dbranes, quivers and ALE
instantons, available as hepth/9603167.

Quote of the Week:
"This thesis is an attempt to show an astonishing relation between
basic objects from different fields in mathematics. Most peculiarly
it turns out that their classification is "the same": the ADE
classification. Altogether these objects and the connections between
them form a coherent web.
The connections are accomplished by direct constructions leading to
bijections between these classes of objects. These constructions
however do not always explain or give satisfactory intuition why these
classifications [exist], or to say it better, why they should be
related in this way. Therefore the deeper reason remains mysterious
and when discovered will have to be of great depth. This gives a
high motivation to look for new concepts and it shows that simple
and since long understood mathematics can still raise very interesting
questions, show paths for new research and give a glance at the
mystery of mathematics. In my opinion to be aware of a certain
truth without having its reason is fundamental to the practise of
mathematics."  Joris van Hoboken

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
obtained at
http://math.ucr.edu/home/baez/
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May 5, 2006
This Week's Finds in Mathematical Physics (Week 230)
John Baez
As we've seen in previous weeks, Mars is a beautiful world, but a
world in a minor key, a world whose glory days  the Hesperian Epoch 
are long gone, whose once grand oceans are now reduced to windy canyons
and icy dunes. Let's say goodbye to it for now... leaving off
with this Martian sunset, photographed by the rover Spirit in
Gusev Crater on May 19th, 2005:
1) A moment frozen in time, NASA Mars Exploration Rover Mission,
http://marsrovers.nasa.gov/gallery/press/spirit/20050610a.html
This week I'll talk about Dynkin diagrams, quivers and Hall algebras.
But first, some cool identities!
My student Mike Stay did computer science before he came to UCR.
When he was applying, he mentioned a result he helped prove, which
relates Goedel's theorem to the Heisenberg uncertainty principle:
2) C. S. Calude and M. A. Stay, From Heisenberg to Goedel via Chaitin,
International Journal of Theoretical Physics, 44 (2005), 10531065.
Also available at http://math.ucr.edu/~mike/
Now, this particular combination of topics is classic crackpot fodder.
People think "Gee, uncertainty sounds like incompleteness, they're
both limitations on knowledge  they must be related!" and go off the
deep end. So I got pretty suspicious until I read his paper and
saw it was CORRECT... at which point I *definitely* wanted him around!
The connection they establish is not as precise as I'd like, but it's
solid math.
So, now Mike is here at UCR working with me on quantum logic and
quantum computation using ideas from category theory. In his spare
time, he sometimes fools around with math identities and tries to
categorify them  see "week184" and "week202" if you don't know what
that means. Anyway, maybe that's how he stumbled on this:
3) Jonathan Sondow, A faster product for pi and a new integral
for ln(pi/2), Amer. Math. Monthly 112 (2005), 729734. Also
available as math.NT/0401406.
In this paper, Sondow gives eerily similar formulas for some of
our favorite math constants. First, one for e:2 1/1 2^2 1/2 2^3 x 4 1/3 2^4 x 4^4 1/4
e = (  ) (  ) (  ) (  ) ...
1 1 x 3 1 x 3^3 1 x 3^6 x 5Then, one for pi/2:pi 2 1/2 2^2 1/4 2^3 x 4 1/8 2^4 x 4^4 1/16
 = (  ) (  ) (  ) ( ) ...
2 1 1 x 3 1 x 3^3 1 x 3^6 x 5Then one for e^{gamma}, where gamma is Euler's constant:gamma 2 1/2 2^2 1/3 2^3 x 4 1/4 2^4 x 4^4 1/5
e = (  ) (  ) (  ) (  ) ...
1 1 x 3 1 x 3^3 1 x 3^6 x 5He also points out Wallis' product for pi and Pippenger's for e:pi 2 1/1 2x4 1/1 4x6x6x8 1/1
 = (  ) (  ) (  ) ...
2 1 3x3 5x5x7x7
2 1/2 2x4 1/4 4x6x6x8 1/8
e = (  ) (  ) (  ) ...
1 3x3 5x5x7x7What does it all mean? I haven't a clue! Another mystery thrown
down to us by the math gods, like a bone from on high... we can merely
choose to chew on it or not, as we wish.
Julie Bergner gave a great talk on "derived Hall algebras" at the
Mac Lane memorial conference. I just want to explain the very
first result she mentioned, due to Ringel  a surprising trick for
constructing certain quantum groups from simplylaced Dynkin diagrams.
It's very different from the *usual* method for getting quantum groups
from Dynkin diagrams, and it's a miracle that it works.
But, I guess I should start near the beginning!
Way back in 1995, in "week62", "week63", "week64" and "week65",
I explained how "Dynkin diagrams"  little gizmos like this:
o

ooooooo
show up all over mathematics. They have a strange way of tying
together subjects that superficially seem completely unrelated.
In one sense people understand how they work, but in another sense
they're very puzzling  their power keeps growing in unexpected ways.
I love mysterious connections, so as soon I understood enough about
Dynkin diagrams to appreciate them, I became fascinated by them,
and I've been studying them ever since. I explained their relation
to geometry in "week178", "week179", "week180", "week181" and "week182",
and their relation to quantum deformation and combinatorics in
"week186" and "week187".
You might think that would be enough  but you'd be wrong, way wrong!
I haven't really talked about the most mysterious aspects of Dynkin
diagrams, like their relation to singularity theory and representations
of quivers. That's because these aspects were too mysterious!
I didn't understand them *at all*. But lately, James Dolan and Todd
Trimble and I have been making some progress understanding these aspects.
First, I should remind you how Dynkin diagrams infest so much of
mathematics. Let's start with a little puzzle mentioned in "week182".
Draw n dots and connect some of them with edges  at most one edge
between any pair of dots, please:
o
/ \
/ \
o ooooo
/
/
o oo
Now, try to find a basis of R^n consisting of one unit vector per dot,
subject to these rules: if two dots are connected by an edge, the angle
between their vectors must be 120 degrees, but otherwise their vectors
must be at right angles.
This sounds like a silly puzzle that only a mathematician could give a
hoot about. It takes a while to see its magnificent depth. But anyway,
it turns out you can solve this problem only for certain special diagrams
called "simplylaced Dynkin diagrams". The basic kinds are called A_n,
D_n, E_6, E_7, and E_8.
The A_n Dynkin diagram is a line of n dots connected by edges like this:
ooooo
The D_n diagram has n dots arranged like this:
o
/
ooooo
\
o
A line of them but then a little fishtail at the end! We should
take n to be at least 4, to make the diagram connected and different
from A_n.
The E_6, E_7, and E_8 diagrams look like this:
o o o
  
ooooo oooooo ooooooo
You're also allowed to take disjoint unions of the above diagrams.
So, a weird problem with a weird answer! Its depth is revealed
only when we see that many DIFFERENT puzzles lead us to the SAME
diagrams. For example:
A) the classification of integral lattices in R^n having a basis of
vectors whose length squared equals 2
B) the classification of simply laced semisimple Lie groups
C) the classification of finite subgroups of the 3d rotation group
D) the classification of simple singularities
E) the classification of tame quivers
Let me run through these problems and say a bit about how they're
connected:
A) An "integral lattice" in R^n is a lattice where the dot product
of any two vectors in the lattice is an integer. There are zillions
of these  but if we demand that they have a basis of vectors whose
length squared is 2, we can only get them from simplylaced Dynkin
diagrams.
It's not very hard to see that finding a lattice like this is equivalent
to the puzzle I mentioned earlier. For example, given a solution of
that puzzle, you can just multiply all your vectors by sqrt(2) and
form the lattice of their integer linear combinations.
So, this connection is not so deep. But it connects to something
a bit deeper...
B) Lie groups are fundamental throughout math and physics: they're
groups of continuous symmetries, like rotations. The nicest of the
lot are the semisimple Lie groups. Some familiar examples are the
group of rotations in ndimensional space, which is called SO(n),
and the group of unitary matrices with determinant 1, which is
called SU(n). There are more, but people know what they all are.
They're classified by Dynkin diagrams!
Why? The key point is that any semisimple Lie group has a "root lattice".
This is an integral lattice spanned by special vectors called "roots".
I won't give the details, since I explained this stuff in "week63"
and "week64", but it turns out that root lattices, and thus semisimple
Lie groups, are classified by Dynkin diagrams.
Not all these Dynkin diagrams look like the A, D and E diagrams listed
above. But, it turns out that the length squared of any root must be
either 1 or 2. If all the roots have length squared equal to 2, we
say our semisimple Lie group is "simply laced". In this case, we're
back to problem B), which we already solved! So then our Lie group
corresponds to a diagram of type A, D, or E  or a disjoint union of
such diagrams.
Here's how it goes:
* The diagram A_n gives the compact Lie group SU(n+1),
consisting of (n+1) x (n+1) unitary matrices with determinant 1.
It's the isometry group of complex projective nspace.
* The diagram D_n gives the compact Lie group SO(2n),
consisting of 2n x 2n orthogonal matrices with determinant 1.
It's the isometry group of real projective (2n)space.
* The diagram E_6 gives a 78dimensional compact Lie group
that people call E_6. It's the isometry group of the bioctonionic
projective plane.
* The diagram E_7 gives a 133dimensional compact Lie group
that people call E_7. It's the isometry group of the quateroctonionic
projective plane.
* The diagram E_8 gives a 248dimensional compact Lie group
that people call E_8. It's the isometry group of the octooctonionic
projective plane.
In short, two regular series and three exotic weirdos.
You may ask where the rotation groups SO(n) with n odd went!
Well, these correspond to fancier Dynkin diagrams that aren't
simply laced, like this:
oooo>o
The funny arrow here indicates that the last two vectors aren't
at a 120degree angle; they're at a 135degree angle, and the last
vector is shorter than the rest: it has length one instead of sqrt(2).
Why are semisimple Lie groups "better" when they're simply laced?
What's the big deal? I don't really understand this, but for one,
when all the roots have the same length, they're all alike 
a certain symmetry group called the Weyl group acts transitively on them.
Anyway, so far our A, D, E Dynkin diagrams have been classifying
things that are clearly related to lattices. But now things get
downright spooky...
C) Take a Platonic solid and look at its group of rotational symmetries.
You get a finite subgroup of the 3d rotation group SO(3). But in
general, finite subgroups of SO(3) are classified by ADE Dynkin
diagrams!
So, Platonic solids turn out to fit into the game we're playing here!
First I'll say which diagram corresponds to which subgroup of SO(3).
Then I'll explain how the correspondence works:
* The diagram A_n corresponds to the group of obvious rotational symmetries
of the regular ngon. This group is called the "cyclic group" Z/n.
* The diagram D_n corresponds to the group of rotational symmetries
of the regular ngon where you can turn it and also flip it over.
By sheer coincidence, this group is called the "dihedral group" D_n.
A cosmic stroke of good luck!
* The diagram E_6 corresponds to the group of rotational symmetries
of the tetrahedron: the "tetrahedral group". This is also the
group of even permutations of 4 elements, the "alternating group" A_4 
not to be confused with the A_n's we were just talking about.
A cosmic stroke of bad luck!
* The diagram E_7 corresponds to the group of rotational symmetries
of the octahedron or cube: the "octahedral group". This is also
the group of all permutations of 4 elements, the "symmetric group" S_4.
* The diagram E_8 corresponds to the group of rotational symmetries
of the icosahedron or dodecahedron: the "icosahedral group". This
is also the group of even permutations of 5 elements, called A_5. Darn!
So, the exceptional Lie groups E_6, E_7 and E_8 correspond to Platonic
solids in a sneaky way.
To understand what's going on here, first we need to switch from SO(3)
to SU(2). The group SU(2) is used to describe rotations in quantum
mechanics: it's the double cover of the rotation group SO(3).
It's really finite subgroups of SU(2) that are classified by ADE Dynkin
diagrams! It just so happens that these correspond, in a slightly
slippery way, to finite subgroups of SO(3).
You'll see how if I list the finite subgroups of SU(2):
* The diagram A_n corresponds to the cyclic subgroup Z/n of SU(2).
This double covers a cyclic subgroup of SO(3) when n is even.
* The diagram D_n corresponds to a subgroup of SU(2) that double
covers the dihedral group D_n.
* The diagram E_6 corresponds to a subgroup of SU(2) that double
covers the rotational symmetries of the tetrahedron. This
subgroup has 24 elements and it's called the "binary tetrahedral group".
* The diagram E_7 corresponds to a subgroup of SU(2) that double
covers the rotational symmetries of the octahedron. This
subgroup has 48 elements and it's called the "binary octahedral group".
* The diagram E_8 corresponds to a subgroup of SU(2) that double
covers the rotational symmetries of the icosahedron. This
subgroup has 120 elements and it's called the "binary icosahedral group".
Now, how does the correspondence work? For this, I'm afraid I have to
raise the sophistication level a bit  I've been trying to keep things
simple, but it's getting tough.
In his book on the icosahedron, Felix Klein noticed it was interesting
to let the icosahedral group act on the Riemann sphere, and look for
rational functions invariant under this group.
It turned out that every such function depends on a single one:
Klein's icosahedral function! The explict formula for it is pretty
disgusting, but it's a beautiful thing: you can pick it so that it
equals 0 at all the vertices of the icosahedron, 1 at the midpoints
of the edges, and infinity at the midpoints of the faces. Even
better, if you write the function like this:
w = f(z)
then Klein showed that knowing how to solve for z as a function of w
lets you solve every quintic equation! The reason is that the Galois
group of the general quintic is a close relative of the icosahedral
group: the former is S_5, the latter is A_5.
Anyway, when I said that "every such function depends on a single one",
what I really meant was this. Let C(z) be the field of rational functions
of one variable; then the icosahedral group acts on this, and the invariant
functions form a subfield C(w) where w is Klein's icosahedral function.
The Galois group of the little field in the big one is the icosahedral
group.
The same kind of thing works for the other finite subgroups of SO(3),
except of course for the connection to the quintic equation.
But, it's actually even better to think about finite subgroups of
SU(2), since SU(2) acts on C^2, and when we *projectivize* C^2 we
get SO(3) acting on the Riemann sphere. This viewpoint fits more
squarely into the worldview of algebraic geometry.
If we take the quotient of C^2 by a finite subgroup G of SU(2), we
don't get a smooth manifold: the quotient has a singularity at 0.
But we can "resolve" the singularity, finding a smooth complex manifold
with a holomorphic map
p: M > C^2/G
that has a holomorphic inverse on a dense open set. There may be
lots of ways to do this, but in the present case there's just one
"minimal" resolution, meaning a resolution that every other resolution
factors through.
Then  and here's the magic part!  the inverse image of 0 in M
turns out to be the union of a bunch of Riemann spheres. And
if we draw a dot for each sphere, and an edge between these dots
whenever their spheres intersect, we get a simply laced Dynkin
diagram on the above list!
Well, almost. We get this diagram with an extra dot thrown in,
connected by some extra edges in a specific way. This is called
the "extended" Dynkin diagram. It also shows up naturally from
the Lie group viewpoint, when we consider central extensions of
loop groups.
That's *one* way the correspondence works. Another way, discovered
by McKay, is to draw a dot for each irrep of G. There's one special
dot, since there's always a 2dimensional irrep coming from the action
of SU(2) on C^2. Let's just call this irrep C^2. Then, draw an edge
from the dot R to the dot S whenever the irrep S shows up in the rep
R tensor C^2. You get the same extended Dynkin diagram as before!
This second way is called the "McKay correspondence". The first way
is sometimes called the "geometric McKay correspondence", though I think
it was discovered earlier.
Now we're well on the road to the next item...
D) Simplylaced Dynkin diagrams also classify the simple critical points
of holomorphic functions
f: C^3 > C
A "critical point" is just a place where the gradient of f vanishes.
We can try to classify critical points up to a holomorphic change of
variables. It's better to classify their "germs", meaning we only
look at what's going on *right near* the critical point. But, even
this is hopelessly complicated unless we somehow limit our quest.
To do this, we can restrict attention to "stable" critical points,
which are those that don't change type under small perturbations.
But we can do better: we can classify "simple" critical points,
namely those that change into only finitely many other types under
small perturbations.
These correspond to simplylaced Dynkin diagrams!
First I'll say which diagram corresponds to which type of critical
point. To do this, I'll give a polynomial f(x,y,z) that has a certain
type of critical point at x = y = z = 0. Then I'll explain how
the correspondence works:
* The diagram A_n corresponds to the critical point of x^{n+1} + y^2 + z^2.
* The diagram D_n corresponds to the critical point of x^{n1} + xy^2 + z^2.
* The diagram E_6 corresponds to the critical point of x^4 + y^3 + z^2.
* The diagram E_7 corresponds to the critical point of x^3 y + y^3 + z^2.
* The diagram E_8 corresponds to the critical point of x^5 + y^3 + z^2.
Here's how the correspondence works. For each of our Dynkin diagrams
we have a finite subgroup of SU(2), thanks to item C). This subgroup
acts on the ring of polynomials on C^2, so we can form the subring of
invariant polynomials. This turns out to be generated by three polynomials
that we will arbitrarily call x, y, and z. But, they satisfy one relation,
given by the polynomial above!
Conversely, we can start with the polynomial
f: C^3 > C
The zero set
{f = 0}
has an isolated singularity at the origin. But, we can resolve
this singularity, finding a smooth complex manifold N with a
holomorphic map
q: N > {f = 0}
that has a holomorphic inverse on a dense open set. There may be
lots of ways to do this, but in the present case there's just one
"minimal" resolution, meaning one that every other resolution
factors through this one.
Then  and here's the magic part!  the inverse image of 0 in N
turns out to be the union of a bunch of Riemann spheres. And
if we draw a dot for each sphere, and an edge between these dots
whenever their spheres intersect, we get back our simply laced
Dynkin diagram!
This whole section should have given you a feeling of deja vu.
It's a lot like section D). If I were smarter, I'd probably see
how it's *exactly* the same stuff, repackaged slightly.
The last item on our list seems different...
E) A quiver is just a category freely generated by some set of
morphisms. To specify a quiver we just write down some dots
and arrows. The dots are the objects of our category; the
arrows are the generating morphisms.
A representation of a quiver Q is just a functor
F: Q > Vect
So, we get a vector space for each dot and a linear map for
each arrow, with no extra restrictions. There's an obvious
category of representations Rep(Q) of a quiver Q.
A guy named Gabriel proved a divine result about these categories
Rep(Q). We say a quiver Q is "tame" if Rep(Q) has finitely
many indecomposable objects  objects that aren't direct sums of
others. And, it turns out the tame quivers are just those coming
from simplylaced Dynkin diagrams!
Actually, for this to make sense, you need to take your Dynkin
diagram and turn it into a quiver by putting arrows along the edges.
If you have an ADE Dynkin diagram, you get a tame quiver no matter
which way you let the arrows point.
There's clearly a lot of mysterious stuff going on here. In
particular, this last item sounds completely unrelated to the
rest. But it's not! There are cool relationships between quivers
and quantum groups, which tie this item to the rest.
I'll just mention one  the one Julie Bergner started her talk with.
For this, you need to know a bit about abelian categories.
Abelian categories are categories like the category of abelian groups,
or more generally the category of modules of any ring, where you can
talk about chain complexes, exact sequences and stuff like that.
You can see the precise definition here:
4) Abelian categories, Wikipedia,
http://en.wikipedia.org/wiki/Abelian_category
and learn more here:
5) Peter Freyd, Abelian Categories, Harper and Row, New York, 1964.
Also available at http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html
It's really interesting to study the "Grothendieck group" K(A) of an
abelian category A. As a set, this consists of formal linear combinations
of isomorphism classes of objects of A, where we impose the relations
[a] + = [a + b]
where a + b on the right side is the direct sum of the objects
a and b. It becomes an abelian group in an obvious way.
For example, if A is the category of representations of some group G,
it's an abelian category and K(A) is called the "representation ring"
of A  it's a ring because we tensor representations. Or, if A is the
category of vector bundles over a space X, it's again abelian, and K(A)
is called the "Ktheory of X".
The Hall ring H(A) of an abelian category is a vaguely similar idea.
As a set, this consists of formal linear combinations of isomorphism
classes of objects of A. No extra relations! It's an abelian group
with the obvious addition. But the cool part is, with a little luck,
we can make it into a *ring* by letting the product [a] be the
sum of all isomorphism classes of objects [x] weighted by the number
of isomorphism classes of short exact sequences
0 > a > x > b > 0
This only works if the number is always finite.
So far when speaking of "formal linear combinations" I've been
implicitly using integer coefficients, but people seem to prefer
complex coefficients in the Hall case, and they get something
called the "Hall algebra" instead of the "Hall ring".
The fun starts when we take the Hall algebra of Rep(Q), where
Q is a quiver. We could look at representations in vector
spaces over any field, but let's use a finite field  necessarily
a field with q elements, where q is a prime power.
Then, Ringel proved an amazing theorem about the Hall algebra
H(Rep(Q)) when Q comes from a Dynkin diagram of type A, D, or E:
5) C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101
(1990), 583592.
He showed this Hall algebra is a quantum group! More precisely,
it's isomorphic to the qdeformed universal enveloping algebra
of a maximal nilpotent subalgebra of the Lie algebra associated
to the given Dynkin diagram.
That's a mouthful, but it's cool. For example, the Lie algebra
associated to A_n is sl(n+1), and the maximal nilpotent subalgebra
consists of strictly upper triangular matrices. We're qdeforming
the universal enveloping algebra of this. One cool thing is that
the "q" of qdeformation gets interpreted as a prime power  something
we've already seen in "week185" and subsequent weeks.
So, it seems that all the ways simplylaced Dynkin diagrams show
up in math are related. But, I don't think anyone understands
what's really going on! It's like black magic.
And, I've just described *some* of the black magic!
For example, you'll notice I portrayed the Hall algebra H(A)
as a kind of evil twin of the more familiar Grothendieck group K(A).
They have some funny relations. For example, if you take the minimal
resolution of C^2/G where G is a finite subgroup of SU(2), you get a
variety whose Ktheory (as defined above) is isomorphic to the
representation ring of G! This was shown here:
6) G. GonzalezSpringberg and J. L. Verdier, Construction geometrique
de la correspondance de McKay, Ann. ENS 16 (1983), 409449.
For further developments, try this paper, which studies the derived
category of coherent sheaves on this minimal resolution of C^2/G:
7) Mikhail Kapranov and Eric Vasserot, Kleinian singularities,
derived categories and Hall algebras, available as math.AG/9812016.
Now let me give a bunch of references for further study. For a
really quick overview of the whole ADE business, try these:
8) Andrei Gabrielov, CoxeterDynkin diagrams and singularities,
in Selected Papers of E. B. Dynkin with Commentary, eds. A. A.
Yushkevich, G. M. Seitz and A. I. Onishchik, AMS, 1999.
9) John McKay, A rapid introduction to ADE theory,
http://math.ucr.edu/home/baez/ADE.html
Here's a more detailed but still highly readable introduction:
10) 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
Here's a really nice, elementary introduction to Klein's work
on the icosahedron and the quintic:
11) Jerry Michael Shurman, Geometry of the Quintic, Wiley, New York, 1997.
This is a bit harder to get ahold of, but it's a classic:
12) M. Hazewinkel, W. Hesselink, D. Siermsa, and F. D. Veldkamp, The
ubiquity of CoxeterDynkin diagrams (an introduction to the ADE problem),
Niew. Arch. Wisk., 25 (1977), 257307.
I haven't seen this book, but I hear it's good:
13) P. Slowody, Simple Singularities and Algebraic Groups,
Lecture Notes in Mathematics 815, Springer, Berlin, 1980.
Here's a bibliography with links to online references:
14) Miles Reid, Links to papers on McKay correspondence,
http://www.maths.warwick.ac.uk/~miles/McKay/
Of those references, I especially like this:
15) Miles Reid, La Correspondence de McKay (in English),
Seminaire Bourbaki, 52eme annee, November 1999, no. 867,
to appear in Asterisque 2000. Also available as math.AG/9911165.
Here you'll also see some material about *generalizations*
of the McKay correspondence. For example, if we take a finite
subgroup G of SU(3), we get a quotient C^3/G, which has
singularities. If we take a "crepant" resolution of
p: M > C^3/G
which is the right generalization of a minimal resolution, then
M is a CalabiYau manifold. This gets string theory into the act!
Around 1985, Dixon, Harvey, Vafa and Witten used this to guess
that the Euler characteristic of M equals the number of irreps of G.
A lot of work has been done on this since then, and Reid's article
summarizes a bunch.
Apparently a "crepant" resolution is one that induces an isomorphism
of canonical bundles; when this fails to happen folks say there's a
discrepancy, so a crepant resolution is one with no discrepancy.
Get it? Since a CalabiYau manifold is one whose canonical bundle
is trivial, it shouldn't be completely shocking that crepant
resolutions yield CalabiYaus. This all works in the original
2d McKay correspondence, too  the minimal resolutions we saw
there are also crepant.
In fact, string theory also sheds light on the original McKay
correspondence. The reason is that the minimal resolution of
C^2/G is a very nice Riemannian 4manifold (when viewed as a *real*
manifold). It's an "asymptotically locally Euclidean" manifold,
or ALE manifold for short. Doing string theory on this gives a
way of seeing how the extended Dynkin diagrams sneak into the McKay
correspondence: they're the Dynkin diagrams for central extensions
of loop groups, which show up as gauge groups in string theory!
I don't really understand this, but it makes a kind of sense.
I guess this is a famous paper about this stuff:
16) Michael R. Douglas and Gregory Moore, Dbranes, quivers and ALE
instantons, available as hepth/9603167.

Quote of the Week:
"This thesis is an attempt to show an astonishing relation between
basic objects from different fields in mathematics. Most peculiarly
it turns out that their classification is "the same": the ADE
classification. Altogether these objects and the connections between
them form a coherent web.
The connections are accomplished by direct constructions leading to
bijections between these classes of objects. These constructions
however do not always explain or give satisfactory intuition why these
classifications [exist], or to say it better, why they should be
related in this way. Therefore the deeper reason remains mysterious
and when discovered will have to be of great depth. This gives a
high motivation to look for new concepts and it shows that simple
and since long understood mathematics can still raise very interesting
questions, show paths for new research and give a glance at the
mystery of mathematics. In my opinion to be aware of a certain
truth without having its reason is fundamental to the practise of
mathematics."  Joris van Hoboken

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
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.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
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