week154

August 12, 2000

This Week's Finds in Mathematical Physics (Week 154)

John Baez

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) Hoi-Kwong Lo, Will quantum cryptography ever become a successful technology in the marketplace?, preprint available as 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.

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.

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 "week44" and "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.

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 "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.

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 "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, 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":

                              R

                     R+R              C



                    R                     H
 


                      C              H+H

                               H

As 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 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 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 7-dimensional 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 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.

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

20) Clifford V. Johnson, D-brane primer, preprint available as hep-th/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 hep-th/9506150.

22) B. S. Acharya, N=1 heterotic-supergravity duality and Joyce manifolds, preprint available as hep-th/9508046.

N=1 heterotic/M-theory duality and Joyce manifolds, preprint available as hep-th/9603033.

N=1 M-theory-heterotic duality in three dimensions and Joyce manifolds, preprint available as hep-th/9604133.

Dirichlet Joyce manifolds, discrete torsion and duality, preprint available as hep-th/9611036.

M theory, Joyce orbifolds and super Yang-Mills, preprint available as hep-th/9812205.

23) Chien-Hao Liu, On the global structure of some natural fibrations of Joyce manifolds, preprint available as 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 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 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 G2 holonomy (just as you need manifolds of 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 S¹/Z₂ 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,"
http://xxx.lanl.gov/abs/hep-th/9506150.

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