
At the 13th International Congress on Mathematical Physics, held at Imperial College in London, I was suprised 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) HoiKwong Lo, Will quantum cryptography ever become a successful technology in the marketplace?, preprint available as quantph/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 Dbranes, but the highly ncategorical flavor of the whole thing  he even presented a picture proof the AtiyahSinger 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 hepth during the year 1999.
Here they are, from the topcited one on down:
3) Juan Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231252, preprint available as hepth/9711200.
This one launched the "AdSCFT" craze, by pointing out an interesting relation between supergravity on antiDeSitter spacetime and conformal field theories on its "boundary at infinity".
4) Nathan Seiberg and Edward Witten, Electricmagnetic duality, monopole condensation, and confinement in N=2 supersymmetric YangMills theory, Nucl. Phys. B426 (1994) 1952, preprint available as hepth/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 SeibergWitten theory as a substitute for Donaldson theory when it comes to the study of 4dimensional smooth manifolds. (See "week44" and "week45".) But for physicists, it highlighted the growing importance of "dualities" relating seemingly different physical theories  of which the AdSCFT craze is a more recent outgrowth.
5) Edward Witten, String theory dynamics in various dimensions, Nucl. Phys. B443 (1995) 85126, preprint available as hepth/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 11dimensional supergravity  reduced to 10 dimensions by curling up one dimension into a very large circle. And as I described in "week118", this helped lead to the search for "Mtheory", of which 11dimensional supergravity is hoped to be a lowenergy limit.
6) Edward Witten, AntiDeSitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253291, preprint available as hepth/9802150.
More on the AdSCFT business.
7) S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B428 (1998) 105114, preprint available as hepth/9802109.
Still more on the AdSCFT business.
8) Joseph Polchinski, Dirichlet branes and RamondRamond charges, Phys. Rev. Lett. 75 (1995) 47244727, preprint available as hepth/9510017.
This helped launch the Dbrane revolution: the realization that when we take nonperturbative effects into account, open strings seem to have their ends "stuck" on higherdimensional surfaces called Dbranes.
9) Nathan Seiberg and Edward Witten, Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B431 (1994) 484550, preprint available as hepth/9408099.
More on what's now called SeibergWitten theory.
10) T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, Mtheory as a matrix model: a conjecture, Phys. Rev. D55 (1997), 51125128, preprint available as hepth/9610043.
This was an attempt to given an explicit formulation for Mtheory in terms of a matrix model.
11) C. M. Hull and P. K. Townsend, Unity of superstring dualities, Nucl. Phys. B438 (1995) 109137, preprint available as hepth/9410167.
More about dualities, obviously! (But also some stuff about the exceptional Lie group E7, which is bound to tickle the fancy of any exceptionologist.)
12) Edward Witten, Bound states of strings and pbranes, Nucl. Phys. B460 (1996), 335350, preprint available as hepth/9510135.
More on Dbranes.
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 "week82", "week93", and "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, SpringerVerlag, 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":
R R+R C R H C H+H HAs I explained in "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 x 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 R+R and H+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 2planes in nspace. 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 7dimensional Riemannian manifolds whose holonomy groups lie in the exceptional Lie group G2. I bet this stuff is gonna be important in string theory someday  if it isn't already. After all, G2 is the automorphism group of the octonions, and it has a 7dimensional irreducible representation on the imaginary octonions; as explained in "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
Cliff(p,q) tensor R(2) = Cliff(q+2,p)
where R(2) is the algebra of 2x2 real matrices. A proof of this (actually wellknown) fact appears in (7.8b) of his book.
In response to my list of mostcited papers, Aaron Bergman suggested the following 261page review article on the AdSCFT 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) 183386, preprint available as hepth/9905111.
For a similarly enormous review article on Dbranes, try:
20) Clifford V. Johnson, Dbrane primer, preprint available as hepth/0007170.
Finally, it turns out that manifolds with G2 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 hepth/9506150.
22) B. S. Acharya, N=1 heteroticsupergravity duality and Joyce manifolds, preprint available as hepth/9508046.
N=1 heterotic/Mtheory duality and Joyce manifolds, preprint available as hepth/9603033.
N=1 Mtheoryheterotic duality in three dimensions and Joyce manifolds, preprint available as hepth/9604133.
Dirichlet Joyce manifolds, discrete torsion and duality, preprint available as hepth/9611036.
M theory, Joyce orbifolds and super YangMills, preprint available as hepth/9812205.
23) ChienHao Liu, On the global structure of some natural fibrations of Joyce manifolds, preprint available as hepth/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 7dimensional Riemannian > manifolds whose holonomy groups lie in the exceptional Lie group G2. > 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 11dimensional supergravity/Mtheory to a theory with N=1 supersymmetry in 4 dimensions, which is what people like for phenomenological reasons, you need a 7dimensional manifold of G2 holonomy (just as you need manifolds of SU(3) holonomy, i.e. CalabiYau 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 11d SUGRA compactifications, which is that they are nonchiral, so these days people seem to be concentrating more on HoravaWitten compactifications, with Mtheory on S^{1}/Z_{2} times a CalabiYau, 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," http://xxx.lanl.gov/abs/hepth/9506150.  Paul Shocklee Graduate Student, Department of Physics, Princeton University Researcher, Science Institute, Dunhaga 3, 107 Reykjavík, Iceland Phone: +3545254429
