- #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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