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In "week101" I claimed to have figured out the real reason for the importance of the number 24 in string theory. Now I'm not so sure - some pieces of the puzzle that I thought would fit together don't seem to be fitting. Maybe if I explain what I know so far, some experts will hand me some of the missing pieces, or tell me where the ones I have go.
Most of the puzzle pieces came from a talk at a conference on homotopy theory that I went to:
1) Ulrike Tillmann, The moduli space of Riemann surfaces - a homotopy theory approach, talk at Northwestern University Algebraic Topology Conference, March 27, 1997
However, some conversations with Andre Joyal during this conference really helped turn my attention towards what might be going on here.
Let's start by recalling some stuff about homotopy groups of spheres. There are often lots of topologically different ways of wrapping an m-dimensional sphere around a k-dimensional sphere. For example, if m = k = 1, we're talking about the ways of wrapping a circle around a circle. These are classified by an integer called the "winding number". We can make this concrete by thinking of the circle as the unit circle in the complex plane. Take your favorite integer and call it n. Then the function
f(z) = zn
maps the unit circle (the complex numbers with |z| = 1) to itself. If n is positive, this function wraps the unit circle around itself n times in the counterclockwise direction. If n is negative, the circle gets wrapped around in the other direction. If n is zero, f(z) = 1, so we have a constant function - no "wrapping around" at all!
It turns out that any continuous function from the circle to itself can be continuously deformed to exactly one of these functions f(z) = zn. Homotopy theory is all about such continuous deformations. In the jargon of homotopy theory, we say two functions from some space to some other space are "homotopic" if we can continuously deform the first function to the second. Another way of putting it is that the two functions lie in the same "homotopy class". Speaking of jargon, real topologists never say "continuous function": instead, they say "map". So, using this jargon: we know the homotopy class of a map from the circle to itself if we know its winding number.
Now: what happens if we go to higher dimensions? What are all the homotopy classes of maps from the m-dimensional sphere to the k-dimensional sphere? Spheres are pretty simple spaces, so one might at first guess there is some simple answer to this question for all m and k.
Unfortunately, it's far from simple. In fact, nobody knows the answer for all m and k! People do know the answer for zillions of particular values of m and k. But there is no simple pattern to it: instead, there is an incredibly complicated and beautiful weave of subtle patterns, which we have not gotten to the bottom of... and perhaps never will.
To get a little feel for this, let's bring in some standard notation: folks use πm(X) to denote the set of homotopy classes of maps from an m-dimensional sphere to the space X. When m > 0, this set is actually a group, called the "mth homotopy group" of X. These groups are of major importance in algebraic topology.
So, what we are talking about is πm(Sk): the set of all homotopy classes of ways of wrapping an m-sphere around an k-sphere. I already implicitly said that
π1(S1) = Z
where Z stands for the integers, since the winding number is an integer. The same thing happens if we go up a dimension:
π2(S2) = Z
In other words: you can wrap a 2-sphere (an ordinary sphere) n times around itself for any integer n. How? Well, say we use spherical coordinates and describe a point on the sphere using its angle φ from the north pole, together with the angle θ saying how far east it is from Greenwich. Then the map
f(φ,θ) = (φ, nθ)
does the job. Any map from S2 to itself is homotopic to exactly one of these.
The same basic idea works in any higher dimension, too:
πk(Sk) = Z for any k ≥ 1
In other words, there is always an integer n that plays the role of the "winding number" of a map from the k-sphere to itself - though only uncouth physicists call it the "winding number"; mathematicians call it the "degree".
So far, so good. Now, what about mapping a sphere to another sphere of higher dimension? This is nice and simple:
πm(Sk) = {0} whenever m < k
The {0} there is just a standard way to write a set with only one element, which we call "zero". So what we mean is that there's only one homotopy class of ways to map a sphere to a sphere of higher dimension. There is always enough "room" to wiggle around one map until it looks like another.
What about mapping a sphere to another sphere of lower dimension? Here is where the trouble starts! - or the fun, depending on your attitude towards complexity. For example, there is only one homotopy class of maps from a 2-sphere to a circle:
π2(S1) = {0}
There is just no way a 2-sphere can get interestingly "stuck" on the "hole" of the circle. This may seem obvious. But it's not really quite as obvious at it seems, because if we move up one dimension, we have:
π3(S2) = Z
This came as a big shock when Heinz Hopf first discovered it in the 1930's; before then, people had no idea how sneaky homotopy groups were!
There is a beautiful way to compute an integer called the "Hopf invariant" that keeps track of the homotopy class of a map from the 3-sphere to the 2-sphere. There are lots of nice ways to compute it, but alas, I only have time to briefly sketch one! Suppose that the map f: S3 →S2 is smooth (otherwise we can always smooth it up). Then most points p in S2 have the property that the points x in S3 with f(x) = p form a "link": a bunch of knots in S3. If we take two different points in S2 with this property, we get two links. From these two links we can compute an integer called the "linking number": for example, we can just draw these two links and count the times one crosses over or under the other (with appropriate plus or minus signs for each crossing). This number turns out not to depend on how we picked the two points! Moreover, it only depends on the homotopy class of f. It's called the Hopf invariant of f.
Moving up one dimension, it turns out that
π4(S3) = Z/2
Here Z/2 is the group with two elements, usually written 0 and 1, with addition mod 2. Why only two homotopy classes of maps from S4 to S3? Well, you can compute something like the Hopf invariant for these maps, exactly as we did before, but the thing is, links in 4 dimensions are easy to unlink. You can unlink something like
\ / \ / \ / \ / \ \ / \ / \ / \ / \and make it look like
| | | | | | | | | | | | | | | | | | | | | |so the linking number in 4 dimensions is only defined mod 2. Thus the "Hopf invariant" is only defined mod 2.
The exact same thing happens in higher dimensions, too, so in fact we have:
πk+1(Sk) = Z/2 for any k >= 3
This illustrates an important general fact: when the dimensions get high enough, there's more room to wiggle things around, and as we keep jacking up the dimension, homotopy groups simplify a bit and settle down after a while. This is the idea behind "stable homotopy theory".
Let's look at some more examples. We have
π3(S1) = {0}
π4(S2) = Z/2
π5(S3) = Z/2
π6(S4) = Z/2
and so on:
πk+2(Sk) = Z/2 for any k ≥ 2
Sadly, I do not understand why this is true. How do you wrap a 4-sphere around a 2-sphere in an interesting way? Dunno.
(Thanks to Dan Christensen, an answer appears at the end of this post.)
Plunging on undeterred, we have:
π4(S1) = {0}
π5(S2) = Z/2
π6(S3) = Z/12
π7(S4) = Z + Z/12
π8(S5) = Z/24
π9(S6) = Z/24
and so on:
πk+3(Sk) = Z/24 for any k ≥ 5.
Here is where the magic number 24 comes in! What the above says is that if k is large enough, there are exactly 24 different homotopy class of maps from an (k+3)-sphere to an k-sphere!
Now I should explain what this has to do with string theory. But first I should say more about the homotopy groups of spheres. There are some simple patterns worth knowing about. First,
πm(S1) = {0} for any m ≥ 2.
Second, there is a nice formula for when the homotopy groups settle down as we jack up the dimension:
πk+n(Sk) is independent of k as long as k ≥ n+2
The homotopy groups can stabilize sooner, as we saw for n = 2, but never later, and often they stabilize right at k = n+2. There is a simple reason for this. We saw that πk+1(Sk) stabilized at k = 3 because it's easy to unlink links in 4 or more dimensions. Similarly, πk+n(Sk) must stabilize by the time k = n+2, because it's easy to untie knotted n-dimensional surfaces in 2n+2 or more dimensions!
For more on stable homotopy groups of spheres, try:
2) Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, Orlando, 1986.
Douglas C. Ravenel, Nilpotence and periodicity in stable homotopy theory, Princeton University Press, Princeton, 1992.
Ravenel also spoke at this conference and is a real expert on stable homotopy groups of spheres. Unfortunately his talk was too high-powered for me. The 2nd book above is a bit more forgiving to the amateur, but the first one has lots of nice tables of stable homotopy groups of spheres.
The relationship between homotopy groups of spheres and higher- dimensional knot theory is a wonderful thing. James Dolan and I are learning a lot about n-categories by pondering it. When I spoke to him at the conference at Northwestern, it became clear that Andre Joyal had also thought about it very deeply. Joyal is famous for his work relating category theory, combinatorics and topology, and his way of thinking about the homotopy groups of spheres reflects these interests. He said a very fascinating thing; he said "really we should call the sphere spectrum the 'true integers'". I would like to explain this... but here things get a bit technical, and I am afraid they will get a lot more technical when I get around to the string theory stuff.
What's the "sphere spectrum"? Well, roughly it's just the list of spheres S0, S1, S2, ..., but the word "spectrum" refers to the way all these spaces are all related, all aspects of one big thing.
Here's a nice way to think of it. Start with the integers. Normally we think of these as just a set, or actually a group, since we can add them. But if we avoid the sin of mistaking isomorphism for equality we can think of them as a category.
I already began to explain this in my parable about the shepherd in "week99". The shepherd started with the category of finite sets and "decategorified" it to obtain the set of natural numbers, associating to each finite set a natural number, its number of elements. Taking disjoint unions of sets corresponds to addition, the empty set corresponds to zero, and so on.
Okay. What are the integers the decategorification of?
Well, we can imagine finite sets that have both "positive" and "negative" elements. The "number of elements" of such a set will be the number of positive elements minus the number of negative elements. This is a bit weird if we're talking about sheep, but perhaps not so weird if we talk about positrons and electrons, which can annihilate each other. (In "week92" I explain what I'm hinting at here: the relation between antiparticles and adjunctions.)
Topologists prefer to speak of "positively and negatively oriented points". We can draw a set of positively and negatively oriented points like this:
- + + + + - -
We can add them by setting them side by side. But how do the positively
and negatively oriented points cancel? Well, remember, we're trying to
get a category! If finite lists of positively and negatively oriented
points are our objects, what are our morphisms? How about tangles, like this:
- + + + + - -
\ / | | \ / |
\ / | | \ / |
\/ | \ \/ /
| \ /
| \ /
| \ /
| \ /
/\ | \ /
/ \ | \ /
/ \ | \/
/ \ |
+ - +
These let us cancel or create positive and negative points in pairs.
Voila! The categorified integers! Just as the integers form a monoid
under addition, these form a monoidal category (see "week89" for these
concepts). The monoidal structure here is disjoint union, which we can
denote with a plus sign if we like. Similarly, we can write the empty
set as 0.
Above it looks like I'm drawing a 1-dimensional tangle in 2-dimensional space. To understand the "commutativity" of the categorified integers we should work with 1-dimensional tangles in higher-dimensional space. If we consider them in 3-dimensional space, we have room to switch things around:
+ +
\ /
\ /
\ /
/
/ \
/ \
/ \
+ +
This gets us commutativity, as I explained in "week100". Technically
speaking, we get a "braided" monoidal category. However, there are two
different ways to switch things around; for example, in addition to the
above way there is
+ +
\ /
\ /
\ /
\
/ \
/ \
/ \
+ +
To get rid of this problem (if you consider it a problem) we can work
with 1-dimensional tangles in 4-dimensional space, where we can deform
the first way of switching things to the second. We get a "symmetric"
monoidal category. Working in higher dimensions doesn't change
anything: things have stabilized.
If we impose the extra condition that the morphisms
/\
/ \
/ \
/ \
+ -
and
+ -
\ /
\ /
\ /
\/
are inverses, as are
/\
/ \
/ \
/ \
- +
and
- +
\ /
\ /
\ /
\/
then all morphisms become invertible, so we have not just a monoidal
category but a monoidal groupoid - a groupoid being a category with all
morphisms invertible (see "week74"). In fact, not only are morphisms
invertible, so are all objects! By this I mean that every object x
has an object -x such that x + -x and -x + x are isomorphic to 0.
For example, if x is the positively oriented point, -x is the negatively
oriented point, and vice versa. So we have not just a monoidal groupoid
but a "groupal groupoid". (I have adopted this charming terminology
from James Dolan.)
Very nice. We seem to have avoided the sin of decategorification, and are no longer treating the integers as a mere set (or group, or commutative group). We are treating them as a category (or groupal groupoid, or braided groupal groupoid, or symmetric groupal groupoid).
On the other hand, it's a bit odd to say that
/\
/ \
/ \
/ \
+ -
and
+ -
\ /
\ /
\ /
\/
are inverses. This amounts to saying that the morphism:
/\
/ \
/ \
/ \
\ /
+\ /-
\ /
\/
is equal to the identity morphism from 0 to 0, which corresponds to the
empty picture:
Hmm. They sure don't look equal. We must be doing something wrong.
What are we doing wrong? We're committing the sin of decategorification: treating isomorphisms as equations! We should treat the integers not as a mere category, but as a 2-category! See "week80" for the precise definition of this concept; for now, it's enough to say that a 2-category has things called 2-morphisms going between morphisms. If we treat the integers as a 2-category, we can say there is a 2-morphism going from
/\
/ \
/ \
/ \
\ /
+\ /-
\ /
\/
to the identity morphism. This 2-morphism has a nice geometrical
description in terms of a 2-dimensional surface: the surface in
3d space that's traced out as the above picture shrinks down to the
empty picture. It's hard to draw, but let me try:
/\
/ \ /\
/ \ / \ /\
/ \ => / \ => / \ => /\ =>
\ / \ / \ / +\/-
+\ /- +\ /- +\/-
\ / \/
\/
Okay, say we do this: treat the integers as a 2-category. We again are
faced with a question: do we make all the 2-morphisms invertible? If we
do, we get a "2-groupoid", or actually a "groupal 2-groupoid". But
again, to do so amounts to committing the sin of decategorification. To
avoid this sin, we should tread the integers as a 3-category. Etc, etc!
You may have noted how worlds of ever higher-dimensional topology are automatically unfolding from our attempt to avoid the sin of decategorification. This is what's so neat about n-categories. I haven't told you how it all works, but let me summarize what's actually happening here. Normally we treat the integers as the free group on one generator, or else the free commutative group on one generator. But groups and commutative groups are just the tip of the iceberg! The following picture is similar to that in "week74":
k-tuply groupal n-groupoids
n = 0 n = 1 n = 2
k = 0 sets groupoids 2-groupoids
k = 1 groups groupal groupal
groupoids 2-groupoids
k = 2 commutative braided braided
groups groupal groupal
groupoids 2-groupoids
k = 3 `' symmetric weakly
groupal involutory
groupoids groupal
2-groupoids
k = 4 `' `' strongly
involutory
groupal
2-groupoids
k = 5 `' `' `'
What are all these things? Well, an n-groupoid is an n-category with
all morphisms invertible, at least up to equivalence. An (k+n)-groupoid
with only trivial j-morphisms for j < k can be seen as a special sort of
n-groupoid, which we call a "k-tuply groupal n-groupoid".
Part of Joyal's point was that we should really think of the integers as the "free k-tuply monoidal n-groupoid on one object". Here the idea is not to fix n and k once and for all - this would only prevent us from understanding the subtleties that show up when we increase n and k! Instead, we should think of them as variable, or perhaps consider the limit as they become large.
The other part of his point was that there's a correspondence between n-groupoids and the information left in topological spaces when we ignore all homotopy groups above dimension n - so-called "homotopy n-types". Using this correspondence, the "free k-tuply monoidal n-groupoid on one object" corresponds to the homotopy (k+n)-type of the k-sphere. Moreover, if we keep jacking up k, this stabilizes when k ≥ n+2. Actually, as the dittos in the above chart suggest, it's a quite general fact that the notion of k-tuply monoidal n-groupoid stabilizes for k ≥ n+2.
Yet another point is that the pictures above explain the relation between higher-dimensional knot theory and the homotopy groups of spheres in a very vivid, direct way.
Okay. What about string theory and the magic number 24? Well, notice that the pictures above started looking a bit like strings! Hmm....
Here's the idea, as far as I understand it. Presumably all but the hardy have stopped reading this article by now, so I will pull out all the stops. The string worldsheet is a Riemann surface so we'll need some stuff about Riemann surfaces from "week28". Let M(g,n) be the moduli space of Riemann surfaces with genus g and n punctures, and let G(g,n) be the corresponding mapping class group. Since M(g,n) is the quotient of Teichmueller space by G(g,n) and Teichmueller space is contractible, we have
M(g,n) = BG(g,n)
where "B" means "classifying space". There's a natural inclusion
G(g,n) → G(g+1,n)
defined by sewing a torus with two punctures onto your genus-g surface with n punctures, which increases the genus by 1. Let's define G(∞,n) to be direct limit as g → ∞, and let
M(∞,n) = BG(∞,n).
Now it turns out M(∞,1) has a kind of product on it. The reason is that there are products
M(g,1) x M(h,1) → M(g+h,1)
given by sewing two surfaces together with a 3-punctured sphere. Using this product we can define the group completion
M(∞,1)+
and the result Tillmann stated which got me so excited was that
π3(M(∞,1)+) = Z/24 + H
for some unknown group H. Since this is really a result about the mapping class groups of surfaces, it must have something to do with how conformal field theories always give projective representations of these mapping class groups, with the "phase ambiguity" being of the form exp(2 π c i / 24), where c is the "central charge". No? I just don't quite see why. Maybe someone will enlighten me.
Anyway, the way she proved this definitely ties right into the stuff about stable homotopy groups of spheres. She used explicit maps between the third stable homotopy group of spheres
πk+3(Sk) = Z/24 for k >= 5
and π3(M(∞,1)+)! And the way she got the map from the latter to the former amounts to working with pictures I was drawing above. To put it more precisely, in
3) Higher-dimensional algebra and topological quantum field theory, by John Baez and James Dolan, Jour. Math. Phys. 36 (1995), 6073-6105.
we argue that framed n-dimensional surfaces embedded in (n+k)-dimensions should be described by the free k-tuply monoidal n-category with duals on one object. This should map down to the free k-tuply groupal n-groupoid on one object, by the usual yoga of "freeness". Taking n = 3 and k sufficiently large, we should obtain a homomorphism from the mapping class group of any Riemann surface to the third stable homotopy group of spheres! Presumably the idea is that in the limit of large genus this homomorphism is onto!
Of course, Tillmann doesn't prove her result using the still-nascent formalism of n-categories, but I think it will eventually be possible. (Also, my rough argument applies to Riemann surfaces with no punctures, while she considers those with one puncture, but various things she said make me think this might not be such a big deal.) The real puzzle is this: what does
π3(M(∞,n)+)
have to do with central extensions of G(g,n) for finite g? If I could figure this out I'd be very happy.
Addendum: Dan Christensen answered a puzzle above. Here's how to get a nontrivial element of
π4(S2).
Take the map f: S3 → S2 generating
π3(S2)
and compose it with the map g: S4 → S3 generating
π4(S3)
(which, by the way, is obtained from f by "suspension") to obtain the desired map from S4 to S2. This is an instance of a very general trick: composing elements of homotopy groups of spheres to get new ones!
Think of one and minus one. Together they add up to zero, nothing, nada, niente, right? Picture them together, then picture them separating, peeling part.... Now you have something, you have two somethings, where you once had nothing. - John Updike, Roger's Version
/\
/ \
/ \
/ \
+ -
© 1997 John Baez
baez@math.removethis.ucr.andthis.edu
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