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

March 30, 2008

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

Posted by John Baez

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In week262 of This Week’s Finds, see the Southern Ring Nebula and the frosty dunes of Mars:

Then read about quantum technology in Singapore, atom chips, graphene transistors, nitrogen-vacancy pairs in diamonds, a new construction of $e_8$ , and a categorification of quantum $sl(2)$ .

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Whenever I write This Week’s Finds, I come up with a huge list of questions that I don’t know the answers to. I just realized I can get help from you! Here are some things I’d love to know:

What’s the coolest thing people have done so far with atom chips? Do all these things involve Bose–Einstein condensates?
What’s the coolest thing people have done so far with graphene? Why is it so much trickier to get carbon to act like a semiconductor than silicon?
What’s the coolest thing people have done so far with nitrogen-vacancy clusters in diamonds? What are “platelets” in diamonds really like?
What’s the state of the art in spintronics? I hear it’s already being used commercially for some applications. Like what, exactly?
Does diamond ever melt, or does it turn to graphite first as you heat it, regardless of the pressure? How much do people know about the phase diagram of carbon at high temperatures and pressures?
What’s the precise relation between Killing spinors and supergravity (or superstring) backgrounds? Does this relation shed light on Figueroa-O’Farrill’s construction of $e_8$ ?
Does Figueroa-O’Farrill’s construction ever give Lie superalgebras when the bracket of Killing spinors is symmetric?
How does Aaron Lauda’s new work fit into the current state of the art in Khovanov homology?

Here’s a more detailed view of the frosty dunes of Mars:

Posted at March 30, 2008 3:12 AM UTC

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37 Comments & 1 Trackback

Re: This Week’s Finds in Mathematical Physics (Week 262)

Can’t offer any intelligent comment on the content of the article(!), but I spotted a couple of typos:

* The paragraph starting “But regardless of whether anyone…” is duplicated, once with links and once without.

* “…you knock ourself on the head…”

Posted by: stuart on March 30, 2008 6:38 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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Thanks for catching those errors!

I can explain “You knock ourself on the head” — it’s because I’m constantly changing my mind about whether a given passage should be phrased in terms of me, you, we or the dull and anonymous one.

I like to use ‘you’ to put the reader on the spot — especially when there’s a little calculation I hope you can do on your own. I like to use ‘we’ to encourage a spirit of ‘working together on a tough problem’. Working out this particular Killing superalgebra is more of a ‘we’ sort of thing, since I don’t expect most readers can do these calculations on their own. But then I realized I needed to build up to the all-important joke: “You could have had a $V_8$ ”. I’ve been waiting for years to use this joke.

(There’s also another version: “You could have had an $E_8$ .”)

Posted by: John Baez on March 30, 2008 11:51 PM | Permalink | Reply to this

Melted Diamonds; Re: This Week’s Finds in Mathematical Physics (Week 262)

Scientists Melt Diamond

By Andrea Thompson, LiveScience Staff Writer

posted: 06 November 2006 03:05 pm ET

So much for “diamonds are forever.” Scientists at Sandia National Laboratories have taken diamond, the hardest known natural material on Earth, and melted it into a puddle.

Diamond isn’t easy to melt, which is why the scientists used Sandia’s Z machine, the world’s largest X-ray generator, to subject tiny squares of diamond, only a few nanometers thick, to pressures more than 10 million times the atmosphere’s pressure at sea level.

“It’s very difficult to reach those pressures,” said Marcus Knudson, a Sandia experimenter.

To create the pressure, the machine’s magnetic fields hurled small plates at the diamond at 34 kilometers per second (21 miles per second), or faster than the Earth orbits the Sun.

Researchers were investigating how the diamond reacted to a range of extreme pressures to see if it could be used to encase BB-sized fuel pellets needed to drive a nuclear fusion reaction.

Nuclear fusion occurs when multiple nuclei combine to make one heavier nucleus. If lighter elements are used, the reaction can create tremendous amounts of energy, but scientists are still learning how to manipulate and control fusion. (All current nuclear reactors harness the energy from fission reactions, where an atom splits into two or more smaller nuclei.)

To get a controlled fusion reaction, whatever material is surrounding the pellet must transmit any pressure applied evenly to the fuel inside to force it to implode. To do this, the material must either stay a solid or melt to a liquid–a mixture would create instabilities that could fail to compress the material enough and therefore “kill” the implosion, Knudson told LiveScience.

Currently, beryllium is being used to encase the pellets, but diamond is being considered as an alternate material because of problems with the beryllium leaking.

Posted by: Jonathan Vos Post on March 30, 2008 7:29 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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What’s the precise relation between Killing spinors and supergravity (or superstring) backgrounds?

It’s a generalization of how in ordinary gravity, a Riemannian manifold solving the equations of motion has one “preserved symmetry” per Killing vector it has.

Generically a solution has no Killing vectors. This can be read as saying that it does not “preserve any of the symmetries”. On the opposite side, if the solution is something like flat Minkowski space, it has a Killing vector for each traslation and each rotation. The brackets of these reproduce the Poincaré algebra. So flat Minkowski space “preserves all the symmetries”.

When in supergravity the Poincaré or Lorentz group is replaced by its super version, it makes sense to ask how much of that superized symmetry algebra a given solution preserves. As before, Killing vectors come from translational and rotational global symmetries. But now there are also the “supersymmetries” and preserving one of them means having a Killing spinor.

The way this is derived is entirely analogous to the bosonic case: you vary the action functional and check for which variation parameters the result vanishes. Some of the possible variations are now indexed by spinorial quantities, and the variation of the action functional by these is typically given by a kind of covariant derivtive of these spinorial quantities. Setting that to zero is tantamount to demanding that these spinorial quantities are actually Killing spinors with respect to the background about which the variation takes place.

Some 20 years ago people started thinking that concentrating on solutions of supergravity which have precisely one supersymmetry preserved (have one Killing spinor) should be helpful for making contact with phenomenology. If in addition one requires that the 10-dimensional manifold is a direct product of 4-dimensjonal Minkowski space with a compact 6-fold, this leads to the requirement that the compact manifold is a Calabi-Yau.

A comparatively useful quick summary of all this is here: Calabi-Yau Compactifications in Perturbative String Theory.

Posted by: Urs Schreiber on March 31, 2008 7:19 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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Thanks for the nice pedagogical reply, Urs!

The idea of ‘Killing vectors and Killing spinors as infinitesimal supersymmetries of a solution of supergravity’ is a nice generalization of ‘Killing vectors as infinitesimal symmetries of a solution of general relativity’.

However, this analogy makes me naively expect that taking all the Killing vectors and Killing spinors together, we automatically get a Lie superalgebra of ‘infinitesimal supersymmetries’.

Is that true in supergravity?

In Figueroa-O’Farrill’s setup, the bracket of Killing spinors can be skew-symmetric (instead of symmetric, as one would expect for a Lie superalgebra). Furthermore, the (super)Jacobi identity is not automatically valid. What’s up?

Of course, Figueroa-O’Farrill is not talking about solutions of supergravity — just Riemannian manifolds equipped with spin structure. That’s enough to define a notion of Killing spinor — or actually (here’s another wrinkle) one notion of Killing spinor for each real constant $k$ . Maybe this is part of the problem.

Is his bracket of Killing spinors the same as the one that shows up in supergravity?

Posted by: John Baez on March 31, 2008 6:55 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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I’d need to remind myself of some details of the computations.

My first guess is that what I called Killing spinors are spinors which are covariantly constant on the nose, $D \psi = 0$

(where, however, the Dirac operator $D$ may contain more contributions than just that of a Levi-Civita connection: it will generally contain torsion terms coming from the Kalb-Ramond field and possibly higher “ $n$ -form” twistings coming from the Ramond-Ramond fields).

You were mentioning the condition $D_v \psi = k v \psi$ , which is more general. In what I said we have $k= 0$ (always, I think).

For $k= 0$ , I’d certainly think that the infinitesimal symmetries of sugra solutions which I mentioned form a super Lie algebra: because that must follow in complete analogy to how the inf. symmetries of a solution in ordinary gravity form a Lie algebra.

But I realize that my supergravity is rusty…

Posted by: Urs Schreiber on March 31, 2008 7:40 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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In Figueroa-O’Farrill’s setup, the bracket of Killing spinors can be skew-symmetric

This particular aspect, I suppose, is an example of the fact that it is not unusual that given a $\mathbb{Z}/2$ -graded vector space, it might carry a grading-preserving skew bracket – in which case it is just a graded Lie algebra – and/or it might carry a “graded-skew” bracket, in which case it is a Lie superalgebra.

To get a feeling for how non-unusual this is I like to look at the survey Polyvector super-Poincaré-algebras, which classifies both kinds of structures in parallel.

And I suppose that it is this aspect which makes $e_8$ look very much like a Lie super-algebra without actually being one.

Maybe remarkably, graded Lie algebras are, as we know, closely related to categorified Lie algebras. I keep having the feeling that there should be a certain convergence of the concepts of “categorification” and “superification” in certain domains.

Still not much more than a feeling so far. But it is noteworthy that branes made their original appearance in supersymmetric field theory, arising from central extensions of superalgebras by “polyvector” parts. A good part of the entire lore about branes is just analysis of super Lie algebras. This is a remarkable deep fact, whose full $n$ -categorical interpretation ought to be better understood eventually.

(Well, as I said elsewhere, I think that Castellani’s result goes a long way towards understanding this: the Polyvector-extended super Lie algebras are Lie algebras of inner derivations of certain $L_\infty$ -algebras…)

Posted by: Urs Schreiber on March 31, 2008 8:12 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

Urs wrote:

given a Z/2-graded vector space, it might carry a grading-preserving skew bracket; in which case it is just a graded Lie algebra; and/or it might carry a graded-skew bracket, in which case it is a Lie superalgebra.

sorry - you are losing me - you seem to be mixing two or more? things
maybe you’ll have to resort to formulas

graded Lie bracket preserves the grading and in skew in the graded sense
why isin’t that Lie super if the grading is Z/2

on the other hand, we might have an odd bracket e.g. shifting the grading by 1
did you have that distinction in mind?

Posted by: jim stasheff on March 31, 2008 10:27 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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I just mean that we may have a $\mathbb{Z}/2$ -graded bracket which is still a properly skew bracket for all arguments (picks up a minus sign when any two of its arguments are exchanged). This is not super, just graded.

Posted by: Urs Schreiber on March 31, 2008 10:51 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

Isn’t that a distinction without a difference?

Posted by: jim stasheff on April 3, 2008 2:35 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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What I mean is that the difference is in whether or not the bracket is antisymmetric everywhere or not.

An ordinary Lie algebra $g$ may happen to be $\mathbb{Z}/2$ -graded in that $g = g_0 \oplus g_1$ and the ordinary skew-symmetric (not graded skew symmetric) $[x,y] = - [y,x] \,, \forall x,y \in g$ Lie bracket respects that grading $[\cdot,\cdot] : g_0 \times g_0 \to g_0$ $[\cdot,\cdot] : g_0 \times g_1 \to g_1$ $[\cdot,\cdot] : g_1 \times g_1 \to g_0 \,.$

You can think of this as a Lie algebra internal to the symmetric braided monoidal category of $\mathbb{Z}_2$ -graded vector spaces, where the symmetric braiding is taken to be the trivial one.

In contrast to that is a Lie superalgebra, for which instead $[x,y] = - (-1)^{|x| |y|}[y,x] \,.$

A Lie superalgebra can be regarded as a Lie algebra internal to the symmetric monoidal category of $\mathbb{Z}_2$ -graded vector spaces, where now the braiding is taken to be the unique non-trivial symmetric one, namely the one which sticks in a minus sign when two odd graded vector spaces are interchanged.

For example, a long list (even a classification) of both kinds (plain $\mathbb{Z}_2$ -graded as well as super) Lie algebras extending the Poincaré Lie algebra is given here.

Other examples are in José Figueroa-O’Farrill’s work, nice reviews of which John posted links to here.

See for instance around slide 31.

Well, you knew all this and I was just expressing myself badly, I guess.

Posted by: Urs Schreiber on April 3, 2008 3:23 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

oh, OK now
“but I’ve got to use words when talking to you” (TSE)

perhaps graded Lie vs Lie graded algebra?

Warning! Similar problem with Lie n-algebra
and n-Lie algebra
I’m not sure which is used for the algebras with just a single n-ary `bracket’ satisfying one of the two generalizations of Jacobi e.g. Nambu

Posted by: jim stasheff on April 4, 2008 2:32 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

John Baez wrote:
> How does Aaron Lauda’s new work fit into
> the current state of the art in Khovanov
> homology?

How does Lauda’s work relate to the paper by Hao Zhang, Categorification of Integrable representations of Quantum groups, ArXiv 0803:3668 which came out last week?

Posted by: Maarten Bergvelt on March 31, 2008 5:37 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

…the paper by Hao Zhang…

is described as “a very recent striking work” in yet another paper by Lauda, this one with Khovanov - A diagrammatic approach to categorification of quantum groups I.

Posted by: David Corfield on March 31, 2008 7:06 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

Goodness me, Aaron has just released three papers in three consecutive days.

Posted by: Bruce Bartlett on March 31, 2008 8:14 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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The sphere packing problem originated in trying to stack cannonballs on ships in such a way so they occupied as little volume as possible since space was at such a premium.

I have just a few things about some previous stuff you’ve written recently.

In the paper called Rosetta stone, and your previous article trying to unite quantum mechanics and general relativity by pointing out they both use categories, all of the manifolds that you use in your illustrations are disjoint unions of various numbers of circles, which can then merge, separate, shrink to nothing, etc.

S^1 U S^1 -> S^1 U S^1 U S^1

To what extent was this choice made for ease of illustration purposes? What other types of manifolds are allowed to change into each other? If you take a circle, and take a radius to infinity, you end up with a line. You can think of a line as a circle with infinite radius.

S^1 -> R^1

You could start with a torus, if you take one of its radii, and take it to zero, you end up with a circle.

T^2 -> S^1

If you take it’s other radius, and take it to infinity, you get an infinite cylinder.

T^2 -> R^1 x S^1

Are you allowed to turn a sphere into a klein bottle? Are you allowed to use the worldsheet operator Omega which is supposed to change the orientation of the worldsheet?

Then later on in the paper when you were talking about logic, it reminded me of something I had seen on television during the impeachment of Bill Clinton. Juanita Broderick signed an affidavit saying that she was not raped by Bill Clinton, and later recanted the affidavit, saying she was pressured to sign it.

http://www.slate.com/id/1002010/

I saw Lanny Davis, of one Clinton’s fanatical supporters, on television, and I swear he used the following chain of reasoning.

Juanita Broderick was raped by Bill Clinton.
Juanita Broderick said she was not raped by Bill Clinton.
Therefore, Juanita Broderick is a liar.
Juanita Broderick is a liar.
Juanita Broderick said she was raped by Bill Clinton.
Therefore, Juanita Broderick was not raped by Bill Clinton.

That is literally the chain of reasoning he used to prove that Juanita Broderick was not raped by Bill Clinton. I am not making this up! The conclusion of the first soliphism was a premise in the second soliphism. The premise of the first soliphism was the exact opposite of the conclusion of the second soliphism. In other words, he literally manages to use a statement to prove its opposite.

In the law of the excluded middle, you assume

not not A = A

You can drop this law of the excluded middle although it’s often makes the proof more transparent since it’s intuitively true. In Lanny Davis logic, you have

not A = A

How would you describe a system of logic based on that statement?

Posted by: Jeffery Winkler on March 31, 2008 7:07 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

not A = A

How about “in inconsistent systems every statement is true”? (try the various cases of A being true or not, excluded middle or not)

Posted by: David Roberts on April 1, 2008 1:38 AM | Permalink | Reply to this

unreliable narrator; Re: This Week’s Finds in Mathematical Physics (Week 262)

From Grace Paley’s last book of poems
(Fidelity, Farrar Strauss and Giroux, 84 pp):

believe me I am
an unreliable
narrator no story
I’ve ever told
was true many people
have said this before
but they were lying

Posted by: Jonathan Vos Post on April 1, 2008 6:24 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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Jeffrey wrote:

In the paper called Rosetta stone, and your previous article trying to unite quantum mechanics and general relativity by pointing out they both use categories, all of the manifolds that you use in your illustrations are disjoint unions of various numbers of circles […] To what extent was this choice made for ease of illustration purposes?

It was partially to remind people of string theory, but mainly because 2d cobordisms are really easy to draw, while higher-dimensional ones require special tricks (e.g. the Kirby calculus).

What other types of manifolds are allowed to change into each other?

The technical way to phrase this puzzle is: which manifolds are cobordant to one another?

All compact oriented 1d manifolds are cobordant to another. Why? Easy: they’re all finite disjoint unions of circles, and there’s a cobordism between the empty set to a single circle.

All compact oriented 2d manifolds are cobordant to another. Why? Because they’re all finite disjoint unions of $n$ -handled tori, and there’s a cobordism between the empty set and the 2-sphere (which is a 0-handled torus), and also a cobordism between the $n$ -handled torus and the $(n+1)$ -handled torus. All this is fun to ponder if you haven’t yet.

All compact oriented 3d manifolds are cobordant to another, but this takes more work to show!

When we hit dimension 4 we really must distinguish between topological manifolds and smooth manifolds (which still are equivalent to piecewise linear manifolds in this dimension). Let’s consider compact oriented smooth manifolds and compact oriented smooth cobordisms between them.

Not all compact oriented smooth 4-manifolds are cobordant! In particular, the empty set is not cobordant to $\mathbb{C}P^2$ . The reason is that there’s a cobordism invariant, the ‘first Pontryagin number’, which differs for these two 4-manifolds.

Thom and Pontryagin founded cobordism theory to tackle questions such as these. A key concept was to take the set of cobordism classes of manifolds and massage it to get a a group. It’s not hard. Cobordism classes of $n$ -manifolds form a commutative monoid with disjoint union as the addition; if we formally throw in inverses, we get a group called a cobordism group.

To compute this group we first need to say precisely what kind of manifolds and what kind of cobordisms we’re talking about; above I’ve been using ‘compact smooth oriented’ manifolds and cobordisms. It then turns out that the group we get, which of course depends on $n$ , is $\pi_n(MSO)$ for some space $MSO$ , called the Thom space of the bundle $ESO \to BSO$ . Using tricks which Jim Stasheff knew back when I was still learning my times tables, these groups can be calculated. For starters, the cobordism group of compact oriented 4-manifolds is $\mathbb{Z}$ , with the class of $\mathbb{C}P^2$ as a generator.

A nice fact is that these groups fit together in a graded ring. Why? Because we can take the cobordism class of an $n$ -manifold $M$ and the class of an $m$ -manifold $N$ and ‘multiply’ them to get the class of the $(n+m)$ -manifold $N \times M$ .

Anyway, there’s much more to say about this business, only some of which I actually know. The first really cool thing to learn is the Thom–Pontryagin construction, which also plays a key role in something I call the Cobordism Hypothesis. A very readable introduction is Modern Geometry: Methods and Applications — Part 3: Introduction to Homology Theory by Dubrovin, Fomenko and Novikov. It has some mistakes in it, but frankly I’d rather have a book I can read with some mistakes in it than a perfectly correct, perfectly unreadable tome.

Posted by: John Baez on April 1, 2008 7:37 PM | Permalink | Reply to this

Killing spinors

Dear Professor Baez,

Can you see any value in trying to do a similar thing as you mention for e8 and f4, but starting with conformal Killing spinors (whose squares preserve the metric up to a scale factor) instead of just “normal” Killing spinors?

Posted by: anon on April 1, 2008 5:33 AM | Permalink | Reply to this

Re: Killing spinors

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I don’t have enough intuition on this subject to know exactly what would happen if you used conformal Killing spinors. But, if you’re the sort who can do calculations like this, it might be good.

After all, it’s hard to go wrong studying the geometry of spheres. The concept of ‘Killing vector’ has been around a long time, and Killing spinors have been around for a while too. So, I’m sort of amazed that only last year did someone study the algebraic structure that Killing vectors and Killing spinors naturally form, work out this structure in detail for spheres, and observe that $E_8$ jumps out when we try a 15-dimensional sphere!

(Perhaps part of the reason is that, as Urs mentioned, Figueroa-O’Farrill needed a slight generalization of the simplest sort of Killing spinor. So: you may need a similar generalization of ‘conformal Killing spinor’ to get something interesting to emerge!)

The group of conformal transformations of the sphere $S^n$ is the connected component of $SO(n+1,1)$ . So, the conformal Killing vector fields on the sphere $S^n$ form the Lie algebra $so(n+1,1)$ . So, if you did the calculation you’re suggesting, you’d find some sort of ‘superalgebra’ including this Lie algebra. That reminds me of these papers:

Gunaydin, Koepsell and Nicolai, Conformal and quasiconformal realizations of exceptional Lie groups, Commun. Math. Phys. 221 (2001), 57-76, also available as hep-th/0008063.
Murat Gunaydin, Generalized conformal and superconformal group actions and Jordan algebras, Mod. Phys. Lett. A8 (1993), 1407-1416. Also available as hep-th/9301050.

I believe the first paper may give some examples of exceptional Lie algebras containing certain conformal Lie algebras $so(n+1,1)$ . I explained a little of that paper in week193.

Happy hunting!

I got a nice email about this stuff from Figueroa-O’Farrill himself, and I’ll ask his permission to post it as a comment here.

Posted by: John Baez on April 1, 2008 9:31 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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Here is an email from José Figueroa-O’Farrill, which he has given me permission to post here:

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.

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

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 $L \times R$ , 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 $K= K_0 \oplus K_1$ denote the Killing superalgebra, then

$K_1 = {Killing spinors}$

and

$K_0 = [K_1,K_1]$

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:

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

Posted by: John Baez on April 3, 2008 5:11 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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Thanks for the links! These pdf notes by José Figueroa-O’Farrill are really nice. I should spend more time thinking about this stuff…

If I understand his notes correctly, they confirm my reply above, but I should check:

while the symmetry Lie superalgebra of any supergravity solution is in fact that: a Lie superalgebra, the potential problem with the Killing spinors on those spheres is that, while they do exist, they might not be the super-symmetries of any supergravity solution (or anything else, for that matter), hence are not guaranteed to form a Lie superalgebra.

In particular, the Killing spinors arising in supergravity, as on slides 9 and 10 are not, as I mentioned, the type of “geometric” Killing spinors appearing on slide 30:

while the sugra Killing spinors may have inhomogenity terms in their defining equations coming from contraction with higher differential forms $F$ , $H$ ,

$\nabla_X \psi = \iota_X F \cdot \psi + \cdots \,,$ (slide 9)

they don’t have the particular dependence $\nabla_X \psi = \lambda X \cdot \psi$ (slide 30)

of “geometric” Killing spinors (unless $\lambda = 0$ and $F = 0$ ).

Put differently: for the Killing spinors arising in supergravity we do know (by construction!) that they are part of the automorphism Lie superalgebra of something, because that’s how we find them.

For the geometric spinors on the sphere we do not know a priori if they are part of the automorphism Lie superalgebra of anything, therefore it is a nontrivial task to check if they indeed do form either a Lie superalgebra or a $\mathbb{Z}_2$ -graded Lie algebra.

I suppose that’s what José Figueroa-O’Farrill means when he says, as quoted by you above:

One satisfactory way to achieve this would be to understand of what in, say, the 15-sphere is E8 the automorphisms [of(?)].

Posted by: Urs Schreiber on April 3, 2008 7:28 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

Urs Schreiber:”I suppose that’s what José Figueroa-O’Farrill means when he says, as quoted by you above:

One satisfactory way to achieve this would be to understand of what in, say, the 15-sphere is E8 the automorphisms [of(?)].”

Urs,John, would you please to explain what such journalistic sentences have to do with math?

Regards, Dany.

P.S.“You could have had a V 8 ”: A. Einstein mentioned that being student he suffered to survive the lectures by H. Minkowski, but run off from his another math teacher A. Hurwitz.

Posted by: Daniel Sepunaru on April 5, 2008 8:40 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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I suppose that’s what José Figueroa-O’Farrill means when he says, as quoted by you above:

One satisfactory way to achieve this would be to understand of what in, say, the 15-sphere is E8 the automorphisms [of(?)].

Urs, John, would you please to explain what such journalistic sentences have to do with math?

I am not sure I understand what you are complaining about, but I can try to say more about what a sentence such as

[…] what in, say, the 15-sphere is $E_8$ the automorphisms of.

is referring to. Namely this:

many groups which one encounters come to us as automorphism groups.

Roughly this means: groups of admissable transformations on a given object which leave this object invariant.

More precisely it means (lest you have to complain about journalistic style again): groups of invertible endomorphisms of a given object in a given category.

The most basic example is the symmetric group $S_n$ , for any integer $n$ : this is the group of autmorphisms of the set with $n$ -elements (where for sets, automorphisms are the bijective maps from the set to itself).

This is an example where a group is defined as the automorphism group of something (a finite set in this case).

The situation which we were talking about is converse to this: it may happen that we have in our hands a group and don’t know what it is the group of automorphisms of.

This happened, famously, for example with the monster group:

that was a group known to exist before anyone knew what it would be the automorphism group of. Then later it was found that the Monster group is the group of automorphisms of the Griess algebra and still later that it is also the automorphis group of a vertex operator algebra called the Monster module.

So far these examples are taken from the finite groups.

A completely analogous discussion can be considered for other kinds of groups, notably for Lie groups. This are the kind of groups that we were talking about here.

Lie groups are to a great extent characterized by their Lie algebras, which are essentially the tangent spaces at the identity element of the manifold of elements of the Lie group.

Now, José Figueroa-O’Farrill finds that to the 15-sphere there is, in a natural way, associated the Lie algebra called $e_8$ , which belongs to the exceptional Lie group called $E_8$ .

While it comes from the 15-sphere “in a natural way” (for more details see his lecture notes), it does not, in his derivation, appear as the automorphism group of any structure related to the 15-sphere.

Therefore it is a natural question: which structure related to the 15-sphere might $E_8$ be the automorphism group of?

I suppose it is clear now that this is a question more of mathematical than of journalistic interest.

Posted by: Urs Schreiber on April 5, 2008 10:02 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

Urs Schreiber:” I am not sure I understand what you are complaining about”

I complaining that you (and JB) apparently support the bizarre style of J. Figueroa-O’Farrill presentation. For example: “2007 will be known as the year where E8 (and Lie groups) went mainstream…” Lie groups became main stream in math-ph about 100 years ago (E.P. Wigner and H. Weyl). Exceptional Lie groups and their automorphisms were intensively investigated about 30 years ago (H. Harari, F. Gursey et al, P.Ramond and I.Bars). You missed the point since I ask to explain how that style of presentation complies with the standard way of doing math-ph: one formulate the assumptions and demonstrate the consequences of his assumptions which must have mathematically supported proof.

Regards, Dany.

P.S. Urs Schreiber:” Therefore it is a natural question: which structure related to the 15-sphere might E 8 be the automorphism group of?”

Without intention to annoy you again, in what sense you use word “natural”?

Posted by: Daniel Sepunaru on April 5, 2008 1:22 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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Without intention to annoy you again, in what sense you use word “natural”?

In the everyday sense. It is the kind of question one feels like asking here in order to understand the situation a bit better. Would you disagree?

the bizarre style of J. Figueroa-O’Farrill presentation

The presentation in question are the slides he used for a talk, right? Or is this about a research article?

I found his slides informative and readable, especially as supplementary reading for John Baez’s latest column that the present discussion is about, and I do not feel that it is at me to approve or disapprove anyone’s choice of presentation for talk slides.

And it seems to be a fact that 2007 had an unprecendented frequency of $E_8$ appearing in mass media. I took it that this was all that was meant. I can’t see anything wrong with lightning up a talk by mentioning some trivia like that, though I am aware that tastes about lecture style do differ greatly.

Posted by: Urs Schreiber on April 5, 2008 3:04 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

Urs Schreiber:”In the everyday sense. It is the kind of question one feels like asking here in order to understand the situation a bit better. Would you disagree?”

I agree completely. However, I feel comfortable with S3 (actually with {+,-,-,-}) and have personal troubles already with S7.

Urs Schreiber:”The presentation in question are the slides he used for a talk, right? Or is this about a research article?”

I meant 0706.2829 as well.

Urs Schreiber:”And it seems to be a fact that 2007 had an unprecendented frequency of E 8 appearing in mass media.”

That is my point. We don’t need help of mass media to do math-ph (compare with EPR story). Would you disagree?

Regards, Dany.

Posted by: Daniel Sepunaru on April 5, 2008 4:42 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

$MathML-enabled post (click for more details).$

We don’t need help of mass media to do math-ph (compare with EPR story). Would you disagree?

I certainly agree that we don’t need it scientifically.

Worse, if any possible connection to phenomenological physics is involved (as partly true in the $E_8$ case and also in other cases which are more “formal hep-th” than math-ph) the mass media treatments tend to be highly misleading or outright absurd.

So now I am getting the point: let me assure you that I didn’t interpret the phrase “2007 will be known as the year when $E_8$ became mainstream” as meant in any way as supportive evidence for any mathematical claim or the like. I took those newspaper articles mentioned to be just a, potentially entertaining, piece of trivia.

(I might have trained a higher tolerance for such things from looking at String theory slides whose authors find it amusing to show even weirder (no bound on weirdness there) mass media articles at the beginning, such as advertisements for underwear :-/)

Posted by: Urs Schreiber on April 5, 2008 5:17 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

If the mass media convey the idea that math-physics is worthwhile
even if they don’t get the story quite right
that may indeed help us to continue our work
except for thsoe who are already independently wealthy

Posted by: jim stasheff on April 6, 2008 4:23 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

I guess I have an obsolete version of “The Crackpot Index”. Perhaps it is worth to develop the

“20 points for talking about how great your theory is, but never actually explaining it.”

20 points for talking about mass media appearance of your theory.

20 points for referring to current/past media dude.

Regards, Dany.

Posted by: Daniel Sepunaru on April 6, 2008 6:12 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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I believe what you should do now is to say which of the statements that the discussion was about (such as statements about Killing spinors on spheres and exceptional Lie algebras I suppose?) you think are problematic, if that’s what it is you think.

Posted by: Urs Schreiber on April 6, 2008 8:41 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

Urs,

I believe that we already understood each other. I read very slow, therefore I should choose what is most interesting for me. I apologize, but I prefer now to look Classical vs Quantum Computation series which I missed entirely.

Regards, Dany.

Posted by: Daniel Sepunaru on April 6, 2008 5:25 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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Urs wrote:

One satisfactory way to achieve this would be to understand of what in, say, the 15-sphere is E8 the automorphisms [of(?)].

I have another irrelevant remark about this sentence. There’s already one ‘of’ in this sentence, Urs — it doesn’t need another. But, I know why you put it in there: there’s a common tendency to stick in another ‘of’ when the first one gets so far away that it gets forgotten.

Example: “Of which state is Baton Rouge the capital of?”

Or, more plausibly: “Of which state is Baton Rouge — that illustrious city whose name is French for ‘red stick’ (for reasons known to few, but easily found on Wikipedia) — the capital of?”

In either case, grammarians would tell you to remove the second ‘of’. Ordinary people would tell you to remove the first!

Posted by: John Baez on April 6, 2008 7:37 PM | Permalink | Reply to this

Propositions on Prepositions; Re: This Week’s Finds in Mathematical Physics (Week 262)

“Of which state is Baton Rouge the capital of?” might be grammatically equivalent to the phrase “in this ever changing world in which we live in” [“Live and Let Die”, Sir Paul McCartney] except that a previous blog comment corrected this line (which is how most Americans heard it) to “in this ever changing world in which we’re livin’”

This also relates to the dispute of whether or not Sir Winston actually said
“This is the sort of English up with which I will not put.”
Churchill on Prepositions

Posted by: Jonathan Vos Post on April 7, 2008 3:02 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

“You could have had a V 8 .”

I have another and final most off topic and irrelevant remark about this sentence.
Indeed above I referred to “Autobiographisches”. Now look on the Eq. (1) there and compare with L.D. Landau and E.M.Lifschitz “Field Theory”. Deduction:
math-ph must study math from the mathematicians but should be careful and don’t believe them since their notion of “natural” is different from ours (otherwise, he may find himself confined inside S15).

Regards, Dany.

P.S. I study that from my spiritual “father” V.A.Fock.

Posted by: Daniel Sepunaru on April 8, 2008 3:53 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 262)

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It’s been a long time, but some of you may remember that José Figueroa-O’Farrill showed the Lie algebra of $E_8$ can be seen as consisting of Killing vectors and Killing spinors on a 15-sphere… without, alas, getting a better proof of the Jacobi identity for this Lie algebra. I got an email today from some authors who seem to have made progress on the latter task:

Andrei Moroianu, Uwe Semmelmann, Invariant 4-forms and symmetric pairs.

Abstract. We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations whose second exterior power is irreducible or has an irreducible summand of codimension one, and we give a conceptual computation-free argument for the construction of the exceptional Lie algebras of compact type.

Posted by: John Baez on February 16, 2012 8:11 AM | Permalink | Reply to this

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