# August 12, 2000 {#week154}

At the 13th International Congress on Mathematical Physics, held at
Imperial College in London, I was surprised at how much energy was
focussed on quantum computation and quantum cryptography. But it makes
perfect sense --- this is one area where fundamental physics still has the
potential to drastically affect everyday life. I'm not sure quantum
computation will ever be practical, but it's certainly worth checking
out. Quantum cryptography is well on its way --- though people are busy
arguing just *how* practical it will be:

1) Hoi-Kwong Lo, "Will quantum cryptography ever become a successful technology in the marketplace?", preprint available as [`quant-ph/9912011`](https://arxiv.org/abs/quant-ph/9912011)

It seems that both quantum computation and quantum cryptography are
becoming part of a bigger subject, perhaps called "quantum information
theory" --- the study of how information can be transmitted and
manipulated in the context of quantum theory. There's certainly a need
for good theorems and definitions in this subject, as well as more
experiments. For example, nobody seems sure how to calculate the
information capacity of a quantum channel --- or even how to define it!

If you're interested in this, it might be good to start with John
Preskill's lecture notes, which are available for free on the web:

2) John Preskill, Lecture notes on quantum computation and quantum information theory, available at `http://www.theory.caltech.edu/people/preskill/ph229`

Also try the references, homework problems, and links on this webpage.

There was also a lot of stuff about quantum gravity and string theory at
the ICMP. I especially enjoyed Robert Dijkgraaf's talk, for example.
Not just the cute animated movies of strings and D-branes, but the
highly $n$-categorical flavor of the whole thing --- he even presented a
picture proof the Atiyah-Singer index theorem! It wasn't clear how
relevant this is to the physics of our particular universe, but at the
end of the talk Dijkgraaf urged us not to worry about that too much:
after all, the math is so pretty in its own right. Insofar as I'm a
physicist this makes me unhappy --- but in my other persona, as a
mathematician, it makes sense.

I prefer to stay one or two trends behind the times when it comes to
string theory, since I'm not actually working on the subject --- so it's
easier for me to learn about stuff after it's been prettied up a bit by
the mathematicians. Dijkgraaf's talk made me feel a vague
responsibility to tell you all about what's been going on lately in
string theory.... but I'm not really up on this stuff, so I will
discharge this duty in the laziest manner possible, by listing the 10
papers most cited by preprints on hep-th during the year 1999.

Here they are, from the top-cited one on down:

3) Juan Maldacena, "The large $N$ limit of superconformal field theories and supergravity", _Adv. Theor. Math. Phys._ **2** (1998) 231--252, preprint
available as [`hep-th/9711200`](https://arxiv.org/abs/hep-th/9711200).

This one launched the "AdS-CFT" craze, by pointing out an interesting
relation between supergravity on anti-DeSitter spacetime and conformal
field theories on its "boundary at infinity".

4) Nathan Seiberg and Edward Witten, "Electric-magnetic duality, monopole condensation, and confinement in $N=2$ supersymmetric Yang-Mills theory", _Nucl. Phys._ **B426** (1994) 19--52, preprint available as [`hep-th/9407087`](https://arxiv.org/abs/hep-th/9407087).

This one is ancient history by now, but it's still near the top of the
list! For mathematicians, this paper marked the birth of Seiberg-Witten
theory as a substitute for Donaldson theory when it comes to the study
of $4$-dimensional smooth manifolds. (See ["Week 44"](#week44) and
["Week 45"](#week45).) But for physicists, it highlighted the
growing importance of "dualities" relating seemingly different
physical theories --- of which the AdS-CFT craze is a more recent
outgrowth.

5) Edward Witten, "String theory dynamics in various dimensions", _Nucl. Phys._ **B443** (1995) 85--126, preprint available as [`hep-th/9503124`](https://arxiv.org/abs/hep-th/9503124).

This paper was also important in the quest to understand dualities:
among other things, it argued that the type IIA superstring in 10
dimensions is related to $11$-dimensional supergravity --- reduced to 10
dimensions by curling up one dimension into a very *large* circle. And
as I described in ["Week 118"](#week118), this helped lead to the
search for "M-theory", of which $11$-dimensional supergravity is hoped
to be a low-energy limit.

6) Edward Witten, "Anti-DeSitter space and holography", _Adv. Theor. Math. Phys._ **2** (1998) 253--291, preprint available as [`hep-th/9802150`](https://arxiv.org/abs/hep-th/9802150).

More on the AdS-CFT business.

7) S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, "Gauge theory correlators from noncritical string theory", _Phys. Lett._ **B428** (1998) 105--114, preprint available as [`hep-th/9802109`](https://arxiv.org/abs/hep-th/9802109).

Still more on the AdS-CFT business.

8) Joseph Polchinski, "Dirichlet branes and Ramond-Ramond charges", _Phys. Rev. Lett._ **75** (1995) 4724--4727, preprint available as [`hep-th/9510017`](https://arxiv.org/abs/hep-th/9510017).

This helped launch the D-brane revolution: the realization that when we
take nonperturbative effects into account, open strings seem to have
their ends "stuck" on higher-dimensional surfaces called D-branes.

9) Nathan Seiberg and Edward Witten, "Monopoles, duality and chiral symmetry breaking in $N=2$ supersymmetric QCD", _Nucl. Phys._ **B431** (1994) 484--550, preprint available as [`hep-th/9408099`](https://arxiv.org/abs/hep-th/9408099).

More on what's now called Seiberg-Witten theory.

10) T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, "M-theory as a matrix model: a conjecture", _Phys. Rev._ **D55** (1997), 5112--5128, preprint available as [`hep-th/9610043`](https://arxiv.org/abs/hep-th/9610043).

This was an attempt to given an explicit formulation for M-theory in
terms of a matrix model.

11) C. M. Hull and P. K. Townsend, "Unity of superstring dualities", _Nucl. Phys._ **B438** (1995) 109--137, preprint available as
[`hep-th/9410167`](https://arxiv.org/abs/hep-th/9410167).

More about dualities, obviously! (But also some stuff about the
exceptional Lie group $\mathrm{E}_7$, which is bound to tickle the fancy of any
exceptionologist.)

12) Edward Witten, "Bound states of strings and $p$-branes", _Nucl. Phys._ **B460** (1996), 335--350, preprint available as [`hep-th/9510135`](https://arxiv.org/abs/hep-th/9510135).

More on D-branes.

By the way: if you do physics, you can look up your *own* top cited
papers on the SPIRES database, at least if someone has cited you 50 or
more times:

13) Searching top cited papers on SPIRES, at `http://www.slac.stanford.edu/spires/hep/topcite.html`

This will allow you to measure your fame in milliwittens.

And now for something completely different:

I've been thinking about Clifford algebras a lot recently, because I'm
writing a review article on the octonions and exceptional Lie groups,
and a good way to undestand this stuff is to use a lot of Clifford
algebras machinery. I talked about Clifford algebras in
["Week 82"](#week82), ["Week 93"](#week93), and
["Week 105"](#week105), but here are some more nice books about
them.

First, when I was giving a little talk on Clifford algebras at
Nottingham University after the ICMP, I needed to look up a few things,
and I bumped into this book:

14) P. Budinich and A. Trautman, _The Spinorial Chessboard_, Springer-Verlag, Berlin, 1988.

Unfortunately it's out of print, but John Barrett happened to have a
copy. Springer should reprint it! It has a nice discussion of the
"Clifford algebra clock":
$$
  \begin{tikzpicture}
    \draw (0,0) circle[radius=2.65cm];
    \node[label=below:{$\mathbb{R}$}] at (90:2.3) {0};
    \node[label=below left:{$\mathbb{C}$}] at (45:2.3) {1};
    \node[label=left:{$\mathbb{H}$}] at (0:2.3) {2};
    \node[label={[label distance=-2mm]above left:{$\mathbb{H}\oplus\mathbb{H}$}}] at (-45:2.3) {3};
    \node[label=above:{$\mathbb{H}$}] at (-90:2.3) {4};
    \node[label=above right:{$\mathbb{C}$}] at (-135:2.3) {5};
    \node[label=right:{$\mathbb{R}$}] at (180:2.3) {6};
    \node[label={[label distance=-2mm]below right:{$\mathbb{R}\oplus\mathbb{R}$}}] at (135:2.3) {7};
    \foreach \a in {0,45,90,135,180,-135,-90,-45}
      \draw (\a:2.5) to (\a:2.65);
  \end{tikzpicture}
$$
As I explained in ["Week 105"](#week105), this clock easily lets
you remember the real Clifford algebras in every dimension and signature
of spacetime. Bott periodicity explains why it loops around after 8
hours. The spinorial chessboard presents the same information in the
form of an $8\times 8$ grid. I won't draw it here, but it's a picture of the
Clifford algebras with $p$ roots of $-1$ and $q$ roots of $1$ for
$p,q =0,1,2,3,4,5,6,7$. The black squares correspond to cases that admit chiral
spinors; the red ones correspond to cases that don't. Black is when $p+q$
is even; red is when it's odd.

By the way, I have a little question: why does the above clock have a
reflection symmetry along the line joining $\mathbb{R}+\mathbb{R}$ and $\mathbb{H}+\mathbb{H}$?

Later, by coincidence, when I was in the library I discovered that
Chevalley's work on spinors has been reprinted:

15) Claude Chevalley, _The Algebraic Theory of Spinors_, Springer, Berlin, 1991.

It has a lot of neat stuff on "pure spinors", which are closely
related to the "simple bivectors" that describe 2-planes in $n$-space.
The latter play an important role in spin foam models of quantum
gravity, so I bet pure spinors will too.

Here's another fundamental text, which really helped get the whole
subject going:

16) Eli Cartan, _The Theory of Spinors_, Dover Press, 1966.

While I'm at it, I should mention this book by the infamous Pertti
Lounesto, which is also good:

17) Pertti Lounesto, _Clifford Algebras and Spinors_, Cambridge U. Press, Cambridge, 1997.

I also saw this book at a book fair:

18) Dominic Joyce, _Compact Manifolds with Special Holonomy_, Oxford U. Press, Oxford, 2000.

There's some incredible stuff here about $7$-dimensional Riemannian
manifolds whose holonomy groups lie in the exceptional Lie group $\mathrm{G}_2$. I
bet this stuff is gonna be important in string theory someday --- if it
isn't already. After all, $\mathrm{G}_2$ is the automorphism group of the
octonions, and it has a $7$-dimensional irreducible representation on the
imaginary octonions; as explained in ["Week 104"](#week104) by
Robert Helling, the octonions are secretly what let you write down the
superstring Lagrangian in 10d spacetime.

------------------------------------------------------------------------

Footnote:

Andrzej Trautman answered my question about reflection symmetry in the
Clifford algebra clock by noting that
$$\mathrm{Cliff}(p,q) \otimes \mathbb{R}(2) = \mathrm{Cliff}(q+2,p)$$
where $\mathbb{R}(2)$ is the algebra of $2\times2$ real matrices. A proof of this
(actually well-known) fact appears in (7.8b) of his book.

In response to my list of most-cited papers, Aaron Bergman suggested the
following 261-page review article on the AdS-CFT correspondence:

19) O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, "Large $N$ field theories, string theory and gravity", _Phys. Rept._ **323** (2000)
183--386, preprint available as [`hep-th/9905111`](https://arxiv.org/abs/hep-th/9905111).

For a similarly enormous review article on D-branes, try:

20) Clifford V. Johnson, "D-brane primer", preprint available as [`hep-th/0007170`](https://arxiv.org/abs/hep-th/0007170).

Finally, it turns out that manifolds with $\mathrm{G}_2$ holonomy *are* important in
superstring theory, where they go by the name of "Joyce manifolds".
Here are some places to read about them:

21) G. Papadopoulos and P. K. Townsend, "Compactification of $D=11$ supergravity on spaces of exceptional holonomy", preprint available as [`hep-th/9506150`](https://arxiv.org/abs/hep-th/9506150).

22) B. S. Acharya, "$N=1$ heterotic-supergravity duality and Joyce manifolds", preprint available as [`hep-th/9508046`](https://arxiv.org/abs/hep-th/9508046).

    "$N=1$ heterotic/M-theory duality and Joyce manifolds", preprint available as [`hep-th/9603033`](https://arxiv.org/abs/hep-th/9603033).

    "$N=1$ M-theory-heterotic duality in three dimensions and Joyce manifolds", preprint available as [`hep-th/9604133`](https://arxiv.org/abs/hep-th/9604133).

    "Dirichlet Joyce manifolds, discrete torsion and duality", preprint available as [`hep-th/9611036`](https://arxiv.org/abs/hep-th/9611036).

    "M theory, Joyce orbifolds and super Yang-Mills", preprint available as [`hep-th/9812205`](https://arxiv.org/abs/hep-th/9812205).

23) Chien-Hao Liu, "On the global structure of some natural fibrations of Joyce manifolds", preprint available as [`hep-th/9809007`](https://arxiv.org/abs/hep-th/9809007).

I learned this thanks to Allen Knutson and Paul Schocklee. Paul also had
the following interesting comments:

> John Baez wrote:

> > There's some incredible stuff here about $7$-dimensional Riemannian
> > manifolds whose holonomy groups lie in the exceptional Lie group $\mathrm{G}_2$.  
> > I bet this stuff is gonna be important in string theory someday --- if it
> > isn't already.  

> They are important!

> If you want to directly compactify $11$-dimensional supergravity/M-theory 
> to a theory with $N=1$ supersymmetry in 4 dimensions, which is what people 
> like for phenomenological reasons, you need a $7$-dimensional manifold
> of $\mathrm{G}_2$ holonomy (just as you need manifolds of $\mathrm{SU}(3)$ holonomy, i.e.
> Calabi-Yau manifolds, in six dimensions).  I have seen these referred 
> to as "Joyce manifolds," after Dominic Joyce, who constructed several 
> examples of such spaces.  (I didn't know there was so much known about
> them.  I'll have to check out the above book; I see that our library 
> in Iceland has a copy.)

> Unfortunately, these models are afflicted by the usual problem of
> 11-d SUGRA compactifications, which is that they are non-chiral,
> so these days people seem to be concentrating more on Horava-Witten
> compactifications, with M-theory on $S^1/\mathbb{Z}_2$ times a Calabi-Yau, or on
> an orbifold. 

> If you're interested, you might want to check out Papadopoulos and Townsend,
> "Compactification of D=11 supergravity on spaces of exceptional holonomy,"
> https://arxiv.org/abs/hep-th/9506150.

> --
> Paul Shocklee
> Graduate Student, Department of Physics, Princeton University
> Researcher, Science Institute, Dunhaga 3, 107 Reykjavk, Iceland
> Phone: +354-525-4429