Recently I've been talking a bit about elliptic cohomology, but I've really just been nibbling around the edges so far. Sometime I want to dig deeper, but not just now. Right now, I instead want to say a bit more about the physics lurking in the space K(Z,2).
But first, here's a cool article on violins:
1) Colin Gough, Science and the Stradivarius, Physics World, vol. 13 no. 4, April 2000, 27-33.
Before reading this, I never knew how a string on a violin vibrates! Lots of well-known European physicists have studied the violin, and in the 19th century, Helmholtz showed that the bow excites a mode of the violin string that is quite unlike the sine waves we all know and love. In this "Helmholtz waveform", the string consists of two straight-line segments separated by a kink:
. . . . . . .As time passes, the kink travels back and forth along the string, being reflected at the ends. The beauty of this becomes apparent as we watch the string right at the point where the bow is rubbing over it, near the bottom end of the string. When the kink is between the bow and the top end of the string:
bow | ^ . | TOP . .| BOTTOM . | . . | . | |this point in the string moves at the same speed and in the same direction as the bow. This is called the "sticking regime", because the static friction of the rosin-coated bow is enough to pull the string along with it. But when the kink moves past the bow:
| ^ | . . | TOP . | . BOTTOM . | . | . |the string slips off the bow and starts moving in the opposite direction to it. This is called the "sliding regime". Since the coefficient of sliding friction is less than the coefficient of static friction, the string can slide against the motion of the bow in this regime.
The really nice thing is that the string is vibrating almost freely: the violinist just needs to apply the right amount of pressure to keep this vibrational mode excited - too much pressure will ruin it! Being able to delicately control the Helmholtz waveform is part of what distinguishes the virtuoso from the blood-curdling amateur.
The full physics of the violin is infinitely more complicated than this, of course. The vibrating string excites the bridge which excites the sound box, and that produces most of the sound we hear. For more information try these:
2) A. H. Benade, Fundamentals of Musical Acoustics, Oxford University Press, Oxford, 1976.
L. Cremer, The Physics of the Violin, MIT Press, Cambridge, Massachusetts, 1984.
N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments, 2nd edition, Springer, New York, 1998.
C. Hutchins and V. Benade, editors, Research Papers on Violin Acoustics 1975-1993, 2 volumes, Acoustical Society of America, New York, 1997.
Okay, now on to K(Z,2)! I explained a bit about this space in "week149", but I've been pondering it a lot lately, so I'd like to say a bit more.
First let me review and elaborate on some basic stuff I said already. If G is any topological group, there is a topological space BG with a basepoint such that the space of loops in BG starting and ending at this point is homotopy equivalent to G. This space BG is unique up to homotopy equivalence. 
BG is important because it's the "classifying space for G-bundles". What this means is that there's a principal G-bundle over BG called the "universal G-bundle", with the marvelous property that any principal G-bundle over any space X is a pullback of this one by some map
f: X → BG.
(I explained in "week149" how to pull back complex line bundles, and pulling back principal G-bundles works the same way.) Even better, two G-bundles that we get this way are isomorphic if and only if the maps they come from are homotopic! So there is a one-to-one correspondence between:
A) isomorphism classes of principal G-bundles over X
B) homotopy classes of maps from X to BG.
Now, suppose G is an abelian topological group. Then BG is better than a topological space with basepoint. It's an abelian topological group!
This means that we can iterate this trick. Starting with an abelian topological group G we can form BG, and BBG, and BBBG, and so on. This is called "delooping", because the loop space of each of these spaces is the previous one.
It's always fun to iterate any process whenever you can - Freud called this "repetition compulsion" - but there's more going on here than just that. In "week149" I said that when we have a list of spaces, each being the loop space of the previous one, it's called a "spectrum". And I said that we can use a spectrum to get a generalized cohomology theory. So we now have a trick for getting a generalized cohomology theory from a topological abelian group!
In particular, suppose we start with a plain old abelian group A. We can think of it as a topological group with the discrete topology - let's call this K(A,0). Then we can define
K(A,1) = B(K(A,0))
K(A,2) = B(K(A,1))
K(A,3) = B(K(A,2))
... and so on. We get a spectrum K(A,n) called an "Eilenberg-MacLane spectrum". The corresponding generalized cohomology theory is just ordinary cohomology with coeffients in the abelian group A! This means that
Hn(X,A) = [X, K(A,n)]
where the right-hand side is the set of homotopy classes of maps from X to K(A,n). In short, K(A,n) knows everything there is to know about the nth cohomology with coefficients in A.
We've seen this trick a couple of times lately, and it's actually a big theme in homotopy theory: whenever we have some interesting invariant of spaces, we try to cook up a space that "represents" this invariant. I could say a LOT more about THIS idea, but that would propel us into further heights of abstraction, when what I really want is to come down to earth a bit. Just a little bit....
So: let's take A to be the integers, Z. As I said in "week149", we then get
K(Z,0) = Z,
K(Z,1) = U(1),
where U(1) is the group of "phases" or unit complex numbers, and
K(Z,2) = CP∞
where CP∞ is infinite-dimensional complex projective space. There are a couple of slightly different versions of this. Topologists like to start with the direct limit of the spaces Cn, which they call C∞. Then they take the space of all 1-dimensional subspaces and call that CP∞. Mathematical physicists prefer to start with a Hilbert space of countable dimension. Then they take the space of unit vectors modulo phase. Both these versions are equally good models of K(Z,2). The first one is a lean, stripped-down version of the second.
Now U(1) is very important in quantum theory, and so are unit vectors modulo phase in a Hilbert space - physicists call these "pure states". So something cool is going on here. For some mysterious reason, it looks like K(Z,n)'s are important quantum physics! This is especially interesting because the abstract definition of the K(Z,n)'s has nothing to do with the complex numbers - just the integers. The complex numbers show up on their own accord. So maybe this hints at some explanation of why the complex numbers are important in quantum mechanics.
Why are K(Z,n)'s connected to quantum theory? I don't really know. But we can get some clues by asking some more specific questions.
First of all, why is K(Z,2) the same as CP∞? In "week149" I just asserted this without proof. That's one of the fun things I'm allowed to do in this column. But let me sketch why it's true.
First I need to remind you of some more basic facts about topology. Suppose G is any topological group, and let P → X be any principal G-bundle. This gives us a long exact sequence of homotopy groups:
... → πn+1(X) → πn(G) → πn(P) → πn(X) → πn-1(G) → ...
Two-thirds of the arrows in this sequence come from the maps
G → P → X
while the less obvious remaining one-third come from the map
LX → G
sending each loop in the base space to the holonomy of some connection on our bundle. Here LX means the space of based loops in X, and we're using the fact that
πn(LX) = πn+1(X)
which is obvious from the definition of the homotopy groups.
But now suppose P is contractible! Then all its homotopy groups vanish, so the above long exact sequence breaks up into lots of puny exact sequences like this:
0 → πn+1(X) → πn(G) → 0
or in other words:
0 → πn(LX) → πn(G) → 0
This says that the map from LX to G induces isomorphisms on all homotopy groups. By the Whitehead theorem, this implies that this map is a homotopy equivalence! So LX is really just G!! So X is just BG!!!
In short: if we have a space X with a principal G-bundle P over it, and P is contractible, X must be BG. 
Now let's use this fact to show that CP∞ is K(Z,2). Remember that by our recursive definition,
K(Z,2) = B(K(Z,1)) = B(U(1))
so to show that CP∞ is K(Z,2), we just need to find a principal U(1)-bundle over it with a contractible total space.
In "week149" we discussed a complex line bundle over CP∞ called the "universal complex line bundle". If you take the space of unit vectors in a complex line bundle you get a principal U(1)-bundle. So let's do this to the universal complex line bundle. What do we get? We get a principal U(1)-bundle like this:
S∞ → CP∞
Being a mathematical physicist, I'm using S∞ here to stand for the unit sphere in some countable-dimensional Hilbert space, and the map sends each unit vector to the corresponding pure state, or unit vector mod phase. Since there's a circle of unit vectors for each pure state, this is indeed a principal U(1)-bundle. But now for the cool part: the unit sphere in an infinite-dimensional Hilbert space is contractible! So we've got a principal U(1)-bundle with a contractible total space sitting over CP∞, proving that CP∞ is K(Z,2). Even better, the bundle
S∞ → CP∞
is the universal principal U(1)-bundle.
I can't resist explaining why the unit sphere in an infinite-dimensional Hilbert space is contractible. It seems very odd that a sphere could be contractible, but this is one of those funny things about infinite dimensions. Take our Hilbert space to be L2[0,1] and consider any function f in the unit sphere of this Hilbert space:
∫ |f(x)|2 dx = 1
For t between 0 and 1, let ft(x) be a function that equals 1 for x < t, and a sped-up version of f for x greater than or equal to t. If you do this right ft will still lie in the unit sphere, and you'll have a way of contracting the whole unit sphere down to a single point, namely the constant function 1.
Next question: how does CP∞ become an abelian topological group? There's a very pretty answer. Consider the space of rational functions of a single complex variable. This is a infinite-dimensional complex vector space, and there's a natural way to give it the topology of C∞. This gives us a nice way to think of CP∞: it's just the nonzero rational functions modulo multiplication by constants.
But nonzero rational functions form an abelian group under multiplication! And this is still true when we mod out by constant factors! So CP∞ becomes an abelian group - and in fact an abelian topological group.
We can visualize CP∞ quite easily this way. A rational function of a single complex variable has a bunch of zeros and poles - think of them as points on the Riemann sphere. We should really stick an integer at each of these points: a positive integer at each zero, and a negative integer at each pole, to tell us the order of that zero or pole. This gives enough information to completely specify the rational function up to a constant factor. So a point in CP∞ is the same as a finite set of points on the sphere labelled by integers - which must add up to zero.
Of course, we have to get the right topology on CP∞. As we move our point in CP∞ around in a continuous way, the corresponding points on the sphere all move around continuously, like a swarm of flies... but when points collide, their numbers add! For example, when a point labelled by the number 7 collides with a point labelled by the number -3, it turns into a point labelled by the number 7 - 3 = 4.
In the lingo of physics, we've got a picture of points in CP∞ as "collections of particles and antiparticles on the sphere". The integer at any point on the sphere tells us the number of particles sitting there - but if it's negative, it means we've got antiparticles there. Particle-antiparticle pairs can be created out of nothing, and they annihilate when they collide... it's very nice!
By the way, there's something called the Thom-Dold theorem that lets us generalize the heck out of this. We just showed that if you take the 2-sphere and consider the space of particle-antiparticle swarms in it, you get K(Z,2). But suppose instead we started with the n-sphere and considered the space of particle-antiparticle swarms in that. Then we'd get K(Z,n)!
More generally, suppose we didn't use integers to say how many particles were at each point in the n-sphere - suppose we used elements of some abelian group A. Then we'd get K(A,n)!
For more tricks like this, try this paper:
3) Dusa McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91-107.
Now let me mention a different picture of K(Z,2), that's also nice, and also related to quantum theory. Take any countable-dimensional Hilbert space H and let U(H) be the group of unitary operators on H. Just like the unit sphere in this Hilbert space is contractible, it turns out that U(H) is contractible if we give it the norm topology or the strong topology.
Anyway, now let PU(H) be the "projective unitary group" of H, meaning the group of unitary operators modulo phase. There's an obvious map
U(H) → PU(H)
sending a circle's worth of points to each point in PU(H). It's easy to check that this is a principal U(1)-bundle. Since the total space U(H) is contractible, it follows that PU(H) is K(Z,2)!
This give a nonabelian group structure on K(Z,2), which may seem kind of weird, given that we just made it into an abelian group a minute ago. But I guess this other product is "abelian up to homotopy" in a very strong sense, so it's just as good as abelian for the purposes of homotopy theory.
Anyway, some people in Australia have figured out an extra trick you can do with this PU(H) group:
4) Alan L. Carey, Diarmuid Crowley and Michael K. Murray, Principal bundles and the Dixmier-Douady class, Comm. Math. Physics 193 (1998) 171-196, preprint available as hep-th/9702147.
Here's how it goes, at least in part. We say a linear operator
A: H → H
is "Hilbert-Schmidt" if the trace of AA* is finite. The space of Hilbert-Schmidt operators is a Hilbert space in its own right, with this inner product:
<A,B> = tr(AB*)
Let's call this Hilbert space X. U(H) acts on X by conjugation, and this gives an action of PU(H) on X, because phases commute with everything. This in turn gives an action of PU(H) on U(X)! Is your brain melting yet? Anyway, it turns out that this makes U(X) into the total space of a principal PU(H)-bundle:
PU(H) → U(X) → U(X)/PU(H)
But X is a countable-dimensional Hilbert space, so U(X) is contractible, so this is the universal principal PU(H)-bundle. And as we've seen, this means that
U(X)/PU(H) = B(PU(H))
but we just saw that
PU(H) = K(Z,2)
U(X)/PU(H) = B(PU(H)) = B(K(Z,2)) = K(Z,3) !
In "week149", I said I'd like K(Z,3) to be some sort of infinite- dimensional manifold closely related to quantum physics. I'm happier now, because here we are getting just that - technically, we're getting it to be a "Banach manifold". Of course, I could still complain that this description doesn't make the abelian group structure on K(Z,3) obvious. But it's definitely a big step towards understanding what K(Z,n)'s have to do with quantum theory.
While I'm at it, I should report some other things people have told me via email. If you ponder what I've said, you can see that CP∞ has 2nd homology equal to Z, and that the generator of this homology group - the "universal cycle" - is given geometrically by the obvious way of sticking the sphere CP1 inside CP∞. This is nice because CP1 is actually a submanifold of the manifold CP∞. But according to email from Mark Goresky, Rene Thom has shown that for n > 6, we cannot make K(Z,n) into a manifold in such a way that the universal cycle is represented by a submanifold!
On the other hand, Michael Murray reports that Pawel Gajer has managed to make K(Z,n) into something called a "differential space", which is not quite a manifold, but good enough to do geometry on. I'm not sure how this relates to Thom's work... but anyway, I should read this stuff:
5) Pawel Gajer, Geometry of Deligne cohomology, Invent. Math. 127 (1997), 155-207, also available as alg-geom/9601025.
Pawel Gajer, Higher holonomies, geometric loop groups and smooth Deligne cohomology, Advances in Geometry, Birkhauser, Boston, 1999, pp. 195-235.
Now, so far I've been restraining myself from talking about "gerbes", but if you've gotten this far you must be pretty comfortable with abstract nonsense, so you'll probably like gerbes. Very roughly speaking, a gerbe is a categorified version of a principal bundle! Actually it's a categorified version of a sheaf, but sometimes we can think of it as analogous to the sheaf of sections of a bundle. And just as K(Z,2) is the classifying space for U(1) bundles, K(Z,3) is the classifying space for a certain sort of gerbe!
I sort of explained how this works in "week25", but you can read the details here:
6) Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhauser, Boston, 1993.
What this means is that as we explore the meaning of these K(Z,n)'s for quantum theory, we are really categorifying familiar ideas from quantum theory. In particular, this story should keep going on forever: K(Z,4) should be the classifying space for a certain sort of 2-gerbe, and so on. But I don't think people have worked out the details beyond the case of 2-gerbes. If you want to learn about 2-gerbes, you have to read this:
7) Lawrence Breen, On the Classification of 2-Gerbes and 2-Stacks, Asterisque 225, 1994.
Finally, for more applications to physics, try these papers:
8) Alan L. Carey and Michael K. Murray, Faddeev's anomaly and bundle gerbes, Lett. Math. Phys. 37 (1996), 29-36.
Jouko Mickelsson, Gerbes and Hamiltonian quantization of chiral fermions, Lie Theory and Its Applications in Physics, World Scientific, Singapore, 1996, pp. 216-225.
Michael K. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996), 403-416.
Alan L. Carey, Jouko Mickelsson and Michael K. Murray, Index theory, gerbes, and Hamiltonian quantization, Comm. Math. Phys. 183 (1997), 707-722, preprint available as hep-th/9511151.
Alan L. Carey, Michael K. Murray and B. L. Wang, Higher bundle gerbes and cohomology classes in gauge theories, J. Geom. Phys. 21 (1997) 183-197, preprint available as hep-th/9511169.
Alan L. Carey, Jouko Mickelsson and Michael K. Murray, Bundle gerbes applied to quantum field theory, Rev. Math. Phys. 12 (2000), 65-90, preprint available as hep-th/9711133.
I thank N. Christopher Phillips of the University of Oregon, Michael K. Murray and Diarmuid Crowley of the University of Adelaide, and Mark Goresky of IHES for educating me about these matters... all remaining errors are mine!
 I'm being sloppy here. Throughout this discussion, when I say "homotopy equivalent", I really mean "weakly homotopy equivalent" - a technical nuance that you can read about in any good book on homotopy theory.
 Moreover, P must be the universal principal G-bundle. Conversely,
for any topological group G the total space of the universal principal
G-bundle is contractible. Everything fits together very neatly! But
I don't need all this stuff now.
© 2000 John Baez