
It's been a while since I've written an issue of This Week's Finds... due to holiday distractions and a bunch of papers that need writing up. But tonight I just can't seem to get any work done, so let me do a bit of catching up.
I'm no string theorist, but I still can't help hearing all the rumbling noises over in that direction: first about all the dualities relating seemingly different string theories, and then about the mysterious "Mtheory" in 11 dimensions which seems to underlie all these developments. Let me try to explain a bit of this stuff... in the hopes that I prompt some string theorists to correct me and explain it better! I will simplify everything a lot to keep people from getting scared of the math involved. But I may also make some mistakes, so the experts should be kind to me and try to distinguish between the simplifications and the mistakes.
Recall that it's hard to get a consistent string theory  one that's not plagued by infinite answers to interesting questions. But this difficulty is generally regarded as a good thing, because it drastically limits the number of different versions of string theory one needs to think about. It's often said that there are only 5 consistent string theories: the type I theory, the type IIA and IIB theory, and the two kinds of heterotic string theory. I'm not sure exactly what this statement means, but certainly it's only meant to cover supersymmetric string theories, which can handle fermions (like the electron and neutrino) in addition to bosons (like the photon).
Type I strings are "open strings"  not closed loops  and they live in 10 dimensional spacetime, meaning that you need the dimension to be 10 to make certain nasty infinities cancel out. Type II strings also live in 10 dimensions, but they are "closed strings". That means that they look like a circle, so there are vibrational modes that march around clockwise and other modes that march around counterclockwise, and these are supposed to correspond to different particles that we see. We can think of these vibrational modes as moving around the circle at the speed of light; they are called "leftmovers" and "rightmovers". Now fermions which move at the speed of light are able to be rather asymmetric and only spin one way (when viewed headon). We say they have a "chirality" or handedness. Ordinary neutrinos, for example, are lefthanded. This asymmetry of nature shocked everyone when first discovered, but it appears to be a fact of life, and it's certainly a fact of mathematics. In the type IIA string theory, the leftmoving and rightmoving fermionic vibrational modes have opposite chiralities, while in the IIB theory, they have the same chirality. When I last checked, the type IIA theory seemed to fit our universe a bit better than the IIB theory.
But lots of people say the heterotic theory matches our universe even better. The name "heterotic" refers to the fact that this theory is supposed to have "hybrid vigor". It's quite bizarre: the leftmovers are purely bosonic  no fermions  and live in 26dimensional spacetime, the way nonsupersymmetric string theories do. The rightmovers, on the other hand, are supersymmetric and live in 10 dimensional spacetime. It sounds not merely heterotic, but downright schizophrenic! But in fact, the 26dimensional spacetime can also thought of as being 10dimensional, with 16 extra "curledup dimensions" in the shape of a torus. This torus has two possible shapes: R^16 modulo the E8 x E8 lattice or the D16* lattice. (For some of the wonders of E8 and other lattices, check out "week64" and "week65". The D16* lattice is related to the D16 lattice described in those Weeks, but not quite the same.)
Now there is still lots of room for toying with these theories depending on how you "compactify": how you think of 10dimensional spacetime as 4dimensional spacetime plus 6 curledup dimensions. That's because there are lots of 6dimensional manifolds that will do the job (the socalled "CalabiYau" manifolds). Different choices give different physics, and there is a lot of work to be done to pick the right one.
However, recently it's beginning to seem that all five of the basic sorts of string theory are beginning to look like different manifestations of the same theory in 11 dimensions... some monstrous thing called Mtheory! Let me quote the following paper:
1) Kelly Jay Davis, MTheory and StringString Duality, 28 pages, available as hepth/9601102, uses harvmac.tex.
The idea seems to be roughly that depending on how one compactifies the 11th dimension, one gets different 10dimensional theories from Mtheory:
"In the past year much has happened in the field of string theory. Old results relating the two Type II string theories and the two Heterotic string theories have been combined with newer results relating the Type II theory and the Heterotic theory, as well as the Type I theory and the Heterotic theory, to obtain a single "String Theory." In addition, there has been much recent progress in interpreting some, if not all, properties of String Theory in terms of an elevendimensional MTheory. In this paper we will perform a selfconsistency check on the various relations between MTheory and String Theory. In particular, we will examine the relation between String Theory and MTheory by examining its consistency with the stringstring duality conjecture of sixdimensional String Theory. So, let us now take a quick look at the relations between MTheory and String Theory some of which we will be employing in this article.In Witten's paper he established that the strong coupling limit of Type IIA string theory in ten dimensions is equivalent to elevendimensional supergravity on a "large" S^1. [Note: S^1 just means the circle  jb.] As the low energy limit of Mtheory is elevendimensional supergravity, this relation states that the strong coupling limit of Type IIA string theory in tendimensions is equivalent to the lowenergy limit of MTheory on a "large" S^1. In the paper of Witten and Horava, they establish that the strong coupling limit of the tendimensional E8 x E8 Heterotic string theory is equivalent to MTheory on a "large" S^1/Z_2.
Recently, Witten, motivated by Dasgupta and Mukhi, examined MTheory on a Z_2 orbifold of the fivetorus and established a relation between MTheory on this orbifold and Type IIB string theory on K3. [Note: most of these undefined terms refer to various spaces; for example, the fivetorus is the 5dimensional version of a doughnut, while K3 is a certain 4dimensional manifold  jb.] Also, Schwarz very recently looked at MTheory and its relation to TDuality.
As stated above, MTheory on a "large" S^1 is equivalent to a strongly coupled Type IIA string theory in tendimensions. Also, Mtheory on a "large" S^1/Z_2 is equivalent to a strongly coupled E8 x E8 Heterotic string theory in ten dimensions. However, the stringstring duality conjecture in six dimensions states that the strongly coupled limit of a Heterotic string theory in six dimensions on a fourtorus is equivalent to a weakly coupled Type II string theory in sixdimensions on K3. Similarly, it states that the strongly coupled limit of a Type II theory in six dimensions on K3 is equivalent to a weakly coupled Heterotic string theory in sixdimensions on a fourtorus. Now, as a strongly coupled Type IIA string theory in tendimensions is equivalent to the low energy limit of MTheory on a "large" S^1, the low energy limit of MTheory on S^1 x K3 should be equivalent to a weakly coupled Heterotic string theory on a fourtorus by way of sixdimensional stringstring duality. Similarly, as a strongly coupled E8 x E8 Heterotic string theory in tendimensions is equivalent to the low energy limit of MTheory on a "large" S^1/Z_2, the low energy limit of MTheory on S^1/Z_2 x T^4 should be equivalent to a weakly coupled Type II string theory on K3. The first of the above two consistency checks on the relation between MTheory and String Theory will be the subject of this article. However, we will comment on the second consistency check in our conclusion."
So, as you can see, there is a veritable jungle of relationships out there. But you must be wondering by now: what's Mtheory? According to
2) Edward Witten, Fivebranes and MTheory on an orbifold, available as hepth/9512219.
the M stands for "magic", "mystery", or "membrane", according to taste. From a mathematical viewpoint a better term might be "murky", since apparently everything known about Mtheory is indirect and circumstantial, except for the classical limit, in which it seems to act as a theory of 2branes and 5branes, where an "nbrane" is an ndimensional analog of a membrane or surface.
Well, here I must leave off, for reasons of ignorance. I don't really understand the evidence for the existence of the Mtheory... I can only await the day when the murk clears and it becomes possible to learn about this stuff a bit more easily. It has been suggested that string theory is a bit of 21stcentury mathematics that accidentally fell into the 20th century. I think this is right, and that eventually much of this stuff will be seen as much simpler than it seems now.
Now let me briefly describe some papers I actually sort of understand.
3) Abhay Ashtekar, Polymer geometry at Planck scale and quantum Einstein equations, available as hepth/9601054.
Roumen Borissov, Seth Major and Lee Smolin, The geometry of quantum spin networks, available as grqc/9512043, 35 Postscript figures, uses epsfig.sty.
Bernd Bruegmann, On the constraint algebra of quantum gravity in the loop representation, available as grqc/9512036.
Kiyoshi Ezawa, Nonperturbative solutions for canonical quantum gravity: an overview, available as grqc/9601050
Kiyoshi Ezawa, A semiclassical interpretation of the topological solutions for canonical quantum gravity, available as grqc/9512017.
Jorge Griego, Extended knots and the space of states of quantum gravity, available as grqc/9601007.
Seth Major and Lee Smolin, Quantum deformation of quantum gravity, available as grqc/9512020.
Work on the loop representation of quantum gravity proceeds apace. The paper by Ashtekar and the first one by Ezawa review various recent developments and might be good to look at if one is just getting interested in this subject. Smolin has been pushing the idea of combining ideas about the quantum group SU_q(2) with the loop representation, and his papers with Borissov and Major are about that. This seems rather interesting but still a bit mysterious to me. I suspect that what it amounts to is thinking of loops as excitations not of the AshtekarLewandowksi vacuum state but the ChernSimons state. I'd love to see this clarified, since these two states are two very important exact solutions of quantum gravity, and the latter has the former as a limit as the cosmological constant goes to zero. In the second paper listed, Ezawa gives semiclassical interpretations of these and other exact solutions of quantum gravity.
4) Thomas Kerler, Genealogy of nonperturbative quantuminvariants of 3Manifolds: the surgical family, available as qalg/9601021.
Kerler brings a bit more order to the study of quantum invariants of 3manifolds, in particular, the ReshetikhinTuraev, HenningsKauffmanRadford, and Lyubashenko invariants. All of these are constructed using certain braided monoidal categories, like the category of (nice) representations of a quantum group. He describes how Lyubashenko's invariant specializes to the ReshetikhinTuraev invariant for semisimple categories and to the HenningsKauffmanRadford invariant for Tannakian categories. People interested in extended TQFTs and 2categories will find his work especially interesting, because he works with these invariants using these techniques. James Dolan and I have argued that it's only this way that one will really understand these TQFTs (see "week49").
In future editions of This Week's Finds I will say more about ncategories and topological quantum field theory. I have a feeling that while I've discussed these a lot, I have never really explained the basic ideas very well. As I gradually understand the basic ideas better, they seem simpler and simpler to me, so I think I should try to explain them.
© 1996 John Baez
baez@math.removethis.ucr.andthis.edu
