- #1

John Baez

Also available at http://math.ucr.edu/home/baez/week262.html

March 29, 2008

This Week's Finds in Mathematical Physics (Week 262)

John Baez

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, nitrogen-vacancy pairs in diamonds, graphene transistors,

a new construction of E8, and a categorification of 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!

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 "week162".

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, a "lattice" is technically

a discrete subgroup of R^n that's isomorphic to Z^n. Dn+ is

only a lattice 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 D3 lattice 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

pi/3 sqrt(2) ~ .74

The D3+ pattern, on the other hand, has a density of just

pi sqrt(3)/16 ~ .34

This is why ice becomes denser when it melts: like diamond,

it's arranged in a D3+ pattern.

(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 x 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,F_{32}) inside E8, and the centralizer of a certain

element of SL(2,F_{32}) 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) Jose 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 psi such that

D_v psi = k v psi

for some constant k for every vector field v. Here D_v psi

is the covariant derivative of psi in the v direction, while

v psi 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 tensor S -> S

has a kind of adjoint

S tensor S -> V

This let's 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 let's 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 tensor 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 tensor 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 S^7 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 S_8^+.

The Jacobi identity holds, and you get a Lie algebra

so(8) + S_8^+

But then, thanks to triality, you knock ourself on the head and

say "I could have had a V_8!" After all, up to an outer

automorphism of so(8), the spinor representation S_8^+ is the

same as the 8-dimensional vector representation V_8. So, your

Lie algebra is just the same as

so(8) + V_8

with its 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 S^8 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 S_9. The Jacobi

identity holds, and you get a Lie algebra structure on

so(9) + S_9

This gives the exceptional Lie algebra f4!

Finally, if you take S^{15} 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

S_{16}^+. The Jacobi identity holds, and you get a Lie algebra

so(16) + S_{16}^+

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 S^7, S^8 and S^{15} all

show up in the Hopf fibration associated to the octonionic projective

line:

S^7 -> S^{15} -> S^8

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) + S_{10} + u(1)

and

e7 = so(12) + S_{12}^+ + su(2)

These would need to be trickier, with the u(1) showing up from

the fact that S_{10} is a complex representation, and the su(2)

showing up from the fact that S_{12}^+ 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:

18) Aaron Lauda, A categorification of quantum sl(2), available as

arXiv:0803.3848.

Aaron Lauda, Categorified quantum sl(2) and equivariant cohomology

of iterated flag varieties, available as arXiv:0803.3652.

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...

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

Previous issues of "This Week's Finds" and other expository articles on

mathematics and physics, as well as some of my research papers, can be

obtained at

http://math.ucr.edu/home/baez/

For a table of contents of all the issues of This Week's Finds, try

http://math.ucr.edu/home/baez/twfcontents.html

A simple jumping-off point to the old issues is available at

http://math.ucr.edu/home/baez/twfshort.html

If you just want the latest issue, go to

http://math.ucr.edu/home/baez/this.week.html

March 29, 2008

This Week's Finds in Mathematical Physics (Week 262)

John Baez

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, nitrogen-vacancy pairs in diamonds, graphene transistors,

a new construction of E8, and a categorification of 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!

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 "week162".

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, a "lattice" is technically

a discrete subgroup of R^n that's isomorphic to Z^n. Dn+ is

only a lattice 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 D3 lattice 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

pi/3 sqrt(2) ~ .74

The D3+ pattern, on the other hand, has a density of just

pi sqrt(3)/16 ~ .34

This is why ice becomes denser when it melts: like diamond,

it's arranged in a D3+ pattern.

(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 x 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,F_{32}) inside E8, and the centralizer of a certain

element of SL(2,F_{32}) 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) Jose 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 psi such that

D_v psi = k v psi

for some constant k for every vector field v. Here D_v psi

is the covariant derivative of psi in the v direction, while

v psi 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 tensor S -> S

has a kind of adjoint

S tensor S -> V

This let's 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 let's 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 tensor 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 tensor 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 S^7 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 S_8^+.

The Jacobi identity holds, and you get a Lie algebra

so(8) + S_8^+

But then, thanks to triality, you knock ourself on the head and

say "I could have had a V_8!" After all, up to an outer

automorphism of so(8), the spinor representation S_8^+ is the

same as the 8-dimensional vector representation V_8. So, your

Lie algebra is just the same as

so(8) + V_8

with its 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 S^8 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 S_9. The Jacobi

identity holds, and you get a Lie algebra structure on

so(9) + S_9

This gives the exceptional Lie algebra f4!

Finally, if you take S^{15} 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

S_{16}^+. The Jacobi identity holds, and you get a Lie algebra

so(16) + S_{16}^+

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 S^7, S^8 and S^{15} all

show up in the Hopf fibration associated to the octonionic projective

line:

S^7 -> S^{15} -> S^8

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) + S_{10} + u(1)

and

e7 = so(12) + S_{12}^+ + su(2)

These would need to be trickier, with the u(1) showing up from

the fact that S_{10} is a complex representation, and the su(2)

showing up from the fact that S_{12}^+ 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:

18) Aaron Lauda, A categorification of quantum sl(2), available as

arXiv:0803.3848.

Aaron Lauda, Categorified quantum sl(2) and equivariant cohomology

of iterated flag varieties, available as arXiv:0803.3652.

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...

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http://math.ucr.edu/home/baez/twfcontents.html

A simple jumping-off point to the old issues is available at

http://math.ucr.edu/home/baez/twfshort.html

If you just want the latest issue, go to

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