# February 1, 1996 {#week72}

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 "M-theory" 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 "left-movers" and "right-movers".
Now fermions which move at the speed of light are able to be rather
asymmetric and only spin one way (when viewed head-on). We say they have
a "chirality" or handedness. Ordinary neutrinos, for example, are
left-handed. 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 left-moving and
right-moving 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 left-movers
are purely bosonic --- no fermions --- and live in $26$-dimensional spacetime,
the way non-supersymmetric string theories do. The right-movers, on the
other hand, are supersymmetric and live in 10- dimensional spacetime. It
sounds not merely heterotic, but downright schizophrenic! But in fact,
the $26$-dimensional spacetime can also thought of as being
10-dimensional, with 16 extra "curled-up dimensions" in the shape of a
torus. This torus has two possible shapes: $\mathbb{R}^16$ modulo the $\mathrm{E}_8 \times \mathrm{E}_8$
lattice or the $D_{16}^*$ lattice. (For some of the wonders of $\mathrm{E}_8$ and other
lattices, check out ["Week 64"](#week64) and
["Week 65"](#week65). The $D_{16}^*$ lattice is related to the $D_{16}$
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 $10$-dimensional spacetime as
4-dimensional spacetime plus 6 curled-up dimensions. That's because
there are lots of $6$-dimensional manifolds that will do the job (the
so-called "Calabi-Yau" 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 M-theory! Let me quote the following paper:

1) Kelly Jay Davis, "M-Theory and String-String Duality", 28 pages, available as [`hep-th/9601102`](https://arxiv.org/abs/hep-th/9601102), uses `harvmac.tex`.

The idea seems to be roughly that depending on how one compactifies the
11th dimension, one gets different $10$-dimensional theories from
M-theory:

> "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 eleven-dimensional M-Theory. In this paper we will perform a
> self-consistency check on the various relations between M-Theory and
> String Theory. In particular, we will examine the relation between
> String Theory and M-Theory by examining its consistency with the
> string-string duality conjecture of six-dimensional String Theory. So,
> let us now take a quick look at the relations between M-Theory 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
> eleven-dimensional supergravity on a "large" $S^1$. \[Note: $S^1$ just
> means the circle --- jb.\] As the low energy limit of M-theory is
> eleven-dimensional supergravity, this relation states that the strong
> coupling limit of Type IIA string theory in ten-dimensions is
> equivalent to the low-energy limit of M-Theory on a "large" $S^1$. In
> the paper of Witten and Horava, they establish that the strong
> coupling limit of the ten-dimensional $\mathrm{E}_8 \times \mathrm{E}_8$ Heterotic string theory
> is equivalent to M-Theory on a "large" $S^1/\mathbb{Z}_2$.
>
> Recently, Witten, motivated by Dasgupta and Mukhi, examined M-Theory
> on a $\mathbb{Z}_2$ orbifold of the five-torus and established a relation between
> M-Theory on this orbifold and Type IIB string theory on $K3$. \[Note:
> most of these undefined terms refer to various spaces; for example,
> the five-torus is the $5$-dimensional version of a doughnut, while $K3$ is
> a certain $4$-dimensional manifold --- jb.\] Also, Schwarz very recently
> looked at M-Theory and its relation to T-Duality.
>
> As stated above, M-Theory on a "large" $S^1$ is equivalent to a
> strongly coupled Type IIA string theory in ten-dimensions. Also,
> M-theory on a "large" $S^1/\mathbb{Z}_2$ is equivalent to a strongly coupled
> $\mathrm{E}_8 \times \mathrm{E}_8$ Heterotic string theory in ten dimensions. However, the
> string-string duality conjecture in six dimensions states that the
> strongly coupled limit of a Heterotic string theory in six dimensions
> on a four-torus is equivalent to a weakly coupled Type II string
> theory in six-dimensions 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
> six-dimensions on a four-torus. Now, as a strongly coupled Type IIA
> string theory in ten-dimensions is equivalent to the low energy limit
> of M-Theory on a "large" $S^1$, the low energy limit of M-Theory on
> $S^1 \times K3$ should be equivalent to a weakly coupled Heterotic string
> theory on a four-torus by way of six-dimensional string-string
> duality. Similarly, as a strongly coupled $\mathrm{E}_8 \times \mathrm{E}_8$ Heterotic string
> theory in ten-dimensions is equivalent to the low energy limit of
> M-Theory on a "large" $S^1/\mathbb{Z}_2$, the low energy limit of M-Theory on
> $S^1/\mathbb{Z}_2 \times T^4$ 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 M-Theory 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 M-theory?* According
to

2) Edward Witten, "Five-branes and M-Theory on an orbifold", available as [`hep-th/9512219`](https://arxiv.org/abs/hep-th/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 M-theory is indirect and
circumstantial, except for the classical limit, in which it seems to act
as a theory of $2$-branes and $5$-branes, where an "n-brane" is an
n-dimensional 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 M-theory... 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 21st-century 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 [`hep-th/9601054`](https://arxiv.org/abs/hep-th/9601054).

    Roumen Borissov, Seth Major and Lee Smolin, "The geometry of quantum spin networks", available as [`gr-qc/9512043`](https://arxiv.org/abs/gr-qc/9512043), 35 Postscript figures, uses `epsfig.sty`.

    Bernd Bruegmann, "On the constraint algebra of quantum gravity in the loop representation", available as [`gr-qc/9512036`](https://arxiv.org/abs/gr-qc/9512036).

    Kiyoshi Ezawa, "Nonperturbative solutions for canonical quantum gravity: an overview", available as [`gr-qc/9601050`](https://arxiv.org/abs/gr-qc/9601050)

    Kiyoshi Ezawa, "A semiclassical interpretation of the topological solutions for canonical quantum gravity", available as [`gr-qc/9512017`](https://arxiv.org/abs/gr-qc/9512017).

    Jorge Griego, "Extended knots and the space of states of quantum gravity", available as [`gr-qc/9601007`](https://arxiv.org/abs/gr-qc/9601007).

    Seth Major and Lee Smolin, "Quantum deformation of quantum gravity", available as [`gr-qc/9512020`](https://arxiv.org/abs/gr-qc/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 Ashtekar-Lewandowksi vacuum state but the Chern-Simons 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 quantum-invariants of 3-Manifolds: the surgical family", available as [`q-alg/9601021`](https://arxiv.org/abs/q-alg/9601021).

Kerler brings a bit more order to the study of quantum invariants of
3-manifolds, in particular, the Reshetikhin-Turaev,
Hennings-Kauffman-Radford, 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 Reshetikhin-Turaev invariant
for semisimple categories and to the Hennings-Kauffman-Radford invariant
for Tannakian categories. People interested in extended TQFTs and
$2$-categories 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 ["Week 49"](#week49)).

In future editions of This Week's Finds I will say more about
$n$-categories 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.