Like many people of a certain age, as a youth my interest in mathematics and physics was fed by the Scientific American, especially Martin Gardner's wonderful column. Since then the magazine seems to have gone downhill. For me, the last straw was a silly article on the "death of proof" in mathematics, written by someone wholly unfamiliar with the subject. The author of that article later wrote a book proclaiming the "end of science", and went on to manage a successful chain of funeral homes.
Recently, however, I was pleased to find a terse rebuttal of this fin-de-siecle pessimism in an article appearing in - none other than Scientific American!
1) Michael J. Duff, The theory formerly known as strings, Scientific American 278 (February 1998), 64-69.
The article begins:
At a time when certain pundits are predicting the End of Science on the grounds that all the important discoveries have already been made, it is worth emphasizing that the two main pillars of 20th-century physics, quantum mechanics and Einstein's general theory of relativity, are mutually incompatible.
To declare the end of science at this point, or even of particle physics (the two are not the same!) would thus be ridiculously premature. It's true that the quest for a unified theory of all the forces and particles in nature is experiencing difficulties. On the one hand, particle accelerators have become very expensive. On the other hand, it's truly difficult to envision a consistent and elegant formalism subsuming both general relativity and the Standard Model of particle physics - much less one that makes new testable predictions. But hey, the course of true love never did run smooth.
Duff's own vision certainly has its charms. He has long been advocating the generalization of string theory to a theory of higher-dimensional "membranes". Nowadays people call these "p-branes" to keep track of the dimension of the membrane: a 0-brane is a point particle, a 1-brane is a string, a 2-brane is a 2-dimensional surface, and so on.
For a long time, higher-dimensional membrane theories were unpopular, even among string theorists, because the special tricks that eliminate infinities in string theory don't seem to work for higher-dimensional membranes. But lately membranes are all the rage: it seems they show up in string theory whether or not you put them in from the start! In fact, they seem to be the key to showing that the 5 different supersymmetric string theories are really aspects of a single deeper theory - sometimes called "M-theory".
Now, I don't really understand this stuff at all, but I've been trying to learn about it lately, so I'll say a bit anyway, in hopes that some real experts will correct my mistakes. Much of what I'll say comes from the following nice review article:
2) M. J. Duff, Supermembranes, preprint available as hep-th/9611203
and also the bible of string theory:
3) Michael B. Green, John H. Schwarz, and Edward Witten, Superstring Theory, two volumes, Cambridge U. Press, Cambridge, 1987.
Let's start with superstring theory. Here the "super" refers to the fact that instead of just strings whose vibrational modes correspond to bosonic particles, we have strings with extra degrees of freedom corresponding to fermionic particles. We can actually think of the superstring as a string wiggling around in a "superspace": a kind of space with extra "fermionic" dimensions in addition to the usual "bosonic" ones. These extra dimensions are described by coordinates that anticommute with each other, and commute with the usual bosonic coordinates (which of course commute with each other). This amounts to taking the boson/fermion distinction so seriously that we build it into our description of spacetime from the very start! For more details on the mathematics of superspace, try:
4) Bryce DeWitt, Supermanifolds, Cambridge U. Press, Cambridge, 2nd edition, 1992.
More deeply, "super" refers to "supersymmetry", a special kind of symmetry transformation that mixes the bosonic and fermionic coordinates. We speak of "N = 1 supersymmetry" if there is one fermionic coordinate for each bosonic coordinate, "N = 2 supersymmetry" if there are two, and so on.
Like all nice physical theories, we can in principle derive everything about our theory of superstrings once we know the formula for the action. For bosonic strings, the action is very simple. As time passes, a string traces out a 2-dimensional surface in spacetime called the "string worldsheet". The action is just the area of this worldsheet.
For superstring theory, we thus want a formula for the "super-area" of a surface in superspace. And we need this to be invariant under supersymmetry transformations. Suprisingly, this is only possible if spacetime has dimension 3, 4, 6, or 10. More precisely, these are the dimensions where N = 1 supersymmetric string theory makes sense as a classical theory.
Note: these dimensions are just 2 more than the dimensions of the four normed division algebras: the reals, complexes, quaternions and octonions! This is no coincidence. Robert Helling recently posted a nice article about this on sci.physics.resarch, which I have appended to "week104". The basic idea is that we can describe the vibrations of a string in n-dimensional spacetime by a field on the string worldsheet with n-2 components corresponding to the n-2 directions transverse to the worldsheet. To get an action that's invariant under supersymmetry, we need some magical cancellations to occur. It only works when we can think of this field as taking values in one of the normed division algebras!
This is one of the curious things about superstring theory: the basic idea is simple, but when you try to get it to work, you run into lots of obstacles which only disappear in certain special circumstances - thanks to a mysterious conspiracy of beautiful mathematical facts. These "conspiracies" are probably just indications that we don't understand the theory as deeply as we should. Right now I'm most interested in the algebraic aspects of superstring theory - and especially its relationships to "exceptional algebraic structures" like the octonions, the Lie group E8, and so on. As I learn superstring theory, I like keeping track of the various ways these structures show up, like remembering the clues in a mystery novel.
Interestingly, the quantum version of superstring theory is more delicate than the classical version. When I last checked, it only makes sense in dimension 10. Thus there's something inherently octonionic about it! For more on this angle, see:
5) E. Corrigan and T. J. Hollowood, The exceptional Jordan algebra and the superstring, Commun. Math. Phys., 122 (1989), 393.
6) E. Corrigan and T. J. Hollowood, A string construction of a commutative nonassociative algebra related to the exceptional Jordan algebra, Phys. Lett. B203 (1988), 47.
and some more references I'll give later.
There are actually 5 variants of superstring theory in dimension 10, as I explained in "week72":
To get 4-dimensional physics out of any of these, we need to think of our 10-dimensional spacetime as a bundle with a little 6-dimensional "Calabi-Yau manifold" sitting over each point of good old 4-dimensional spacetime. But there's another step that's very useful when trying to understand the implications of superstring theory for ordinary particle physics. This is to look at the low-energy limit. In this limit, only the lowest-energy vibrational modes of the string contribute, each mode acting like a different kind of massless particle. Thus in this limit superstring theory acts like an ordinary quantum field theory.
What field theory do we get? This is a very important question. The field theory looks simplest in 10-dimensional Minkowski spacetime; it gets more complicated when we curl up 6 of the dimensions and think of it as a 4-dimensional field theory, so let's just talk about the simple situation.
No matter what superstring theory we start with, the low-energy limit looks like some form of "supergravity coupled to super-Yang-Mills fields". What's this? Well, supergravity is basically what we get when we generalize Einstein's equations for general relativity to superspace. Similarly, super-Yang-Mills theory is the supersymmetric version of the Yang-Mills equations - which are important in particle physics because they describe all the forces except gravity. So superstring theory has in it the seeds of general relativity and also the other forces of nature --- or at least their supersymmetric analogues.
Like superstring theory, super-Yang-Mills theory only works in spacetime dimensions 3, 4, 6, and 10. (See "week93" for more on this.) Different forms of supergravity make sense in different dimensions, as explained in:
7) Y. Tanii, Introduction to supergravities in diverse dimensions, preprint available as hep-th/9802138.
In particular highest dimension in which supergravity makes sense is 11 dimensions (where one only has N = 1 supergravity). Note that this is one more than the favored dimension of superstring theory! This puzzled people for a long time. Now it seems that M-theory is beginning to resolve these puzzles. Another interesting discovery is that 11-dimensional supergravity is related to the exceptional Lie group E8:
8) Stephan Melosch and Hermann Nicolai, New canonical variables for d = 11 supergravity, preprint available as hep-th/9709277.
But I'm getting ahead of myself here! Right now I'm talking about the low-energy limit of 10-dimensional superstring theory. I said it amounts to "supergravity coupled to super-Yang-Mills fields", and now I'm attempting to flesh that out a bit. So: starting from N = 1 supergravity in 11 dimensions we can get a theory of supergravity in 10 dimensions simply by requiring that all the fields be constant in one direction - a trick called "dimensional reduction". This is called "type IIA supergravity", because it appears in the low-energy limit of type IIA superstrings. There are also two other forms of supergravity in 10 dimensions: "type IIB supergravity", which appears in the low-energy limit of type IIB superstrings, and a third form which appears in the low-energy limit of the type I and heterotic superstrings. These other two forms of supergravity are chiral - that is, they have a built-in "handedness".
Now let's turn to higher-dimensional supersymmetric membranes, or "supermembranes". Duff summarizes this subject in a chart he calls the "brane scan". This chart lists the known classical theories of supersymetric p-branes. Of course, a p-brane traces out a (p+1)-dimensional surface as time passes, so from a spacetime point of view it's p+1 which is more interesting. But anyway, here's Duff's chart of which supersymmetric p-brane theories are possible in which dimensions d of spacetime:
d 11 X X ? 10 X X X X X X X X X X 9 X X 8 X 7 X X 6 X X X X X X 5 X X 4 X X X X 3 X X X 2 X 1 0 1 2 3 4 5 6 7 8 9 10 pWe immediately notice some patterns. First, we see horizontal stripes in dimensions 3, 4, 6, and 10: all the conceivable p-brane theories exist in these dimensions. I don't know why this is true, but it must be related to the fact that superstring and super-Yang-Mills theories make sense in these dimensions. Second, there are four special p-brane theories:
the 2-brane in dimension 4
the 3-brane in dimension 6
the 5-brane in dimension 10
the 2-brane in dimension 11
which are related to the real numbers, the complex numbers, the quaternions and the octonions, respectively. Duff refers us to the following papers for more information on this:
9) G. Sierra, An application of the theories of Jordan algebras and Freudenthal triple systems to particles and strings, Class. Quant. Grav. 4 (1987), 227.
10) J. M. Evans, Supersymmetric Yang-Mills theories and division algebras, Nucl. Phys. B298 (1988), 92-108.
From these four "fundamental" theories of p-branes in d dimensions we can get theories of (p-k)-branes in d-k dimensions by dimensional reduction of both the spacetime and the p-brane. Thus we see diagonal lines slanting down and to the left starting from these "fundamental" theories. Note that these diagonal lines include the superstring theories in dimensions 3, 4, 6, and 10!
I'll wrap up by saying a bit about how M-theory, superstrings and supergravity fit together. I've already said that: 1) type IIA supergravity in 10 dimensions is the dimensional reduction of 11-dimensional supergravity; and 2) the type IIA superstring has type IIA supergravity coupled to super-Yang-Mills fields as a low-energy limit. This suggests the presence of a theory in 11 dimensions that fills in the question mark below:
low-energy limit ? ----------------------------> 11-dimensional | supergravity | | dimensional| |dimensional reduction| |reduction | | V low-energy limit V type IIA -------------------------> type IIA supergravity in superstrings 10 dimensionsThis conjectured theory is called "M-theory". The actual details of this theory are still rather mysterious, but not surprisingly, it's related to the theory of supersymmetric 2-branes in 11 dimensions - since upon dimensional reduction these give superstrings in 10 dimensions. More surprisingly, it's also related to the theory of 5-branes in 11 dimensions. The reason is that supergravity in 11 dimensions admits "soliton" solutions - solutions that hold their shape and don't disperse - which are shaped like 5-branes. These are now believed to be yet another shadow of M-theory.
While the picture I'm sketching may seem baroque, it's really just a small part of a much more elaborate picture that relates all 5 superstring theories to M-theory. But I think I'll stop here! Maybe later when I know more I can fill in some more details. By the way, I thank Dan Piponi for pointing out that Scientific American article.
For more on this business, check out the following review articles:
11) W. Lerche, Recent developments in string theory, preprint available as hep-th/9710246.
12) John Schwarz, The status of string theory, preprint available as hep-th/9711029.
13) M. J. Duff, M-theory (the theory formerly known as strings), preprint available as hep-th/9608117.
The first one is especially nice if you're interested in a nontechnical survey; the other two are more detailed.
Okay. Now, back to my tour of homotopy theory! I had promised to talk about classifying spaces of groups and monoids, but this post is getting pretty long, so I'll only talk about something else I promised: the foundations of homological algebra. So, remember:
As soon as we can squeeze a simplicial set out of something, we have all sorts of methods for studying it. We can turn the simplicial set into a space and then use all the methods of topology to study this space. Or we can turn it into a chain complex and apply homology theory. So it's very important to have tricks for turning all sorts of gadgets into simplicial sets: groups, rings, algebras, Lie algebras, you name it! And here's how....
N. Simplicial objects from adjunctions. Remember from section D of "week115" that a "simplicial object" in some category is a contravariant functor from Δ to that category. In what follows, I'll take Δ to be the version of the category of simplices that contains the empty simplex. Topologists don't usually do this, so what I'm calling a "simplicial object", they would call an "augmented simplicial object". Oh well.
Concretely, a simplicial object in a category amounts to a bunch of objects x0, x1, x2,... together with morphisms like this:
---i1-> ---i0-> ---i0-> x0 <--d0-- x1 <--d0-- x2 <--d0-- x3 ... <--d1-- <--d1-- <--d2--The morphisms dj are called "face maps" and the morphisms ij are called "degeneracies". They are required to satisfy some equations which I won't bother writing down here, since you can figure them out yourself if you read section B of "week114".
Now, suppose we have an adjunction, that is, a pair of adjoint functors:
---L--> C D <--R---This means we have natural transformations
e: LR => 1D
i: 1C => RL
satisfying a couple of equations, which I again won't write down, since I explained them in "week79" and "week83".
Then an object d in the category D automatically gives a simplicial object as follows:
--LRL.i.R-> --L.i.R-> --L.i.RLR-> d <--e-- LR(d) <--e.LR-- LRLR(d) <--e.LRLR-- LRLRLR(d) ... <--LR.e-- <--LR.e.LR- <--LRLR.e--where . denotes horizontal composition of functors and natural transformations.
For example, if Gp is the category of abelian groups, we have an adjunction
---L--> Set AbGp <--R---where L assigns to each set the free group on that set, and R assigns to each group its underlying set. Thus given a group, the above trick gives us a simplicial object in Gp - or in other words, a simplicial group. This has an underlying simplicial set, and from this we can cook up a chain complex as in section H of "week116". This lets us study groups using homology theory! One can define the homology (and cohomology) of lots other algebraic gadgets in exactly the same way.
Note: I didn't explain why the equations in the definition of adjoint functors - which I didn't write down - imply the equations in the definition of a simplicial object - which I also didn't write down!
The point is, there's a more conceptual approach to understanding why this stuff works. Remember from section K of last week that Δ is "the free monoidal category on a monoid object". This implies that whenever we have a monoid object in a monoidal category M, we get a monoidal functor
F: Δ → M.
This gives a functor
G: Δop → Mop
So: a monoid object in M gives a simplicial object in Mop.
Actually, if M is a monoidal category, Mop becomes ne too, with the same tensor product and unit object. So it's also true that a monoid object in Mop gives a simplicial object in M!
Another name for a monoid object in Mop is a "comonoid object in M". Remember, Mop is just like M but with all the arrows turned around. So if we've got a monoid object in Mop, it gives us a similar gadget in M, but with all the arrows turned around. More precisely, a comonoid object in M is an object, say m, with "coproduct"
c: m → m x m
e: m → 1
morphisms, satisfying "coassociativity" and the left and right "counit laws". You get these laws by taking associativity and the left/right unit laws, writing them out as commutative diagrams, and turning all the arrows around.
So: a comonoid object in a monoidal category M gives a simplicial object in M. Now let's see how this is related to adjoint functors. Suppose we have an adjunction, so we have some functors
---L--> C D <--R---and natural transformations
e: LR => 1D
i: 1C => RL
satisfying the same equations I didn't write before.
Let hom(C,C) be the category whose objects are functors from C to itself and whose morphisms are natural transformations between such functors. This is a monoidal category, since we can compose functors from C to itself. In "week92" I showed that hom(C,C) has a monoid object in it, namely RL. The product for this monoid object is
R.e.L: RLRL => RL
and the unit is
i: 1C => RL
Folks often call this sort of thing a "monad".
Similarly, hom(D,D) is a monoidal category containing a comonoid object, namely LR. The coproduct for this comonoid object is
L.i.R: LR => LRLR
and the counit is
e: LR => 1D
People call this thing a "comonad". But what matters here is that we've seen this comonoid object automatically gives us a simplicial object in hom(D,D)! If we pick any object d of D, we get a functor
hom(D,D) → D
hom(D,D) x D → D
and plugging in d in the second argument. This functor lets us push our simplicial object in hom(D,D) forwards to a simplicial object in D. Voila!