- #1
John Baez
Also available at http://math.ucr.edu/home/baez/week264.html
May 18, 2008
This Week's Finds in Mathematical Physics (Week 264)
John Baez
Here's a puzzle. Guess the next term of this sequence:
1, 1, 2, 3, 4, 5, 6, ...
and then guess the *meaning* of this sequence! I'll give away the
answer after telling you about Coleman's videos on quantum field
theory and an amazing result on the homotopy groups of spheres.
But first... the astronomy picture of the day.
The Eaton Collection at UC Riverside may be the world's best
library of science fiction:
1) The Eaton Collection of Science Fiction, Fantasy, Horror
and Utopian Literature, http://eaton-collection.ucr.edu/
Right now my wife Lisa Raphals is attending a conference there
on the role of Mars in SF, called "Chronicling Mars". Gregory
Benford, Frederik Pohl, Greg Bear, David Brin, Kim Stanley Robinson
and even Ray Bradbury are all there! But for some reason I'm staying
home working on This Week's Finds. I'd say that shows true devotion -
or maybe just stupidity.
Anyway, in honor of the occasion, here's an incredible closeup of a
crater on Mars' moon Phobos:
2) Astronomy Picture of the Day, Stickney Crater
http://apod.nasa.gov/apod/ap080410.html
It's another great example of how machines in space now deliver
many more thrills per buck than the old-fashioned approach
using canned primates. This photo was taken by HiRISE, the
High Resolution Imaging Science Experiment - the same satellite
that took the stunning photos of Martian dunes which graced "week262".
Mars has two moons, Phobos and the even tinier Deimos. Their
names mean "fear" and "dread" in Greek, since in Greek mythology
they were sons of Mars (really Ares), the god of war.
Interestingly, Kepler predicted that Mars had two moons before
they were seen. This sounds impressive, but it was simple
interpolation, since Earth has 1 moon and Jupiter has 4. Or at
least Galileo saw 4 - now we know there are a lot more.
Phobos is only 21 kilometers across, and the big crater you
see here - Stickney Crater - is about 9 kilometers across.
That's almost half the size of the whole moon! The collision
that created it must have almost shattered Phobos.
Phobos is so light - just twice the density of water - that people
once thought it might be hollow. This now seems unlikely, though
it's been the premise of a few SF stories. It's more likely that
Phobos is a loosely packed pile of carbonaceous chondrites captured
from the asteroid belt.
Phobos orbits so close to Mars that it zips around once every
8 hours, faster than Mars itself rotates! Oddly, in 1726
Jonathan Swift wrote about two moons of Mars in his novel
"Gulliver's Travels" - and he guessed that the inner one orbited
Mars every 10 hours.
Gravitational tidal forces are dragging Phobos down, so in only 10
million years it'll either crash or - more likely - be shattered by
tidal forces and form a ring of debris.
So, enjoy it while it lasts.
Anyone who's seriously struggled to master quantum field theory is
likely to have profited from this book:
3) Sidney Coleman, Aspects of Symmetry: Selected Erice Lectures,
Cambridge U. Press, Cambridge, 1988.
It's brimming with wisdom and humor. You should have already
encountered quantum field theory before trying it: what you'll
get are deeper insights.
But what if you're just getting started?
Sidney Coleman, recently deceased, was one of the best quantum field
theorists from the heyday of particle physics. As a grad student I
took a course on quantum field theory from Eddie Farhi, who said he
based his class on the notes from Coleman's class at Harvard. So,
I've always been curious about these notes. Now they're available
online in handwritten form:
4) Sidney Coleman, lecture notes on quantum field theory,
http://www.damtp.cam.ac.uk/user/dt281/qft/col1.pdf
and
http://www.damtp.cam.ac.uk/user/dt281/qft/col2.pdf
Someone should LaTeX them up!
Even more fun, you can now see *videos* of Coleman teaching quantum
field theory:
5) Sidney Coleman, Physics 253: Quantum Field Theory, 50 lectures
recorded 1975-1976, http://www.physics.harvard.edu/about/Phys253.html
This is a younger, hipper Coleman than I'd ever seen: long-haired,
sometimes puffing on a cigarette between sentences. He begins by
saying "Umm... this is Physics 253, a course in relativistic quantum
mechanics. My name is Sidney Coleman. The apparatus you see around
you is part of a CIA surveillance project."
I wish I'd had access to these when I was a kid!
Now for some miraculous math. Daniel Moskovich kindly pointed out a
paper that describes all the homotopy groups of the 2-sphere, and
I want to summarize the main result.
I explained the idea of homotopy groups back in "week102". Very
roughly, the nth homotopy group of a space X, usually denoted pi_n(X),
is the set of ways you can map an n-sphere into that space, where we
count two ways as the same if you can continuously deform one to the
other. If a space has holes, homotopy groups are one way to detect
those holes.
Homotopy groups are notoriously hard to compute - so even for so humble
a space as the 2-sphere, S^2, there's a sense in which "nobody knows"
all its homotopy groups. People know the first 64, though. Here are a
few:
pi_1(S^2) = 0
pi_2(S^2) = Z
pi_3(S^2) = Z
pi_4(S^2) = Z/2
pi_5(S^2) = Z/2
pi_6(S^2) = Z/4 x Z/3
pi_7(S^2) = Z/2
pi_8(S^2) = Z/2
pi_9(S^2) = Z/3
pi_10(S^2) = Z/3 x Z/5
pi_11(S^2) = Z/2
pi_12(S^2) = Z/2 x Z/2
pi_13(S^2) = Z/2 x Z/2 x Z/3
pi_14(S^2) = Z/2 x Z/2 x Z/4 x Z/3 x Z/7
pi_15(S^2) = Z/2 x Z/2
Apart from the fact that they're all finite abelian groups, it's
hard to spot any pattern!
In fact there's a majestic symphony of patterns in the homotopy
groups of spheres, starting from ones that are easy to explain
and working on up to those that push the frontiers of mathematics,
like elliptic cohomology. But, many of these patterns are too
complex for present-day mathematics until we use some tricks to
"water down" or simplify the homotopy groups.
So, what people often do first is take the limit of pi_{n+k}(S^n)
as n -> infinity, getting what's called the kth "stable" homotopy
group of spheres. It's a wonderful but well-understood fact that
these limits really exist. But so far, even these are too
complicated to understand until we work "at a prime p".
This means that we take the kth stable homotopy group of spheres
and see which groups of the form Z/p^n show up in it. For example,
pi_14(S^2) = Z/2 x Z/2 x Z/4 x Z/3 x Z/7
but if we work "at the prime 2" we just see the Z/2 x Z/2 x Z/4.
After all this data processing, we get some astounding pictures:
6) Allen Hatcher, Stable homotopy groups of spheres,
http://www.math.cornell.edu/~hatcher/stemfigs/stems.html
Order teetering on the brink of chaos! If you're brave, you can
learn more about this stuff here:
7) Douglas C. Ravenel, Complex Cobordism and Stable Homotopy Groups
of Spheres, AMS, Providence, Rhode Island, 2003.
If you're less brave, I strongly suggest starting here:
8) Wikipedia, Homotopy groups of spheres,
http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres
But now, I want to talk about an amazing paper that pursues a
very different line of attack. It gives a beautiful description
of *all* the homotopy groups of S^2, in terms of braids:
9) A. Berrick, F. R. Cohen, Y. L. Wong and J. Wu, Configurations,
braids and homotopy groups, J. Amer. Math. Soc., 19 (2006), 265-326.
Also available at http://www.math.nus.edu.sg/~matwujie/BCWWfinal.pdf
For this you need to realize that for any n, there's a group B_n
whose elements are n-strand braids. For example, here's an element
of B_3:
| | |
\ / |
/ |
/ \ |
| \ /
| /
| / \
\ / |
/ |
/ \ |
| \ /
| /
| / \
\ / |
/ |
/ \ |
| \ /
| /
| / \
| | |
I actually talked about this specific braid back in "week233".
But anyway, we count two braids as the same if you can wiggle one
around until it looks like the other without moving the ends at
the top and bottom - which you can think of as nailed to the
ceiling and floor.
How do braids become a group? Easy: we multiply them by putting
one on top of the other. For example, this braid:
| | |
\ / |
A = / |
/ \ |
| | |
times this one:
| | |
| \ /
B = | /
| / \
| | |
equals this:
| | |
\ / |
/ |
/ \ |
| | |
AB = | | |
| \ /
| /
| / \
| | |
and in fact the big one I showed you earlier is (AB)^3.
As you let your eye slide from the top to the bottom of a braid, the
strands move around. We can visualize their motion as a bunch of points
running around the plane, never bumping into each other. This gives
an interesting way to generalize the concept of a braid! Instead
of points running around the plane, we can have points running around
S^2, or some other surface X. So, for any surface X and any number n
of strands, we get a "surface braid group", called B_n(X).
As I hinted in "week261", these surface braid groups have cool
relationships to Dynkin diagrams. I urged you to read this paper,
and I'll urge you again:
10) Daniel Allcock, Braid pictures for Artin groups, available as
arXiv:math.GT/9907194.
But for now, we just need the "spherical braid group" B_n(S^2)
together with the usual braid group B_n.
Let's say a braid is "Brunnian" if when you remove anyone strand,
the remaining braid becomes the identity: you can straighten out
all the remaining strands to make them vertical. It's a fun little
exercise to check that Brunnian braids form a subgroup of all braids.
So, we have an n-strand Brunnian braid group BB_n.
The same idea works for braids on other surface, like the 2-sphere.
So, we also have an n-strand *spherical* Brunnian braid group BB_n(S^2).
Now, there's obvious map
B_n -> B_n(S^2)
Why? An element of B_n describes the motion of a bunch of points
running around the plane, but the plane sits inside the 2-sphere:
the 2-sphere is just the plane with an extra point tacked on. So,
an ordinary braid gives a spherical braid.
This map clearly sends Brunnian braids to spherical Brunnian braids,
so we get a map
f: BB_n -> BB_n(S^2)
And now we're ready for the shocking theorem of Berrick, Cohen,
Wong and Wu:
Theorem: For n > 3, BB_n(S^2) modulo the image of f is the nth
homotopy group of S^2.
In something more like plain English: when n is big enough, the
nth homotopy group of the 2-sphere consists of spherical Brunnian
braids modulo ordinary Brunnian braids!
Zounds! What do the homotopy groups of S^2 have to do with braids?
It's not supposed to be obvious! The proof of this result is long and
deep, making use of flows on metric spaces, and also the fact that all
the Brunnian braid groups BB_n fit together into a "simplicial group"
whose nth homology is the nth homotopy group of S^2. I'd love to
understand all this stuff, but I don't yet.
This result doesn't instantly help us "compute" the homotopy groups of
S^2 - at least not in the sense of writing them down as a product of
groups like Z/p^n. But, it gives a new view of these homotopy groups,
and there's no telling where this might lead.
When I was first getting ready to write this article, I was also
going to tell you about some amazing descriptions of the homotopy
groups of the *3-sphere*, due to Wu.
However, I later realized - first to my shock, and then my embarrassment
for not having known it already - that the nth homotopy group of S^3
is *the same* as the nth homotopy group of S^2, at least for n > 2.
Do you see why?
Given this, it turns out that Wu's results are predecessors of the
theorem just stated, a bit more combinatorial and less "geometric".
Wu's results appeared here:
11) Jie Wu, On combinatorial descriptions of the homotopy groups of
certain spaces, Math. Proc. Camb. Phil. Soc. 130 (2001), 489-513.
Also available at http://www.math.nus.edu.sg/~matwujie/newnewpis_3.pdf
Jie Wu, A braided simplicial group, Proc. London Math. Soc. 84
(2002), 645-662. Also available at
http://www.math.nus.edu.sg/~matwujie/Research2.html
and there's a nice summary of these results on his webpage:
12) Jie Wu, 2.1 Homotopy groups and braids, halfway down the page at
http://www.math.nus.edu.sg/~matwujie/Research2.html
See also this expository paper:
13) Fred R. Cohen and Jie Wu, On braid groups and homotopy groups,
Geometry & Topology Monographs 13 (2008), 169-193. Also available at
http://www.math.nus.edu.sg/~matwujie/cohen.wu.GT.revised.29.august.2007.pdf
Next I want to talk about puzzle mentioned at the start of this
Week's Finds... but first I should answer the puzzle I just raised.
Why do the homotopy groups of S^2 match those of S^3 after a while?
Because of the Hopf fibration! This is a fiber bundle with S^3 as
total space, S^2 as base space and S^1 as fiber:
S^1 -> S^3 -> S^2
Like any fiber bundle, it gives a long exact sequence of homotopy
groups as explained in "week151":
... -> pi_n(S^1) -> pi_n(S^3) -> pi_n(S^2) -> pi_{n-1}(S^1) -> ...
but the homotopy groups of S^1 vanishes after the first, so we get
... -> 0 -> pi_n(S^3) -> pi_n(S^2) -> 0 -> ...
for n > 2, which says that
pi_n(S^3) = pi_n(S^2)
Okay, now for this mysterious sequence:
1, 1, 2, 3, 4, 5, 6, ...
The next term is obviously 7. If you guessed anything else, you
were over-analyzing. So the real question is: why the funny
"hiccup" at the beginning?
You'll find two explanations of this sequence in Sloane's Online
Encyclopedia of Integer Sequences, but neither of them is the reason
James Dolan and I ran into it. We were studying theta functions...
Say you have a torus. Then the complex line bundles over it
are classified by an integer called the "first Chern number".
In some sense, this integer this measures how "twisted" the
bundle is. For example, you can put any connection on the bundle,
compute its curvature 2-form, and integrate it over the torus:
up to some constant factor, you'll then get the first Chern number.
A torus is a 2-dimensional manifold, but we can also make it
into a 1-dimensional *complex* manifold, often called an
"elliptic curve". In fact we can do this in infinitely many
fundamentally different ways, one for each point in the "moduli
space of elliptic curves". I've explained this repeatedly here -
try "week125" for a good starting-point - so I won't do so again.
The details don't really matter here.
Back to line bundles. If we pick an elliptic curve, we can
try to classify the *holomorphic* complex line bundles over it -
that is, those where the transition functions are holomorphic
(or in other words, complex-analytic). Here the classification
is subtler! It turns out you need, not just the first Chern
number, which is discrete, but another parameter which can vary
in a *continuous* way.
Interestingly, after you pick a basepoint for your elliptic
curve, this other parameter can be thought of as just a point
on the elliptic curve! So, the elliptic curve becomes the
space that classifies holomorphic line bundles over itself -
at least, those with fixed first Chern number. Curiously
circular, eh? This is just one of several curiously circular
classification theorems that happen in this game...
But I'm actually digressing a bit - I'm having trouble
resisting the temptation to explain everything I know, since
it's so simple and beautiful, and I just learned it. Don't
worry - all you need to know is that holomorphic line bundles
over an elliptic curve are classified by an integer and some
other continuous parameter.
The puzzle then arises: how many holomorphic sections do
these line bundles have? More precisely: what's the *dimension*
of the space of holomorphic sections?
Before I answer this, I can't resist adding that these holomorphic
sections have a long and illustrious history - they're called
"theta functions", and you can learn about them here:
14) Jun-ichi Igusa, Theta Functions, Springer, Berlin, 1972.
15) David Mumford, Tata Lectures on Theta, 3 volumes, Birkhauser,
Boston, 1983-1991.
They're important in geometric quantization, where holomorphic
sections of line bundles describe states of quantum systems, and the
reciprocal of the first Chern number is proportional to Planck's
constant. In fact, I first ran into theta functions years ago,
when trying to quantize a black hole - see the end of "week112"
for more details.
But anyway, here's the answer to the puzzle. The dimension turns
out not to depend on the continuous parameter labelling our line
bundle, but only on its first Chern number. If that number is
negative, the dimension is 0. But if it's 0,1,2,3,4,5,6 and so on,
the dimension goes like this:
1,1,2,3,4,5,6,...
Now, this sequence is fairly weird, because of the extra "1" at
the beginning. I hadn't noticed this back when I was quantizing
black holes, because the extra "1" happens for first Chern number
zero, which would correspond to Planck's constant being *infinite*.
But now that I'm just thinking about math, it sticks out like a
sore thumb!
It's got to be right, since the line bundle with first Chern number
zero is the trivial bundle, its sections are just functions, and
the only holomorphic functions on a compact complex manifold are
constants - so there's a 1-dimensional space of them. But, it's
weird.
Luckily, Jim figured out the explanation for this sequence.
First of all, we can encode it into a power series:
1 + x + 2x^2 + 3x^3 + 4x^4 + ...
which we can rewrite as a rational function:
(1-x^6)
1 + x + 2x^2 + 3x^3 + 4x^4 + ... = --------------------
(1-x)(1-x^2)(1-x^3)
Now, the reason for doing this is that we can pick a line
bundle of first Chern number 1, say L, and get a line bundle
of any Chern number n by taking the nth tensor power of L - let's
call that L^n. We can multiply a section of L^n and a section of
L^m to get a section of L^{n+m}. So, all these spaces of sections
we're studying fit together to form a commutative graded ring!
And, whenever you have a graded ring, it's a good idea to write
down a power series that encodes the dimensions of each grade,
just as we've done above. This is called a "Poincare series".
And, when you have a commutative graded ring with one generator
of degree 1, one generator of degree 2, one generator of degree 3,
one relation of degree 6, and no "relations between relations"
(or "syzygies"), its Poincare series will be
(1-x^6)
--------------------
(1-x^1)(1-x^2)(1-x^3)
That's how it always works - think about it.
So, it's natural to hope that our ring built from holomorphic
sections of all the line bundles L^n will have one generator
of degree 1, one of degree 2, one of degree 3, and one relation
of degree 6.
And, this seems to be true!
As I mentioned, people usually call these holomorphic sections
"theta functions". So, it seems we're getting a description of
the ring of theta functions in terms of generators and relations.
How does it work, exactly? Well, I must admit I'm not quite sure.
Jim has some ideas, but it seems I need to do something a bit
different to get his story to work for me. Maybe it goes
something like this. We can write any elliptic curve as the
solutions of this equation:
y^2 = x^3 + Bx + C
for certain constants B and C that depend on the elliptic curve.
(See "week13" and "week261" for details.) Now, this equation is
not homogeneous in the variables y and x, but we can think of it
as homogeneous in a sneaky sense if we throw in an extra variable
like this:
y^2 = x^3 + Bxz^5 + Cz^6
and decree that:
y has grade 3
x has grade 2
z has grade 1
Then all the terms in the equation have grade 6. So, we're
getting a commutative graded ring with generators of degree
1, 2, and 3 and a relation of grade 6. And, I'm hoping this
ring consists of algebraic functions on the total space of
some line bundle L* over our elliptic curve. z should be a
function that's linear in the fiber directions, hence a
section of L. x should be quadratic in the fiber directions,
hence a section of L^2. And y should be cubic, hence a
section of L^3. If L has first Chern number 1, I think we're
in business.
If anybody knows about this stuff, I'd appreciate corrections
or references.
There's a *lot* more to say about this business... because it's
all part of a big story about elliptic curves, theta functions
and modular forms. But, I want to quit here for now.
-----------------------------------------------------------------------
Addenda: I thank David Corfield for pointing out how to get ahold of
Wu's papers free online - and earlier, for telling me Wu's
combinatorial description of pi_3(S^2).
Martin Ouwehand told me that some of Coleman's lecture notes on quantum
field theory are available in TeX here:
17) Sidney Coleman, Quantum Field Theory, first 11 lectures notes
TeXed by Bryan Gin-ge Chen, available at
http://www.physics.upenn.edu/~chb/phys253a/coleman/
James Dolan pointed out that this article:
18) Wikipedia, Riemann-Roch theorem,
http://en.wikipedia.org/wiki/Riemann-Roch
has some very relevant information on the sequence
1, 1, 2, 3, 4, 5, 6, ...
though it's phrased not in terms of "sections of line bundles", but
instead in terms of "divisors" (secretly another way of talking about
the same thing). Let me quote a portion, just to whet your interest:
We start with a connected compact Riemann surface of genus g, and a fixed
point P on it. We may look at functions having a pole only at P. There is an
increasing sequence of vector spaces: functions with no poles (i.e.,
constant functions), functions allowed at most a simple pole at P,
functions allowed at most a double pole at P, a triple pole, ... These
spaces are all finite dimensional. In case g = 0 we can see that the
sequence of dimensions starts
1, 2, 3, ...
This can be read off from the theory of partial fractions. Conversely if
this sequence starts
1, 2, ...
then g must be zero (the so-called Riemann sphere).
In the theory of elliptic functions it is shown that for g = 1 this
sequence is
1, 1, 2, 3, 4, 5 ...
and this characterises the case g = 1. For g > 2 there is no set initial
segment; but we can say what the tail of the sequence must be. We can also
see why g = 2 is somewhat special.
The reason that the results take the form they do goes back to the
formulation (Roch's part) of the [Riemann-Roch] theorem: as a difference
of two such dimensions. When one of those can be set to zero, we get an
exact formula, which is linear in the genus and the degree (i.e. number of
degrees of freedom). Already the examples given allow a reconstruction in
the shape
dimension - correction = degree - g + 1.
For g = 1 the correction is 1 for degree 0; and otherwise 0. The full
theorem explains the correction as the dimension associated to a
further, 'complementary' space of functions.
You can see more discussion of this Week's Finds at the n-Category Cafe:
http://golem.ph.utexas.edu/category/2008/05/this_weeks_finds_in_mathematic_25.html
-----------------------------------------------------------------------
Quote of the Week:
The career of a young theoretical physicist consists of treating the
harmonic oscillator in ever-increasing levels of abstraction. - Sidney Coleman
-----------------------------------------------------------------------
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 jumping-off 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
May 18, 2008
This Week's Finds in Mathematical Physics (Week 264)
John Baez
Here's a puzzle. Guess the next term of this sequence:
1, 1, 2, 3, 4, 5, 6, ...
and then guess the *meaning* of this sequence! I'll give away the
answer after telling you about Coleman's videos on quantum field
theory and an amazing result on the homotopy groups of spheres.
But first... the astronomy picture of the day.
The Eaton Collection at UC Riverside may be the world's best
library of science fiction:
1) The Eaton Collection of Science Fiction, Fantasy, Horror
and Utopian Literature, http://eaton-collection.ucr.edu/
Right now my wife Lisa Raphals is attending a conference there
on the role of Mars in SF, called "Chronicling Mars". Gregory
Benford, Frederik Pohl, Greg Bear, David Brin, Kim Stanley Robinson
and even Ray Bradbury are all there! But for some reason I'm staying
home working on This Week's Finds. I'd say that shows true devotion -
or maybe just stupidity.
Anyway, in honor of the occasion, here's an incredible closeup of a
crater on Mars' moon Phobos:
2) Astronomy Picture of the Day, Stickney Crater
http://apod.nasa.gov/apod/ap080410.html
It's another great example of how machines in space now deliver
many more thrills per buck than the old-fashioned approach
using canned primates. This photo was taken by HiRISE, the
High Resolution Imaging Science Experiment - the same satellite
that took the stunning photos of Martian dunes which graced "week262".
Mars has two moons, Phobos and the even tinier Deimos. Their
names mean "fear" and "dread" in Greek, since in Greek mythology
they were sons of Mars (really Ares), the god of war.
Interestingly, Kepler predicted that Mars had two moons before
they were seen. This sounds impressive, but it was simple
interpolation, since Earth has 1 moon and Jupiter has 4. Or at
least Galileo saw 4 - now we know there are a lot more.
Phobos is only 21 kilometers across, and the big crater you
see here - Stickney Crater - is about 9 kilometers across.
That's almost half the size of the whole moon! The collision
that created it must have almost shattered Phobos.
Phobos is so light - just twice the density of water - that people
once thought it might be hollow. This now seems unlikely, though
it's been the premise of a few SF stories. It's more likely that
Phobos is a loosely packed pile of carbonaceous chondrites captured
from the asteroid belt.
Phobos orbits so close to Mars that it zips around once every
8 hours, faster than Mars itself rotates! Oddly, in 1726
Jonathan Swift wrote about two moons of Mars in his novel
"Gulliver's Travels" - and he guessed that the inner one orbited
Mars every 10 hours.
Gravitational tidal forces are dragging Phobos down, so in only 10
million years it'll either crash or - more likely - be shattered by
tidal forces and form a ring of debris.
So, enjoy it while it lasts.
Anyone who's seriously struggled to master quantum field theory is
likely to have profited from this book:
3) Sidney Coleman, Aspects of Symmetry: Selected Erice Lectures,
Cambridge U. Press, Cambridge, 1988.
It's brimming with wisdom and humor. You should have already
encountered quantum field theory before trying it: what you'll
get are deeper insights.
But what if you're just getting started?
Sidney Coleman, recently deceased, was one of the best quantum field
theorists from the heyday of particle physics. As a grad student I
took a course on quantum field theory from Eddie Farhi, who said he
based his class on the notes from Coleman's class at Harvard. So,
I've always been curious about these notes. Now they're available
online in handwritten form:
4) Sidney Coleman, lecture notes on quantum field theory,
http://www.damtp.cam.ac.uk/user/dt281/qft/col1.pdf
and
http://www.damtp.cam.ac.uk/user/dt281/qft/col2.pdf
Someone should LaTeX them up!
Even more fun, you can now see *videos* of Coleman teaching quantum
field theory:
5) Sidney Coleman, Physics 253: Quantum Field Theory, 50 lectures
recorded 1975-1976, http://www.physics.harvard.edu/about/Phys253.html
This is a younger, hipper Coleman than I'd ever seen: long-haired,
sometimes puffing on a cigarette between sentences. He begins by
saying "Umm... this is Physics 253, a course in relativistic quantum
mechanics. My name is Sidney Coleman. The apparatus you see around
you is part of a CIA surveillance project."
I wish I'd had access to these when I was a kid!
Now for some miraculous math. Daniel Moskovich kindly pointed out a
paper that describes all the homotopy groups of the 2-sphere, and
I want to summarize the main result.
I explained the idea of homotopy groups back in "week102". Very
roughly, the nth homotopy group of a space X, usually denoted pi_n(X),
is the set of ways you can map an n-sphere into that space, where we
count two ways as the same if you can continuously deform one to the
other. If a space has holes, homotopy groups are one way to detect
those holes.
Homotopy groups are notoriously hard to compute - so even for so humble
a space as the 2-sphere, S^2, there's a sense in which "nobody knows"
all its homotopy groups. People know the first 64, though. Here are a
few:
pi_1(S^2) = 0
pi_2(S^2) = Z
pi_3(S^2) = Z
pi_4(S^2) = Z/2
pi_5(S^2) = Z/2
pi_6(S^2) = Z/4 x Z/3
pi_7(S^2) = Z/2
pi_8(S^2) = Z/2
pi_9(S^2) = Z/3
pi_10(S^2) = Z/3 x Z/5
pi_11(S^2) = Z/2
pi_12(S^2) = Z/2 x Z/2
pi_13(S^2) = Z/2 x Z/2 x Z/3
pi_14(S^2) = Z/2 x Z/2 x Z/4 x Z/3 x Z/7
pi_15(S^2) = Z/2 x Z/2
Apart from the fact that they're all finite abelian groups, it's
hard to spot any pattern!
In fact there's a majestic symphony of patterns in the homotopy
groups of spheres, starting from ones that are easy to explain
and working on up to those that push the frontiers of mathematics,
like elliptic cohomology. But, many of these patterns are too
complex for present-day mathematics until we use some tricks to
"water down" or simplify the homotopy groups.
So, what people often do first is take the limit of pi_{n+k}(S^n)
as n -> infinity, getting what's called the kth "stable" homotopy
group of spheres. It's a wonderful but well-understood fact that
these limits really exist. But so far, even these are too
complicated to understand until we work "at a prime p".
This means that we take the kth stable homotopy group of spheres
and see which groups of the form Z/p^n show up in it. For example,
pi_14(S^2) = Z/2 x Z/2 x Z/4 x Z/3 x Z/7
but if we work "at the prime 2" we just see the Z/2 x Z/2 x Z/4.
After all this data processing, we get some astounding pictures:
6) Allen Hatcher, Stable homotopy groups of spheres,
http://www.math.cornell.edu/~hatcher/stemfigs/stems.html
Order teetering on the brink of chaos! If you're brave, you can
learn more about this stuff here:
7) Douglas C. Ravenel, Complex Cobordism and Stable Homotopy Groups
of Spheres, AMS, Providence, Rhode Island, 2003.
If you're less brave, I strongly suggest starting here:
8) Wikipedia, Homotopy groups of spheres,
http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres
But now, I want to talk about an amazing paper that pursues a
very different line of attack. It gives a beautiful description
of *all* the homotopy groups of S^2, in terms of braids:
9) A. Berrick, F. R. Cohen, Y. L. Wong and J. Wu, Configurations,
braids and homotopy groups, J. Amer. Math. Soc., 19 (2006), 265-326.
Also available at http://www.math.nus.edu.sg/~matwujie/BCWWfinal.pdf
For this you need to realize that for any n, there's a group B_n
whose elements are n-strand braids. For example, here's an element
of B_3:
| | |
\ / |
/ |
/ \ |
| \ /
| /
| / \
\ / |
/ |
/ \ |
| \ /
| /
| / \
\ / |
/ |
/ \ |
| \ /
| /
| / \
| | |
I actually talked about this specific braid back in "week233".
But anyway, we count two braids as the same if you can wiggle one
around until it looks like the other without moving the ends at
the top and bottom - which you can think of as nailed to the
ceiling and floor.
How do braids become a group? Easy: we multiply them by putting
one on top of the other. For example, this braid:
| | |
\ / |
A = / |
/ \ |
| | |
times this one:
| | |
| \ /
B = | /
| / \
| | |
equals this:
| | |
\ / |
/ |
/ \ |
| | |
AB = | | |
| \ /
| /
| / \
| | |
and in fact the big one I showed you earlier is (AB)^3.
As you let your eye slide from the top to the bottom of a braid, the
strands move around. We can visualize their motion as a bunch of points
running around the plane, never bumping into each other. This gives
an interesting way to generalize the concept of a braid! Instead
of points running around the plane, we can have points running around
S^2, or some other surface X. So, for any surface X and any number n
of strands, we get a "surface braid group", called B_n(X).
As I hinted in "week261", these surface braid groups have cool
relationships to Dynkin diagrams. I urged you to read this paper,
and I'll urge you again:
10) Daniel Allcock, Braid pictures for Artin groups, available as
arXiv:math.GT/9907194.
But for now, we just need the "spherical braid group" B_n(S^2)
together with the usual braid group B_n.
Let's say a braid is "Brunnian" if when you remove anyone strand,
the remaining braid becomes the identity: you can straighten out
all the remaining strands to make them vertical. It's a fun little
exercise to check that Brunnian braids form a subgroup of all braids.
So, we have an n-strand Brunnian braid group BB_n.
The same idea works for braids on other surface, like the 2-sphere.
So, we also have an n-strand *spherical* Brunnian braid group BB_n(S^2).
Now, there's obvious map
B_n -> B_n(S^2)
Why? An element of B_n describes the motion of a bunch of points
running around the plane, but the plane sits inside the 2-sphere:
the 2-sphere is just the plane with an extra point tacked on. So,
an ordinary braid gives a spherical braid.
This map clearly sends Brunnian braids to spherical Brunnian braids,
so we get a map
f: BB_n -> BB_n(S^2)
And now we're ready for the shocking theorem of Berrick, Cohen,
Wong and Wu:
Theorem: For n > 3, BB_n(S^2) modulo the image of f is the nth
homotopy group of S^2.
In something more like plain English: when n is big enough, the
nth homotopy group of the 2-sphere consists of spherical Brunnian
braids modulo ordinary Brunnian braids!
Zounds! What do the homotopy groups of S^2 have to do with braids?
It's not supposed to be obvious! The proof of this result is long and
deep, making use of flows on metric spaces, and also the fact that all
the Brunnian braid groups BB_n fit together into a "simplicial group"
whose nth homology is the nth homotopy group of S^2. I'd love to
understand all this stuff, but I don't yet.
This result doesn't instantly help us "compute" the homotopy groups of
S^2 - at least not in the sense of writing them down as a product of
groups like Z/p^n. But, it gives a new view of these homotopy groups,
and there's no telling where this might lead.
When I was first getting ready to write this article, I was also
going to tell you about some amazing descriptions of the homotopy
groups of the *3-sphere*, due to Wu.
However, I later realized - first to my shock, and then my embarrassment
for not having known it already - that the nth homotopy group of S^3
is *the same* as the nth homotopy group of S^2, at least for n > 2.
Do you see why?
Given this, it turns out that Wu's results are predecessors of the
theorem just stated, a bit more combinatorial and less "geometric".
Wu's results appeared here:
11) Jie Wu, On combinatorial descriptions of the homotopy groups of
certain spaces, Math. Proc. Camb. Phil. Soc. 130 (2001), 489-513.
Also available at http://www.math.nus.edu.sg/~matwujie/newnewpis_3.pdf
Jie Wu, A braided simplicial group, Proc. London Math. Soc. 84
(2002), 645-662. Also available at
http://www.math.nus.edu.sg/~matwujie/Research2.html
and there's a nice summary of these results on his webpage:
12) Jie Wu, 2.1 Homotopy groups and braids, halfway down the page at
http://www.math.nus.edu.sg/~matwujie/Research2.html
See also this expository paper:
13) Fred R. Cohen and Jie Wu, On braid groups and homotopy groups,
Geometry & Topology Monographs 13 (2008), 169-193. Also available at
http://www.math.nus.edu.sg/~matwujie/cohen.wu.GT.revised.29.august.2007.pdf
Next I want to talk about puzzle mentioned at the start of this
Week's Finds... but first I should answer the puzzle I just raised.
Why do the homotopy groups of S^2 match those of S^3 after a while?
Because of the Hopf fibration! This is a fiber bundle with S^3 as
total space, S^2 as base space and S^1 as fiber:
S^1 -> S^3 -> S^2
Like any fiber bundle, it gives a long exact sequence of homotopy
groups as explained in "week151":
... -> pi_n(S^1) -> pi_n(S^3) -> pi_n(S^2) -> pi_{n-1}(S^1) -> ...
but the homotopy groups of S^1 vanishes after the first, so we get
... -> 0 -> pi_n(S^3) -> pi_n(S^2) -> 0 -> ...
for n > 2, which says that
pi_n(S^3) = pi_n(S^2)
Okay, now for this mysterious sequence:
1, 1, 2, 3, 4, 5, 6, ...
The next term is obviously 7. If you guessed anything else, you
were over-analyzing. So the real question is: why the funny
"hiccup" at the beginning?
You'll find two explanations of this sequence in Sloane's Online
Encyclopedia of Integer Sequences, but neither of them is the reason
James Dolan and I ran into it. We were studying theta functions...
Say you have a torus. Then the complex line bundles over it
are classified by an integer called the "first Chern number".
In some sense, this integer this measures how "twisted" the
bundle is. For example, you can put any connection on the bundle,
compute its curvature 2-form, and integrate it over the torus:
up to some constant factor, you'll then get the first Chern number.
A torus is a 2-dimensional manifold, but we can also make it
into a 1-dimensional *complex* manifold, often called an
"elliptic curve". In fact we can do this in infinitely many
fundamentally different ways, one for each point in the "moduli
space of elliptic curves". I've explained this repeatedly here -
try "week125" for a good starting-point - so I won't do so again.
The details don't really matter here.
Back to line bundles. If we pick an elliptic curve, we can
try to classify the *holomorphic* complex line bundles over it -
that is, those where the transition functions are holomorphic
(or in other words, complex-analytic). Here the classification
is subtler! It turns out you need, not just the first Chern
number, which is discrete, but another parameter which can vary
in a *continuous* way.
Interestingly, after you pick a basepoint for your elliptic
curve, this other parameter can be thought of as just a point
on the elliptic curve! So, the elliptic curve becomes the
space that classifies holomorphic line bundles over itself -
at least, those with fixed first Chern number. Curiously
circular, eh? This is just one of several curiously circular
classification theorems that happen in this game...
But I'm actually digressing a bit - I'm having trouble
resisting the temptation to explain everything I know, since
it's so simple and beautiful, and I just learned it. Don't
worry - all you need to know is that holomorphic line bundles
over an elliptic curve are classified by an integer and some
other continuous parameter.
The puzzle then arises: how many holomorphic sections do
these line bundles have? More precisely: what's the *dimension*
of the space of holomorphic sections?
Before I answer this, I can't resist adding that these holomorphic
sections have a long and illustrious history - they're called
"theta functions", and you can learn about them here:
14) Jun-ichi Igusa, Theta Functions, Springer, Berlin, 1972.
15) David Mumford, Tata Lectures on Theta, 3 volumes, Birkhauser,
Boston, 1983-1991.
They're important in geometric quantization, where holomorphic
sections of line bundles describe states of quantum systems, and the
reciprocal of the first Chern number is proportional to Planck's
constant. In fact, I first ran into theta functions years ago,
when trying to quantize a black hole - see the end of "week112"
for more details.
But anyway, here's the answer to the puzzle. The dimension turns
out not to depend on the continuous parameter labelling our line
bundle, but only on its first Chern number. If that number is
negative, the dimension is 0. But if it's 0,1,2,3,4,5,6 and so on,
the dimension goes like this:
1,1,2,3,4,5,6,...
Now, this sequence is fairly weird, because of the extra "1" at
the beginning. I hadn't noticed this back when I was quantizing
black holes, because the extra "1" happens for first Chern number
zero, which would correspond to Planck's constant being *infinite*.
But now that I'm just thinking about math, it sticks out like a
sore thumb!
It's got to be right, since the line bundle with first Chern number
zero is the trivial bundle, its sections are just functions, and
the only holomorphic functions on a compact complex manifold are
constants - so there's a 1-dimensional space of them. But, it's
weird.
Luckily, Jim figured out the explanation for this sequence.
First of all, we can encode it into a power series:
1 + x + 2x^2 + 3x^3 + 4x^4 + ...
which we can rewrite as a rational function:
(1-x^6)
1 + x + 2x^2 + 3x^3 + 4x^4 + ... = --------------------
(1-x)(1-x^2)(1-x^3)
Now, the reason for doing this is that we can pick a line
bundle of first Chern number 1, say L, and get a line bundle
of any Chern number n by taking the nth tensor power of L - let's
call that L^n. We can multiply a section of L^n and a section of
L^m to get a section of L^{n+m}. So, all these spaces of sections
we're studying fit together to form a commutative graded ring!
And, whenever you have a graded ring, it's a good idea to write
down a power series that encodes the dimensions of each grade,
just as we've done above. This is called a "Poincare series".
And, when you have a commutative graded ring with one generator
of degree 1, one generator of degree 2, one generator of degree 3,
one relation of degree 6, and no "relations between relations"
(or "syzygies"), its Poincare series will be
(1-x^6)
--------------------
(1-x^1)(1-x^2)(1-x^3)
That's how it always works - think about it.
So, it's natural to hope that our ring built from holomorphic
sections of all the line bundles L^n will have one generator
of degree 1, one of degree 2, one of degree 3, and one relation
of degree 6.
And, this seems to be true!
As I mentioned, people usually call these holomorphic sections
"theta functions". So, it seems we're getting a description of
the ring of theta functions in terms of generators and relations.
How does it work, exactly? Well, I must admit I'm not quite sure.
Jim has some ideas, but it seems I need to do something a bit
different to get his story to work for me. Maybe it goes
something like this. We can write any elliptic curve as the
solutions of this equation:
y^2 = x^3 + Bx + C
for certain constants B and C that depend on the elliptic curve.
(See "week13" and "week261" for details.) Now, this equation is
not homogeneous in the variables y and x, but we can think of it
as homogeneous in a sneaky sense if we throw in an extra variable
like this:
y^2 = x^3 + Bxz^5 + Cz^6
and decree that:
y has grade 3
x has grade 2
z has grade 1
Then all the terms in the equation have grade 6. So, we're
getting a commutative graded ring with generators of degree
1, 2, and 3 and a relation of grade 6. And, I'm hoping this
ring consists of algebraic functions on the total space of
some line bundle L* over our elliptic curve. z should be a
function that's linear in the fiber directions, hence a
section of L. x should be quadratic in the fiber directions,
hence a section of L^2. And y should be cubic, hence a
section of L^3. If L has first Chern number 1, I think we're
in business.
If anybody knows about this stuff, I'd appreciate corrections
or references.
There's a *lot* more to say about this business... because it's
all part of a big story about elliptic curves, theta functions
and modular forms. But, I want to quit here for now.
-----------------------------------------------------------------------
Addenda: I thank David Corfield for pointing out how to get ahold of
Wu's papers free online - and earlier, for telling me Wu's
combinatorial description of pi_3(S^2).
Martin Ouwehand told me that some of Coleman's lecture notes on quantum
field theory are available in TeX here:
17) Sidney Coleman, Quantum Field Theory, first 11 lectures notes
TeXed by Bryan Gin-ge Chen, available at
http://www.physics.upenn.edu/~chb/phys253a/coleman/
James Dolan pointed out that this article:
18) Wikipedia, Riemann-Roch theorem,
http://en.wikipedia.org/wiki/Riemann-Roch
has some very relevant information on the sequence
1, 1, 2, 3, 4, 5, 6, ...
though it's phrased not in terms of "sections of line bundles", but
instead in terms of "divisors" (secretly another way of talking about
the same thing). Let me quote a portion, just to whet your interest:
We start with a connected compact Riemann surface of genus g, and a fixed
point P on it. We may look at functions having a pole only at P. There is an
increasing sequence of vector spaces: functions with no poles (i.e.,
constant functions), functions allowed at most a simple pole at P,
functions allowed at most a double pole at P, a triple pole, ... These
spaces are all finite dimensional. In case g = 0 we can see that the
sequence of dimensions starts
1, 2, 3, ...
This can be read off from the theory of partial fractions. Conversely if
this sequence starts
1, 2, ...
then g must be zero (the so-called Riemann sphere).
In the theory of elliptic functions it is shown that for g = 1 this
sequence is
1, 1, 2, 3, 4, 5 ...
and this characterises the case g = 1. For g > 2 there is no set initial
segment; but we can say what the tail of the sequence must be. We can also
see why g = 2 is somewhat special.
The reason that the results take the form they do goes back to the
formulation (Roch's part) of the [Riemann-Roch] theorem: as a difference
of two such dimensions. When one of those can be set to zero, we get an
exact formula, which is linear in the genus and the degree (i.e. number of
degrees of freedom). Already the examples given allow a reconstruction in
the shape
dimension - correction = degree - g + 1.
For g = 1 the correction is 1 for degree 0; and otherwise 0. The full
theorem explains the correction as the dimension associated to a
further, 'complementary' space of functions.
You can see more discussion of this Week's Finds at the n-Category Cafe:
http://golem.ph.utexas.edu/category/2008/05/this_weeks_finds_in_mathematic_25.html
-----------------------------------------------------------------------
Quote of the Week:
The career of a young theoretical physicist consists of treating the
harmonic oscillator in ever-increasing levels of abstraction. - Sidney Coleman
-----------------------------------------------------------------------
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 jumping-off 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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