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I'm done with teaching until fall, and now I'll be travelling a lot. I just got back from Singapore. It's an incredibly diverse place. I actually had to buy a book to understand all the foods! I'm now acquainted with the charms of appam, kaya toast, and babi buah keluak. But I didn't get around to trying a chendol, a bandung, or a Milo dinosaur, even though they're all available in every hawker center.
Today I'll talk about quantum technology in Singapore, atom chips, graphene transistors, nitrogen-vacancy pairs in diamonds, a new construction of e8, and a categorification of quantum sl(2).
But first - the astronomy pictures of the week!
First another planetary nebula - the "Southern Ring Nebula":
1) Hubble Heritage Project, Planetary Nebula NGC 3132, http://heritage.stsci.edu/1998/39/index.html
This bubble of hot gas is .4 light years in diameter. You can see two stars near its center. The faint one is the white dwarf remnant of the star that actually threw off the gas forming this nebula. The gas is expanding outwards at about 20 kilometers per second. The intense ultraviolet radiation from the white dwarf is ionizing this gas and making it glow.
The Southern Ring Nebula is 2000 light years from us. Much closer to home, here's a new shot of the frosty dunes of Mars:
2) HiRISE (High Resolution Imaging Science Experiment), Defrosting polar sand dunes, http://hirise.lpl.arizona.edu/PSP_007043_2650
These horn-shaped dunes are called "barchans"; you can read more about them at "week228". The frost is carbon dioxide, evaporating as the springtime sun warms the north polar region. Here's another photo, taken in February:
3) HiRISE (High Resolution Imaging Science Experiment), Defrosting northern dunes, http://hirise.lpl.arizona.edu/PSP_007193_2640
The dark stuff pouring down the steep slopes reminds me of water, but they say it's dust!
(If you click on these Mars photos, you'll get some amazing larger views.)
Meanwhile, down here on Earth, I had some good conversations with mathematicians and physicists at the National University of Singapore (NUS), and also with Artur Ekert and Valerio Scarani, who work here:
4) Centre for Quantum Technologies, http://www.quantumlah.org/
I like the name "quantumlah". "Lah" is perhaps the most famous word in Singlish: you put it at the end of a sentence for emphasis, to convey "acceptance, understanding, lightness, jest, and a medley of other positive feelings". Unfortunately I didn't get to hear much Singlish during my visit.
The Centre for Quantum Technologies is hosted by NUS but is somewhat independent. It reminds me a bit of the Institute for Quantum Computing - see "week235" - but it's smaller, and still getting started. They hope to take advantage of the nearby semiconductor fabrication plants, or "fabs", to build stuff.
They've got theorists and experimentalists. Being overly theoretical myself, I asked: what are the most interesting real-life working devices we're likely to see soon? Ekert mentioned "quantum repeaters" - gadgets that boost the power of a beam of entangled photons while still maintaining quantum coherence, as needed for long-distance quantum cryptography. He also mentioned "atom chips", which use tiny wires embedded in a silicon chip to trap and manipulate cold atoms on the chip's surface:
5) Atomchip Group, http://www.atomchip.org/
6) Atom Optics Group, Laboratoire Charles Fabry, Atom-chip experiment, http://atomoptic.iota.u-psud.fr/research/chip/chip.html
There's also a nanotech group at NUS:
7) Nanoscience and Nanotechnology Initiative, National University of Singapore, http://www.nusnni.nus.edu.sg/
who are doing cool stuff with "graphene" - hexagonal sheets of carbon atoms, like individual layers of a graphite crystal:
Graphene is closely related to buckyballs (see "week79") and polycyclic aromatic hydrocarbons (see "week258").
Some researchers believe that graphene transistors could operate in the terahertz range, about 1000 times faster than conventional silicon ones. The reason is that electrons move much faster through graphene. Unfortunately the difference in conductivity between the "on" and "off" states is less for graphene. This makes it harder to work with. People think they can solve this problem, though:
8) Kevin Bullis, Graphene transistors, Technology Review, January 28, 2008, http://www.technologyreview.com/Nanotech/20119/
Duncan Graham-Rowe, Better graphene transistors, Technology Review, March 17, 2008, http://www.technologyreview.com/Nanotech/20424/
Ekert also told me about another idea for carbon-based computers: "nitrogen-vacancy centers". These are very elegant entities. To understand them, it helps to know a bit about diamonds. You really just need to know that diamonds are crystals made of carbon. But I can't resist saying more, because the geometry of these crystals is fascinating.
A diamond is made of carbon atoms arranged in tetrahedra, which then form a cubical structure, like this:
9) Steve Sque, Structure of diamond, http://newton.ex.ac.uk/research/qsystems/people/sque/diamond/structure/
Here you see 4 tetrahedra of carbon atoms inside a cube. Note that there's one carbon at each corner of the cube, and also one in the middle of each face. If that was all, we'd have a "face-centered cubic". But there are also 4 more carbons inside the cube - one at the center of each tetrahedron!
If you look really carefully, you can see that the full pattern consists of two interpenetrating face-centered cubic lattices, one offset relative to the other along the cube's main diagonal!
While the math of the diamond crystal is perfectly beautiful, nature doesn't always get it quite right. Sometimes a carbon atom will be missing. In fact, sometimes a cosmic ray will knock a carbon out of the lattice! You can also do it yourself with a beam of neutrons or electrons. The resulting hole is called a "vacancy". If you heat a diamond to about 900 kelvin, these vacancies start to move around like particles.
Diamonds also have impurities. The most common is nitrogen, which can form up 1% of a diamond. Nitrogen atoms can take the place of carbon atoms in the crystal. Sometimes these nitrogen atoms are isolated, sometimes they come in pairs.
When a lone nitrogen encounters a vacancy, they stick together! We then have a "nitrogen-vacancy center". It's also common for 4 nitrogens to surround a vacancy. Many other combinations are also possible - and when we get enough of these nitrogen-vacancy combinations around, they form larger structures called "platelets".
10) R. Jones and J. P. Goss, Theory of aggregation of nitrogen in diamond, in Properties, Growth and Application of Diamond, eds. Maria Helena Nazare and A. J. Neves, EMIS Datareviews Series, 2001, 127-130.
A nice thing about nitrogen-vacancy centers is that they act like spin-1 particles. In fact, these spins interact very little with their environment, thanks to the remarkable properties of diamond. So, they might be a good way to store quantum information: they can last 50 microseconds before losing coherence, even at room temperature. If we could couple them to each other in interesting ways, maybe we could do some "spintronics", or even quantum computation:
11) Sankar das Sarma, Spintronics, American Scientist 89 (2001), 516-523. Also available at http://www.physics.umd.edu/cmtc/earlier_papers/AmSci.pdf
Lone nitrogens are even more robust carriers of quantum information: their time to decoherence can be as much as a millisecond! The reason is that, unlike nitrogen-vacancy centers, lone nitrogens have "dark spins" - their spin doesn't interact much with light. But this can also makes them harder to manipulate. So, it may be easier to use nitrogen-vacancy centers. People are busy studying the options:
12) R. J. Epstein, F. M. Mendoza, Y. K. Kato and D. D. Awschalom, Anisotropic interactions of a single spin and dark-spin spectroscopy in diamond, Nature Physics 1 (2005), 94-98. Also available as arXiv:cond-mat/0507706.
13) Ph. Tamarat et al, The excited state structure of the nitrogen-vacancy center in diamond, available as arXiv:cond-mat/0610357.
14) R. Hanson, O. Gywat and D. D. Awschalom, Room-temperature manipulation and decoherence of a single spin in diamond, Phys. Rev. B74 (2006) 161203. Also available as arXiv:quant-ph/0608233
But regardless of whether anyone can coax them into quantum computation, I like diamonds. Not to own - just to contemplate! I told you about the diamond rain on Neptune back in "week160". And in "week193", I explained how diamonds are the closest thing to the E8 lattice you're likely to see in this 3-dimensional world.
The reason is that in any dimension you can define a checkerboard lattice called Dn, consisting of all n-tuples of integers that sum to an even integer. Then you can define a set called Dn+ by taking two copies of the Dn lattice: the original and another shifted by the vector (1/2,...,1/2). D8+ is the E8 lattice, but D3 is the face-centered cubic, and D3+ is the pattern formed by carbons in a diamond!
In case you're wondering: in math, unlike crystallography, we reserve the term "lattice" for a discrete subgroup of Rn that's isomorphic to Zn. The set Dn+ is only closed under addition when n is even. So, the carbons in a diamond don't form a lattice in the strict mathematical sense. On the other hand, the face-centered cubic really is a lattice, the D3 lattice - and this is secretly the same as the A3 lattice, familiar from stacking oranges. It's one of the densest ways to pack spheres, with a density of
π /(3√2) ≈ 0.74
The D3+ pattern, on the other hand, has a density of just
(π√3)/16 ≈ 0.34
This is why ice becomes denser when it melts: it's packed in a close relative of the D3+ pattern, with an equally low density.
(Do diamonds become denser when they melt? Or do they always turn into graphite when they get hot enough, regardless of the pressure? Inquiring minds want to know. These days inquiring minds use search engines to answer questions like this... but right now I'd rather talk about E8.)
As you probably noticed, Garrett Lisi stirred up quite a media sensation with his attempt to pack all known forces and particles into a theory based on the exceptional Lie group E8:
15) Garrett Lisi, An exceptionally simple theory of everything, available as arXiv:0711.0770
Part of his idea was to use Kostant's triality-based description of E8 to explain the three generations of leptons - see "week253" for more. Unfortunately this part of the idea doesn't work, for purely group-theoretical reasons:
16) Jacques Distler, A little group theory,
http://golem.ph.utexas.edu/~distler/blog/archives/001505.html
A little more group theory,
http://golem.ph.utexas.edu/~distler/blog/archives/001532.html
There would also be vast problems trying get all the dimensionless constants in the Standard Model to pop out of such a scheme - or to stick them in somehow.
Meanwhile, Kostant has been doing new things with E8. He's mainly been using the complex form of E8, while Lisi needs a noncompact real form to get gravity into the game. So, the connection between their work is somewhat limited. Nonetheless, Kostant enjoys the idea of a theory of everything based on E8.
He recently gave a talk here at UCR:
17) Bertram Kostant, On some mathematics in Garrett Lisi's "E8 theory of everything", February 12, 2008, UCR. Video and lecture notes at http://math.ucr.edu/home/baez/kostant/
He did some amazing things, like chop the 248-dimensional Lie algebra of E8 into 31 Cartan subalgebras in a nice way, thus categorifying the factorization
248 = 8 × 31
To do this, he used a copy of the 32-element group (Z/2)5 sitting in E8, and the 31 nontrivial characters of this group.
Even more remarkably, this copy of (Z/2)5 sits inside a copy of SL(2,F32) inside E8, and the centralizer of a certain element of SL(2,F32) is a product of two copies of the gauge group of the Standard Model! What this means - if anything - remains a mystery.
Indeed, pretty much everything about E8 seems mysterious to me, since nobody has exhibited it as the symmetry group of anything more comprehensible than E8 itself. This paper sheds some new light this puzzle:
17) José Miguel Figueroa-O'Farrill, A geometric construction of the exceptional Lie algebras F4 and E8, available as arXiv:0706.2829.
The idea here is to build the Lie algebra of E8 using Killing spinors on the unit sphere in 16 dimensions.
Okay - what's a Killing spinor?
Well, first I need to remind you about Killing vectors. Given a Riemannian manifold, a "Killing vector" is a vector field that generates a flow that preserves the metric! A transformation that preserves the metric is called an "isometry", and these form a Lie group. Killing vector fields form a Lie algebra if we use the ordinary Lie bracket of vector fields, and this is the Lie algebra of the group of isometries.
Now, if our manifold has a spin structure, a "Killing spinor" is a spinor field ψ such that
Dvψ = k vψ
for some constant k for every vector field v. Here Dvψ is the covariant derivative of ψ in the v direction, while vψ is defined using the action of vectors on spinors. Only the sign of the constant k really matters, since rescaling the metric rescales this constant.
It's a cute equation, but what's the point of it? Part of the point is this: the action of vectors on spinors
V ⊗ S → S
has a kind of adjoint
S ⊗ S → V
This lets us take a pair of spinor fields and form a vector field. This is what people mean when they say spinors are like the "square root" of vectors. And, if we do this to two Killing spinors, we get a Killing vector! You can prove this using that cute equation - and that's the main point of that equation, as far as I'm concerned.
Under good conditions, this fact lets us define a "Killing superalgebra" which has the Lie algebra of Killing vectors as its even part, and the Killing spinors as its odd part.
In this superalgebra, the bracket of two Killing vectors is just their ordinary Lie bracket. The bracket of a Killing vector and a Killing spinor is defined using a fairly obvious notion of the "Lie derivative of a spinor field". And, the bracket of two Killing spinors is defined using the map
S ⊗ S → V
which, as explained, gives a Killing vector.
Now, you might think our "Killing superalgebra" should be a Lie superalgebra. But in some dimensions, the map
S ⊗ S → V
is skew-symmetric. Then our Killing superalgebra has a chance at being a plain old Lie algebra! We still need to check the Jacobi identity. And this only works in certain special cases:
If you take S7 with its usual round metric, the isometry group is SO(8), so the Lie algebra of Killing vectors is so(8). There's an 8-dimensional space of Killing spinors, and the action of so(8) on this gives the real left-handed spinor representation S8+. The Jacobi identity holds, and you get a Lie algebra structure on
so(8) ⊕ S8+
But then, thanks to triality, you knock yourself on the head and say "I could have had a V8!" After all, up to an outer automorphism of so(8), the spinor representation S8+ is the same as the 8-dimensional vector representation V8. So, your Lie algebra is the same as
so(8) ⊕ V8
with a certain obvious Lie algebra structure. This is just so(9). So, it's nothing exceptional, though you arrived at it by a devious route.
If you take S8 with its usual round metric, the Lie algebra of Killing vector fields is so(9). Now there's a 16-dimensional space of Killing spinor fields, and the action of so(9) on this gives the real (non-chiral) spinor representation S9. The Jacobi identity holds, and you get a Lie algebra structure on
so(9) ⊕ S9
This gives the exceptional Lie algebra f4!
Finally, if you take S15 with its usual round metric, the Lie algebra of Killing vector fields is so(16). Now there's a 128-dimensional space of Killing spinor fields, and the action of so(16) on this gives the left-handed real spinor representation S16+. The Jacobi identity holds, and you get a Lie algebra structure on
so(16) ⊕ S16+
This gives the exceptional Lie algebra e8!
In short, what Figueroa-O'Farrill has done is found a nice geometrical interpretation for some previously known algebraic constructions of f4 and e8. Unfortunately, he still needs to verify the Jacobi identity in the same brute-force way. It would be nice to find a slicker proof. But his new interpretation is suggestive: it raises a lot of new questions. He lists some of these at the end of the paper, and mentions a really big one at the beginning. Namely: the spheres S7, S8 and S15 all show up in the Hopf fibration associated to the octonionic projective line:
S7 → S15 → S8
Does this give a nice relation between so(9), f4 and e8? Can someone guess what this relation should be? Maybe e8 is built from so(9) and f4 somehow.
I also wonder if there's a Killing superalgebra interpretation of the Lie algebra constructions
e6 = so(10) ⊕ S10 ⊕ u(1)
and
e7 = so(12) ⊕ S12+ ⊕ su(2)
These would need to be trickier, with the u(1) showing up from the fact that S10 is a complex representation, and the su(2) showing up from the fact that S12+ is a quaternionic representation. The algebra is explained here:
18) John Baez, The octonions, section 4.3: the magic square, available at http://math.ucr.edu/home/baez/octonions/node16.html
A geometrical interepretation would be nice!
Finally - my former student Aaron Lauda has been working with Khovanov on categorifying quantum groups, and their work is starting to really take off. I'm just beginning to read his new papers, but I can't resist bringing them to your attention:
19) Aaron Lauda, A categorification of quantum sl(2), available as arXiv:0803.3652.
Aaron Lauda, Categorified quantum sl(2) and equivariant cohomology of iterated flag varieties, available as arXiv:0803.3848.
He's got a 2-category that decategorifies to give the quantized universal enveloping algebra of sl(2)! And similarly for all the irreps of this algebra!
There's more to come, too....
Addenda: Starting this Week, you can see more discussion and also questions I'm dying to know the answers to over at the n-Category Café. Whenever I write This Week's Finds, I come up with lots of questions. If you can help me with some of these, I'll be really grateful.
José Figueroa-O'Farrill sent an email saying:
About the geometric constructions of exceptional Lie algebras, you are totally spot on in that what is missing is a more conceptual understanding of the construction which would render the odd-odd-odd component of the Jacobi identity 'trivial', as is the case for the remaining three components. One satisfactory way to achieve this would be to understand of what in, say, the 15-sphere is E8 the automorphisms. I'm afraid I don't have an answer.These comments by Thomas Fischbacher should also fit into the big picture somehow:As for E6 and E7, there is a similar geometric construction for E6 and one for E7 is in the works as part of a paper with Hannu Rajaniemi, who was a student of mine. The construction is analogous, but for one thing. One has to construct more than just the Killing vectors out of the Killing spinors: in the case of E6, it is enough to construct a Killing 0-form (i.e., a constant) which then acts on the Killing spinors via a multiple of the Dirac operator. (This is consistent with the action of 'special Killing forms' a.k.a. 'Killing-Yano tensors' on spinors.) The odd-odd-odd Jacobi identity here is even more mysterious: it does not simply follow from representation theory (i.e., absence of invariants in the relevant representation where the 'jacobator' lives), but follows from an explicit calculation. The case of E7 should work in a similar way, but we still have not finished the construction. (Hannu has a real job now and I've been busy with other projects of a less 'recreational' nature.) In
20) José Figueroa-O'Farrill, A geometrical construction of exceptional Lie algebras, talk at Leeds, February 13, 2008, available at http://www.maths.ed.ac.uk/~jmf/CV/Seminars/Leeds.pdf
you'll find the PDF version of a Keynote file I used for a geometry seminar I gave recently on this topic in Leeds. This geometric construction has its origin, as does the notion of Killing spinor itself, in the early supergravity literature. Much of the early literature on supergravity backgrounds was concerned with the so-called Freund-Rubin backgrounds: product geometries L × R, with L a lorentzian constant curvature spacetime and R a riemannian homogeneous space and the only nonzero components of the flux were proportional to the volume forms of L and/or R. For such backgrounds, supergravity Killing spinors, which are in bijective correspondence with the supersymmetries of a (bosonic) background, reduce to geometric Killing spinors.
To any supersymmetric supergravity background one can associate a Lie superalgebra, called the Killing superalgebra. This is the superalgebra generated by the Killing spinors; that is, if we let K = K0 ⊕ K1 denote the Killing superalgebra, then
K1 = {Killing spinors}
and
K0 = [K1,K1]
This is a Lie superalgebra, due to the odd-odd Lie bracket being symmetric, as is typical in lorentzian signature in the physically interesting dimensions.
I gave a triangular seminar in London about this topic and you can find slides here:
21) José Figueroa-O'Farrill, Killing superalgebras in supergravity, talk at University of London, February 27, 2008, available at http://www.maths.ed.ac.uk/~jmf/CV/Seminars/KSA.pdf
There is some overlap with the one in Leeds, but not too much.
Cheers, José
As you know, there is a nice triality symmetric construction of E8 that starts from SO(8)×SO(8). But, considering the maximally split real form E8(8), did you also know that this SO(8)×SO(8) is best regarded as SO(8,C+), with C+ being the split-complex numbers with i2=+1? There also are 56-dimensional real subgroups such as SO(8,C) (2 different embeddings - "IIA" and "IIB") - and there also is SO(8,C0).Basically, the way this works is that you can extend SO(8)×SO(8) to SO(16) or SO(8,8) - depending on whether you add the V×V or S×S 8×8-block. But if you take diagonal SO(8) subgroups, then the 8×8 all split into 28+35+1, and you can play nice games with these 28's...
See:
22) T. Fischbacher, H. Nicolai and H. Samtleben, Non-semisimple and complex gaugings of N = 16 supergravity, available as arXiv:hep-th/0306276.
© 2008 John Baez
baez@math.removethis.ucr.andthis.edu
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