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July 13, 2007

This Week's Finds in Mathematical Physics (Week 254)

John Baez

This week I'd like to talk about exceptional Lie algebras and the

Standard Model, Witten's new paper on the Monster group and black

holes in 3d gravity, and Connes and Marcolli's new book! Then

I want to continue the Tale of Groupoidification.

However, I don't have the energy to do this all now. And even

if I did, you wouldn't have the energy to read it.

So, I'll just point you towards Connes and Marcolli's new book,

which you can download for free:

1) Alain Connes and Mathilde Marcolli, Noncommutative Geometry,

Quantum Fields and Motives, available at

http://www.alainconnes.org/downloads.html

I hope to discuss it sometime, especially since it tackles a

question I've been mulling lately: is there a good "explanation"

for the Standard Model of particle physics?

For now, I'll start by discussing Witten's latest paper:

2) Edward Witten, Three-dimensional gravity revisited, available

as arXiv:0706.3359.

This is a bold piece of work, which seeks to relate the entropy of

black holes in 3d quantum gravity to representations of the

Monster group - the largest sporadic finite simple group, with

about 10^{54} elements.

If the main idea is right, this gives a whole new view of

"Monstrous Moonshine" - the bizarre connection between the Monster

and fundamental concepts in complex analysis like the j-function.

(See "week66" for a quick intro to Monstrous Moonshine.)

As the title hints, Witten had already tackled quantum gravity in

3 spacetime dimensions. In this earlier work, he argued it was an

exactly soluble problem: a topological field theory called

Chern-Simons theory. However, this theory is really an

*extension* of gravity to the case of "degenerate" metrics:

roughly speaking, geometries of spacetime where certain regions

get squashed down to zero size. Degenerate metrics are weird.

So, what happens if we try to quantize 3d gravity while insisting

that the metric be nondegenerate?

It's hard to say. So, Witten takes a few clues and cleverly fits

them together to make a surprising guess. He considers 3d general

relativity with negative cosmological constant. This has 3d

anti-DeSitter space as a solution. Anti-DeSitter space has a

"boundary at infinity": a 2d cylinder with a conformal structure.

The "AdS-CFT" idea, also known as "holography", suggests that in

this sort of situation, 3d quantum gravity should be completely

described by a field theory living on this boundary at infinity -

a field theory theory with all conformal transformations as

symmetries.

Which conformal field theory should correspond to 3d quantum

gravity with negative cosmological constant? It depends on the

value of the cosmological constant! Some topological arguments

suggest that the Chern-Simons description of 3d quantum gravity

is only gauge-invariant when the cosmological constant Lambda

takes certain special values, namely

Lambda = -1/(16 k)^2

where k is an integer, known as the "level" in Chern-Simons theory.

By the way: I'm working in Planck units here, and I'm assuming

our Chern-Simons theory is left-right symmetric, just to keep

things simple. I may also be making some small numerical errors.

This quantization of the cosmological constant must seem strange

if you've never seen it before, but it's not really so weird.

What's weird is that Witten is using Chern-Simons theory to

determine the allowed values of the cosmological constant even

though he wants to study what happens if gravity is *not*

described by Chern-Simons theory!

Witten knows this is weird: later he says "we used the gauge

theory approach to get some hints about the right values of the

cosmological constant simply because it was the only tool

available."

Indeed, the whole paper seems designed to refute the notion that

mathematicians get less daring as they get older. He writes: "We

make at each stage the most optimistic possible assumption."

Perhaps he has some evidence for his guesses that he's not

revealing yet. Or perhaps he's decided it takes courage verging

on recklessness to track the Monster to its lair.

Anyway: next Witten relates the level k to something called the

"central charge" of the conformal field theory living at the

boundary at infinity.

What's the "central charge"? This is a standard concept in

conformal field theory. Perhaps the simplest explanation is that

in a conformal field theory, the total energy of the vacuum state

is -c/24, where c is the central charge. So, naively you'd expect

c = 0, but quantum effects make nonzero values of the vacuum energy

possible, and even typical. A closely related cool fact is that

the partition function of a conformal field theory is only a

well-defined number up to multiples of

exp(2 pi i c / 24)

This means the partition function is a well-defined number when

c is a multiple of 24. This happens in certain especially nice

conformal field theories which are said to have "holomorphic

factorization".

The appearance of the magic number 24 here is the first taste of

Monstrous Moonshine! For more on the importance of this number

in string theory, see "week124", "week125" and "week126".

As you can see, there are lots of subtleties here, which I really

don't want to get into, but feel guilty about glossing over.

Here's another. There are really *two* conformal field theories

in this game: one that describes ripples of the gravitational

field moving clockwise around the boundary at infinity, and

another for ripples moving counterclockwise. Our simplifying

assumption about left-right symmetry lets us describe these

"right-movers" and "left-movers" with the same theory. So, both

have the same central charge.

In this case, the relation between central charge and level

is simple:

c = 24 k

Next, Witten considers the situation where k takes its smallest

interesting value: k = 1, so c = 24. It just so happens that

c = 24 conformal field theories with holomorphic factorization

have been classified, at least modulo a certain conjecture:

2) A. N. Schellekens, Meromorphic c = 24 conformal field theories,

Comm. Math. Phys. 153 (1993) 159-196. Also available as

hep-th/9205072.

It's believed there are 71 of them. Which one could describe

3d quantum gravity?

Of these 71, ALL BUT ONE have gauge symmetries! Now, Witten

is assuming 3d quantum gravity is *not* described by Chern-Simons

theory, which is a gauge theory. So, he guesses that the one

exceptional theory is the right one!

And this is a very famous conformal field theory. It's a theory

of a bosonic string wiggling around in a 26-dimensional spacetime

curled up in clever way with the help of a 24-dimensional lattice

called the Leech lattice. This theory is famous because its

symmetry group is the Monster group! It is, in fact, the simplest

thing we know that has the Monster group as symmetries.

For more details, try these - in rough order of increasing

thoroughness:

3) Terry Gannon, Postcards from the edge, or Snapshots

of the theory of generalised Moonshine, available as

arXiv:math/0109067.

Terry Gannon, Monstrous Moonshine: the first twenty-five years,

available as arXiv:math/0402345.

4) Richard Borcherds, online papers, available at

http://math.berkeley.edu/~reb/papers/

5) Igor Frenkel, James Lepowsky, Arne Meurman, Vertex operator

algebras and the Monster, Academic Press, New York, 1988.

Now, if this monstrous conformal field theory turned out to be

3d quantum gravity in disguise - viewed from infinity, so to

speak - it might someday give a much better understanding of

Monstrous Moonshine. However, Witten gives no explanation as to

*why* this theory should be 3d gravity, except for the indirect

argument I just sketched. The precise relation between 3d

quantum gravity and the bosonic string wiggling around in

26 dimensions remains obscure.

However, while Witten leaves this mysterious, he does offer a

tantalizing extra tidbit of evidence that the relation is real!

The partition function of the monstrous conformal field theory

I just mentioned is the j-function, or more precisely:

J(q) = q^{-1} + 196884 q + 21493760 q^2 + ...

As I mentioned, this function shows up naturally in complex

analysis. More precisely, it parametrizes the moduli space of

elliptic curves (see "week125"). But, its bizarre coefficients

turn out to be dimensions of interesting representations of the

Monster group. For example, the smallest nontrivial representation

of the Monster has dimension 196883; adding the trivial

representation, we get 196884. This was one of several strange

clues leading to the discovery of Monstrous Moonshine.

What Witten does is assume that the monstrous conformal field

theory describes 3d quantum gravity for k = 1, and then use

properties of the j-function to compute the entropy of black

holes!

I won't attempt to explain the calculation. Suffice it to say

that the lightest possible black hole turns out to have

196883 quantum states - its space of states is the smallest

nontrivial representation of the Monster group. So, its entropy

is:

ln(196883) ~ 12.19

On the other hand, Hawking's semiclassical calculation gives

4 pi ~ 12.57

The match is not perfect - but it doesn't need to be, since we

expect quantum corrections to Hawking's formula for small black

holes.

What's more impressive is that Witten can guess the entropy of

the lightest possible black hole for other values of k - meaning,

other values of the cosmological constant. The space of states

of these black holes are always representations of the Monster

group, so we get logarithms of weird-looking integers. For

example, for k = 4 the entropy is

ln(81026609426) ~ 25.12

while Hawking's formula gives

8 pi ~ 25.13

Much better! And, using a formula of Petersson and Rademacher

for asymptotics of coefficients of the j-function, together with

some facts about Hecke operators, he shows that as k -> infinity,

the agreement becomes perfect!

In short, there are some fascinating hints of a relation between

the Monster group and black hole entropies in 3d gravity, but the

details of Witten's hoped-for "AdS-CFT correspondence" between 3d

gravity and the monstrous conformal field theory remain obscure.

Indeed, there are lots of problems with Witten's proposal:

6) Jacques Distler, Witten on 2+1 gravity,

http://golem.ph.utexas.edu/~distler/blog/archives/001335.html

But, time will tell. In fact, if history is any guide, we can

expect to see armies of string theorists marching into this

territory any day now. So, I'll just pose one question.

There's a well-known route from 2d rational conformal field

theories (or "RCFTs") to 3d topological quantum field theories

(or "TQFTs"), which passes through modular tensor categories.

For example, an RCFT called the Wess-Zumino-Witten model gives

the TQFT called Chern-Simons theory.

But, now Witten is saying 3d quantum isn't Chern-Simons theory;

instead, it's something related to the monstrous CFT.

So: is the monstrous conformal field theory known to be an RCFT?

If so, what 3d TQFT does it give? Could this TQFT be the 3d

quantum gravity theory Witten is seeking?

Even though Witten is now claiming 3d quantum gravity *can't be*

a TQFT, I think this is an interesting question. At the very

least, I'd like to know more about this "Monster TQFT" - if it

exists.

Now let's move from 3d quantum gravity to real-world particle

physics...

Last week I described some mathematical relations between the

Standard Model of particle physics, the most famous grand unified

theories, and some "exceptional" structures in mathematics: the

exceptional Lie group E6, the complexified octonionic projective

plane, and the exceptional Jordan algebra.

This week I want to go a bit further, and talk about the work

of Kac, Larsson and others on the exceptional Lie superalgebras

E(3|6), E(3|8) and E(5|10).

As before, my goal is to point out some curious relations between

the messy pack of particles we see in nature and the "exceptional"

structures we find in mathematics. By this, I mean structures

that show up when you classify algebraic gadgets, but don't fit

into nice systematic infinite families. Right now the Monster

is the king of all exceptional structures, the biggest of the

26 sporadic finite simple groups. But, there are lots of other

such structures, and they all seem to be related.

As mentioned back in "week66", Edward Witten once suggested that

the correct theory of our universe could be an exceptional

structure of some sort. There's even a fun hand-wavy argument

for this idea. It goes like this: the theory of our universe

*must* be incredibly special, since out of all the theories we can

write down, only one describes the universe that actually exists!

In particular, lots of very simple theories do *not* describe our

universe. So there must be some principle besides simplicity that

picks out the theory of our universe.

Unfortunately, when we try to think about these issues seriously,

we're quickly led into very deep waters. In practice, people

quickly muddy these waters and create a quagmire. It's very hard

to discuss this stuff without uttering nonsense. If you want

to see my try, look at "week146".

But right now, I prefer to act like a sober, serious mathematical

physicist. So, I'll tell you a bit about exceptional Lie

superalgebras and how they could be related to the Standard Model.

First, some history. In 1887, Wilhelm Killing sent a letter to

Friedrich Engel saying he'd classified the simple Lie algebras.

Besides the "classical" ones - namely the infinite series sl(n,C),

so(n,C) and sp(n,C) - he found 6 exceptions: a 14-dimensional one,

two 52-dimensional ones, a 78-dimensional one, a 133-dimensional

one and a 248-dimensional one.

In 1894, Eli Cartan finished a doctoral thesis in which he cleaned

up Engel's work. In the process, he noticed that Engel's two

52-dimensional Lie algebras were actually the same. Whoops!

So, we now have just 5 "exceptional" simple Lie algebras. In

order of increasing size, they're called G2, F4, E6, E7 and E8.

In 1914, Cartan realized that the smallest exceptional Lie algebra,

G2, comes from the symmetry group of the octonions! Later it was

realized that all 5 are connected to the octonions. I've written

a lot about this in previous Weeks, but most of that material can

be found here:

6) John Baez, Exceptional Lie algebras,

http://math.ucr.edu/home/baez/octonions/node13.html

Now, whenever mathematicians do something fun, they want to keep

doing it, which means *generalizing* it.

One way to generalize Cartan's work is to study "symmetric spaces",

which I defined last week. Briefly, a symmetric space is a

manifold equipped with a geometrical structure that's very

symmetrical: so much so that every point is just like every other,

and the view in any direction is the same as the view in the

opposite direction.

In fact, it was Cartan himself who invented the concept of

symmetric space, and after he classified the simple Lie algebras

he went ahead and classified these.

More precisely, I think he classified the "compact Riemannian"

symmetric spaces. Every simple Lie algebra gives one of these,

namely a compact simple group. But, there are others too. So,

compact Riemannian symmetric spaces are a nice generalization

of simple Lie algebras - and I believe Cartan succeeded in

classifying them all.

Again, there are some infinite series, but also some exceptions

coming from the octonions. I talked about one of these last week,

namely EIII, the complexified octonionic projective plane. You

can see a list here:

7) Wikipedia, Riemannian symmetric space,

http://en.wikipedia.org/wiki/Riemannian_symmetric_space

For a quick intro to the classification of simple Lie algebras and

compact Riemannian symmetric spaces, try this great book:

8) Daniel Bump, Lie Groups, Springer, Berlin, 2004.

For a slower, more thorough introduction, try the book by Helgason

mentioned last Week.

A second way to generalize Cartan's work is to consider simple

Lie *superalgebras*.

Lie superalgebras are just like Lie algebras, except they're split

into an "even" or bosonic and "odd" or fermionic part. The idea

is that we stick minus signs in the usual Lie algebra formulas

whenever we switch two "odd" elements.

This is very natural from a physics viewpoint, since whenever

you switch two identical fermions, the wavefunction of the

universe gets multiplied by -1. (Take my word for it - I've seen

it happen!)

It's also very natural from a math viewpoint, since "super vector

spaces" form a symmetric monoidal category with almost all the

nice properties of plain old vector spaces. This lets crazed

mathematicians and physicists systematically generalize pretty

much all of linear algebra to the "super" world. So, why not Lie

algebras?

The simple Lie superalgebras were classified by Victor Kac in

1977:

9) Victor Kac, Lie superalgebras, Adv. Math. 26 (1977), 8-96.

Not counting the ordinary simple Lie algebras, there are 8

series of simple Lie superalgebras and a few exceptional ones.

At least some of these exceptions come from the octonions:

10) Anthony Sudbery, Octonionic description of exceptional

Lie superalgebras, Jour. Math. Phys. 24 (1983), 1986-1988.

Do they all? I don't know! Someone please tell me!

A third way to generalize Cartan's work is to classify

*infinite-dimensional* simple Lie algebras - or for that matter,

Lie superalgebras.

So far I've implicitly assumed all our algebraic gadgets are

finite-dimensional, but we can lift that restriction. If you

try to classify infinite-dimensional gadgets without *any*

restrictions, it can get really hairy. It turns out the nice

thing is to classify "linearly compact" infinite-dimensional

simple Lie algebras. I won't define the quoted phrase, since

it's technical and it's explained near the beginning

of this paper:

11) Victor Kac, Classification of infinite-dimensional simple

linearly compact Lie superalgebras, Erwin Schroedinger Institut

preprint, 1998.

Available at http://www.esi.ac.at/Preprint-shadows/esi605.html

Anyway, back in 1880 Lie himself made a guess about

infinite-dimensional Lie algebras that would solve the problem

I'm talking about now, though he didn't phrase it in the modern

way. And, Cartan proved Lie's guess in 1909! Actually, there

was a hole in Cartan's proof, which was only noticed much later.

It was filled by Guillemin, Quillen and Sternberg in 1966.

So, here's the answer: there are 4 families of linearly compact

infinite-dimensional simple Lie algebras, and no exceptions.

Ignoring an important nuance I'll explain later, these are:

A) The Lie algebra of all complex vector fields on C^n.

B) The Lie algebra of all complex vector fields v on C^n

that are "divergence-free":

div v = 0

C) The Lie algebra of all complex vector fields v on C^{2n}

that are "symplectic":

L_v omega = 0

where omega is the usual symplectic structure on C^{2n},

and L means "Lie derivative"

D) The Lie algebra of all complex vector fields v on C^{2n+1}

that are "contact":

L_v alpha = f alpha

for some function f depending on v, where alpha is the usual

contact structure on C^{2n+1}.

If you don't know about symplectic structures or contact

structures, don't worry - we won't need them now. The main point

is that they're differential forms that show up throughout

classical mechanics. So, this classification theorem is

surprisingly nice.

Notice: no exceptions! That's a kind of exception in its own

right.

In 1998, Victor Kac proved the "super" version of this result.

In other words, he classified linearly compact infinite-dimensional

Lie superalgebras! This result is Theorem 6.3 of his paper above.

There turn out to be 10 families and 6 exceptions, which are

called E(1|6), E(2|2), E(3|6), E(3|8), E(4|4) and E(5|10).

Many of the families are straightforward "super" generalizations

of the 4 families I just showed you. Some are stranger. Most

important for us today are the exceptions discovered by Irina

Shchepochkina in 1983:

12) Irena Shchepochkina, New exceptional simple Lie superalgebras,

C. R. Bul. Sci. 36 (1983), 313-314.

The easiest to explain is E(5|10). And, you'll soon see that the

number 5 here is related to the math of the SU(5) grand unified

theory, which I explained last Week!

The even part of E(5|10) is the Lie algebra of divergence-free complex

vector fields on C^5.

The odd part of E(5|10) consists of closed complex 2-forms on C^5.

The bracket of two even guys is the usual Lie bracket of vector

fields.

The bracket of an even guy and an odd guy is the usual "Lie

derivative" of a differential form with respect to a vector field.

The only tricky bit is the bracket of two odd guys! So, suppose

mu and nu are closed complex 2-forms on C^5. Their wedge product

is a 4-form mu ^ nu. But, we can identify this with a vector

field v by demanding:

i_v vol = mu ^ nu

Here vol is the volume form:

vol = dx_1 ^ dx_2 ^ dx_3 ^ dx_4 ^ dx_5

and i_v vol is the "interior product", which feeds v into vol and

leaves us with a 4-form. You can check that this vector field v

is divergence-free. So, we define the bracket of mu and nu to be

v.

sl(5,C) sits inside the even part of E(5|10) in a nice way, as

the divergence-free vector fields whose coefficients are *linear*

functions on C^5. So, since su(5) sits inside sl(5,C), we get a

tempting relation to SU(5).

(Now I'll come clean now and explain the "important nuance" I

ignored earlier. For the classification theorems I mentioned

earlier, we must use vector fields and differential forms with

*formal power series* as coefficients. But for the purposes

of mathematical physics, we should keep a more flexible attitude.)

Next, what about E(3|6)? This is contained in E(5|10). To define

it, we give E(5|10) a clever grading where x_1, x_2, x_3 are

treated differently from the other two variables. Then we take

the subalgebra of degree-zero guys. The details are explained in

the above papers - or more simply, here:

13) Victor Kac, Classification of infinite-dimensional simple

groups of supersymmetries and quantum field theory, available as

math.QA/9912235.

All this is reminiscent of how SU(5) contains the gauge group of

the Standard Model, namely S(U(3) x U(2)). In particular, the

even part of E(3|6) contains the Lie algebra

sl(3,C) + sl(2,C) + gl(1,C)

in a canonical way. So, any representation of E(3|6) automatically

gives a representation of the Standard Model Lie algebra

su(3) + su(2) + u(1)

And in the above paper Kac goes even further! He defines a fairly

natural class of representations of E(3|6), and proves something

remarkable: these restrict to representations of

su(3) + su(2) + u(1)

that correspond precisely to the gluon, the photon and the W and

Z bosons, and the quarks and leptons in one generation...

... together with one other particle, which is *not* the Higgs

boson, but instead acts like a gluon with electric charge +-1.

Darn.

One nice thing is how these Lie superalgebras get both bosons and

fermions into the game in a natural way without forcing the

existence of a bunch of unseen "superpartners". One unfortunate

thing is that the above result gives no hint as to why there

should be three generations of quarks and leptons. However, Kac

and Rudakov develop some mathematics to address that question

here:

14) Victor Kac and Alexi Rudakov, Representations of the

exceptional Lie superalgebra E(3,6): I. Degeneracy conditions.

Available as math-ph/0012049.

Representations of the exceptional Lie superalgebra E(3,6): II.

Four series of degenerate modules. Available as math-ph/0012050.

Representations of the exceptional Lie superalgebra E(3,6)

III: Classification of singular vectors. Available as

math-ph/0310045.

Their results are summarize at the end of this review article:

15) Victor Kac, Classification of supersymmetries, Proceedings

of the ICM, Beijing, 2002, vol. 1, 319-344. Available as

math-ph/0302016.

Here Kac writes that "three generations of leptons occur in

the stable region [whatever that means], but the situation with

quarks is more complicated: this model predicts a complete fourth

generation of quarks and an incomplete fifth generation (with

missing down type triplets)."

So, while I don't understand "this model", it seems tantalizingly

close to capturing the algebraic patterns in the Standard Model...

without quite doing so.

Some more nice explanations and references can be found here:

16) Irina Shchepochkina, The five exceptional simple Lie

superalgebras of vector fields. Available as hep-th/9702.3121.

17) Pavel Grozman, Dimitry Leites and Irina Shchepochkina,

Defining relations for the exceptional Lie superalgebras of

vector fields pertaining to The Standard Model, available as

math-ph/0202025.

18) Pavel Grozman, Dimitry Leites and Irina Shchepochkina,

Invariant operators on supermanifolds and Standard Models,

available as math.RT/0202193.

Thomas Larsson has been working on similar ideas, mainly

using E(3|8) instead. This also contains

sl(3,C) + sl(2,C) + gl(1,C)

in a canonical way.

19) Thomas A. Larsson, Symmetries of everything, available as

math-ph/0103013.

Exceptional Lie superalgebras, invariant morphisms, and a

second-gauged Standard Model, available as math-ph/020202.

Thomas A. Larsson, Maximal depth implies su(3)+su(2)+u(1),

available as hep-th/0208185.

Alas, E(3|8) gets the hypercharges of some fermions wrong.

Larsson seems to say this problem also occurs for E(3|6),

which would appear to contradict what Kac claims - but I

could be misunderstanding.

I'll end with few questions. First, is there any relation

between the exceptional Lie superalgebras E(5|10), E(3|6) or

E(5|10) and the exceptional Lie algebra e6? Last week I

explained some relations between e6 and the Standard Model;

are those secretly connected to what I'm discussing this week?

Second, has anyone tried to unify all three generalizations of

Cartan's classification of simple Lie algebras? Starting from

simple Lie algebras, we've seen three ways to generalize:

A) go to symmetric spaces,

B) go the "super" version,

C) go to the infinite-dimensional case.

So: has anyone tried to classify infinite-dimensional super

versions of symmetric spaces? Or even finite-dimensional ones?

(Maybe the super version of a symmetric space should be

called a "supersymmetric space", just for the sake of a nice pun.)

Next, the Tale of Groupoidification! I'll keep this week's

episode short, since you're probably exhausted already.

I want to work my way to the concept of "Hecke operator"

through a series of examples. The examples I'll use are a bit

trickier than the concept I'm really interested in, since the

examples involve integrals, where the Hecke operators I ultimately

want to discuss involve sums. But, the examples are nice if

you like to visualize stuff...

In these examples we'll always have a relation between two sets

X and Y. We'll use this to get an operator that turns functions

on X into functions on Y - a "Hecke operator".

A) The Radon transform in 2 dimensions

Suppose you're trying to do a CAT scan. You want to obtain

a 3d image of someone's innards. Unfortunately, all you do is

take lots of 2d X-ray photos of them. How can you assemble all

this information into the picture you want?

Who better to help you out than a guy named after a

radioactive gas: Radon!

In 1917, the Viennese mathematician Johann Radon tackled a related

problem one dimension down. You could call it a "CAT scan for

flatlanders".

Suppose you want to obtain a complete image of the insides of a

2-dimensional person, but all you can do is shine beams of X-rays

through them and see how much each beam is attenuated.

So, mathematically: you have a real-valued function on the plane -

roughly speaking, the density of your flatlander. You're trying

to recover this function from its integrals along all possible

lines. Someone hands you this function on the space of *lines*,

and you're trying to figure out the original function on the space

of *points*.

(Points lying on lines! If you've been following the Tale of

Groupoidification, you'll know this "incidence relation" is

connected to Klein's approach to geometry, and ultimately to

spans of groupoids. But pretend you don't notice, yet.)

Now, it's premature to worry about this tricky "inverse problem"

before we ponder what it's the inverse of: the "Radon transform".

This takes our original function on the space of *points* and

gives a function on the space of *lines*.

Let's call the Radon transform T. It takes a function f on the

space of points and gives a function Tf on the space of lines,

as follows. Given a line y, (Tf)(y) is the integral of f(x) over

the set of all points x lying on y.

What Radon did is figure out a nice formula for the inverse of

this transform. But that's not what I'm mainly interested in now.

It's the Radon transform itself that's a kind of Hecke operator!

Next, look at another example.

B) The X-ray transform in n dimensions

This is an obvious generalization to higher dimensions of what

I just described. Before we had a space

X = {points in the plane}

and a space

Y = {lines in the plane}

and an incidence relation

S = {(x,y): x is a point lying on the line y}

If we go to n dimensions, we can replace all this with

X = {points in R^n}

Y = {lines in R^n}

S = {(x,y): x is a point lying on the line y}

Again, the X-ray transform takes a function f on the space of

points and gives a function Tf on the space of lines. Given a

line y, (Tf)(y) is the integral of f(x) over the set of all x

with (x,y) in S.

Next, yet another example!

C) The Radon transform in n dimensions

Radon actually considered a different generalization of the

2d Radon transform, using hyperplanes instead of lines. Using

hyperplanes is nicer, because it gives a very simple relationship

between the Radon transform and the Fourier transform. But never

mind - that's not the point here! The point is how similar

everything is. Now we take:

X = {points in R^n}

Y = {hyperplanes in R^n}

S = {(x,y): x is a point lying on the hyperplane y}

And again, the Radon transform takes a function f on X

and gives a function Tf on Y. Given y in Y, (Tf)(y) is the

integral of f(x) over the set of all x with (x,y) in S.

We're always doing the same thing here. Now I'll describe

the general pattern a bit more abstractly.

We've always got three spaces, and maps that look like this:

S

/ \

/ \

P/ \Q

/ \

v v

X Y

In our examples so far these maps are given by

P(x,y) = x

Q(x,y) = y

But, they don't need to be.

Now, how do we get a linear operator in this situation?

Easy! We start with a real-valued function on our space X:

f: X -> R

Then we take f and "pull it back along P" to get a function on S.

"Pulling back along P" is just impressive jargon for composing

with P:

f o P: S -> R

Next, we take f o P and "push it forwards along Q" to get a

function on Y. The result is our final answer, some function

Tf: Y -> R

"Pushing forwards along Q" is just impressive jargon for

integrating: Tf(y) is the integral over all s in S with Q(s) = y.

For this we need a suitable measure, and we need the integral

to converge.

This is the basic idea: we define an operator T by pulling

back and then pushing forward functions along a "span", meaning

a diagram shaped like a bridge:

S

/ \

/ \

P/ \Q

/ \

v v

X Y

But, the reason this operator counts as a "Hecke operator"

is that it gets along with a symmetry group G that's acting

on everything in sight. In the examples so far, this is

the group of Euclidean symmetries of R^n. But, it could be

anything.

This group G acts on all 3 spaces: X, Y, and S. This makes the

space of functions on X into a representation of G! And, ditto

for the space of function on Y and S.

Furthermore, the maps P and Q are "equivariant", meaning

P(gs) = gP(s)

and

Q(gs) = gQ(s)

This makes "pulling back along P" into an intertwining operator

between representations of G. "Pushing forwards along Q" will

also be an intertwining operator if the measure we use is

G-invariant in a suitable sense. In this case, our transform T

becomes an intertwining operator between group representations!

Let's call an intertwining operator constructed this way a "Hecke

operator".

If you're a nitpicky person, e.g. a mathematician, you may wonder

what I mean by "a suitable sense". Well, each "fiber" Q^{-1}(y)

of the map

Q: S -> Y

needs a measure on it, so we can take a function on S and

integrate it over these fibers to get a function on Y. We need

this family of measures to be invariant under the action of G,

for pushing forwards along Q be an intertwining operator.

This stuff about invariant families of measures is mildly

annoying, and so is the analysis involved in making precise

*which* class of functions on X we can pull back to S and then

push forward to Y - we need to make sure the integrals converge,

and so on. When I really get rolling on this Hecke operator

business, I'll often focus on cases where X, Y, and S are

*finite* sets... and then these issues go away!

Hmm. I'm getting tired, but I can't quit until I say one more

thing. If you try to read about Hecke operators, you *won't*

see anything about the examples I just mentioned! You're most

likely to see examples where X and Y are spaces of lattices in

the complex plane. This is the classic example, which we're

trying to generalize. But, this example is more sophisticated

than the ones I've mentioned, in that the "functions" on X and

Y become "sections of vector bundles" over X and Y. The same

sort of twist happens when we go from the Radon transform to the

more general "Penrose transform".

Anyway, next time I'll talk about some really easy examples,

where X, Y, and S are finite sets and G is a finite group.

These give certain algebras of Hecke operators, called "Hecke

algebras".

In the meantime, see if you can find *any* reference in the

literature which admits that "Hecke algebras" are related

to "Hecke operators". It ain't easy!

It's a great example of a mathematical cover-up - one we're

gonna bust wide open.

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Addendum: for more discussion, go to the n-Category Cafe:

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# This Week's Finds in Mathematical Physics (Week 254)

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