
This Week I'd like to talk about math books in Shanghai, and Urs Schreiber's blog entry on the gauge 3group of Mtheory. But first:
1) Greg Egan, Klein's quartic equation, http://gregegan.customer.netspace.net.au/SCIENCE/KleinQuartic/KleinQuarticEq.html
I discussed Klein's quartic curve in "week214" and "week215". The idea is to take the nontrivial complex solutions of
u^{3} v + v^{3} w + w^{3} u = 0
and "projectivize" them  in other words, count two as the same if one is just a multiple of the other:
(u',v',w') = c(u,v,w)
The result is a 3holed Riemann surface with the maximum number of symmetries! Here by a "symmetry" I mean a conformal transformation mapping the surface to itself. Back in 1893 Hurwitz proved something quite bizarre: an nholed Riemann surface can't have more than 84(n1) symmetries if n > 1. So, a 3holed Riemann surface can't have more than 168 symmetries  and Klein's quartic curve has exactly that many!
These 168 symmetries were constructed by Klein way back in 1879, but Egan gives an elementary proof that uses only algebra and a bit of calculus... and a lot of cleverness. And, his page has a wonderful spinning picture of the real solutions of
u^{3} v + v^{3} w + w^{3} u = 0.
This is what you see above.
As you can see, it consists of lots of lines through the origin, including the u, v, and w axes. When we "projectivize", we get one point for each of these lines, so we get a curve which is the real version of Klein's quartic curve. This curve has an obvious 3fold symmetry, from cyclically permuting the coordinate axes. The rest of the 168 symmetries are only easy to visualize when we go to the complex version  as Egan explains.
It's great that Egan can draw this thing in Perth and I can easily see it my apartment here in Shanghai  I feel like I'm living in a futuristic world, and I'm only 45. What it'll be like when I'm 64?
Another pleasant thing about life in Shanghai, at least for a welloff visitor from America, is how cheap everything is. It's clear why the US is running an enormous trade deficit: there's a vast economic differential. Stopping the flow of goods one way and dollars the other would be like damming the Niagara Falls.
For example, last night I saw this excellent hardcover book on sale for 78 yuan, or about $10:
2) Yu. I. Manin and A. A. Panchishkin, Introduction to Modern Number Theory, second edition, Science Press, 2005.
It's a great overview of number theory, from the basics through class field theory to Lfunctions, modular forms and the Langlands program! It's wisely divided into three sections: "problems and tricks", "ideas and theories", and "analogies and visions". Back when I used to hate number theory, I thought it was all problems and tricks. Now I'm beginning to learn some of the ideas and theories, and I hope eventually to grasp the analogies and visions discussed here  for example, the analogy between Arakelov geometry and noncommutative geometry.
$10 is a nice price for a math book. If you buy this one from Springer Verlag, you'll pay ten times that. Illegal knockoffs of Western books are common in China, but I think the one I saw is legal, since Springer has signed an agreement with Science Press, which is run by the Chinese Academy of Sciences. In exchange for letting Science Press publish Springer books in China at affordable prices, Springer gets to publish translations of Chinese journals in the West at unaffordable prices.
By the way  after checking out the bookstore, I went out to the street vendors and bought an excellent dinner of rice, sausage and vegetables for 35 yuan  about 40 cents US. It was cooked by a husband and wife in a wok on a cart.
Just after I bought it, someone yelled the Chinese equivalent of "cops!", and all the street vendors suddenly dashed away with their carts, leaving only the woman, who kindly handed me my dinner in a styrofoam pack before walking off. They clearly had this down to a fine art: it all happened faster than my brain could process. I guess the cops don't allow street vendors there.
I only wish I'd noticed: did the street vendors turn off their gas stoves before running, or run while still cooking?
Anyway, on to some serious math and physics.
You've probably heard of some mysterious thing called "Mtheory" that lives in 11 dimensions. Back in "week158" and "week159" I took a stab at understanding this. Now I'll try again, with a lot of help from Urs Schreiber:
3) Urs Schreiber, Castellani on free differential algebras in supergravity: gauge 3group of Mtheory, http://golem.ph.utexas.edu/string/archives/000840.html
Calling Mtheory a "theory" is a bit misleading, because nobody knows what this theory is! There's just a lot of clues pointing to its existence. It seems to be the quantum version of a welldefined classical field theory called "11dimensional supergravity". And, it seems to involve 2branes and 5branes: 2 and 5dimensional membranes that trace out 3 and 6dimensional surfaces in spacetime, just like strings trace out 2dimensional surfaces.
Back in "week158" I wrote down a Lagrangian for 11d supergravity. This is a truly monstrous thing involving three fields:
A) a frame field e  the "graviton",
B) a field ψ taking values in the real spin3/2 representation of the 11d Lorentz group  the "gravitino",
C) a 3form A.
When it was discovered back in 1978, people were interested in 11d supergravity mainly because it was the highestdimensional theory they could concoct that includes general relativity and supersymmetry  a symmetry that interchanges bosons and fermions, in this case gravitons and gravitinos  without including any particles of spin > 2. So, the fact that it looked like a mess wasn't such a big deal. But now that some people are taking it very seriously, it's worth trying to understand the math behind it more deeply, to see what makes it tick.
For example: what's so great about 11 dimensions? And: what's the reason for that 3form?
I'm not a huge fan of string theory, but I like puzzles of this sort  finding patterns that make certain things work only in certain dimensions, and stuff like that. So, I got intrigued when I learned that superYangMills theory and superstring theory are nice in dimension 10 because of special properties of the octonions  see "week104". Maybe a little extra stretch could bring us to dimension 11?
I got even more intrigued when I ran across two competing explanations for that 3form in 11d supergravity. One was that it's a connection on a twice categorified version of a U(1) bundle. The other was that it's the ChernSimons form for an E_{8} gauge theory.
Let me say a bit about what these means. I talked about categorified U(1) bundles in "week210", so I'll be sort of brief about those....
A connection on a U(1)bundle looks locally like a 1form, so we can integrate it along a path and compute how the phase of charged particle changes when we move it along that path:
f x>y a path f from the point x to the point y: we write this as f: x → yBelieve it or not, this is the basis of all modern ideas on electromagnetism!
If we categorify this whole idea once, we get a kind of connection that looks locally like a 2form. Folks call this a "connection on a U(1) gerbe", but don't let the use of French here intimidate you: they just do that so they can charge more for the wine. It's just a gadget that you can integrate over a surface, to compute how the phase of a charged string moves when we slide it along that surface:
f > /  \ x F y a pathofpaths F from the path f to the path g: \ \/ / we write this as F: f => g > gAnd, if we categorify once more, we get a "connection on a U(1) 2gerbe". This is something that looks locally like a 3form, which describes what happens when we move 2branes around!
If you're wondering why I'm talking about "categorifying", it's because this:
f x>yis also a picture of a morphism in a category, while this:
f > /  \ x F y \ \/ / > gis a picture of a 2morphism in a 2category and so on. We're talking about processes between processes between processes... so we're climbing up the ladder of ncategories.
Anyway: since 11d supergravity has a 3form in it, and Mtheory apparently has 2branes in it, maybe we need to categorify the concept of a U(1) bundle twice to understand what's going on here!
I came up with this crazy idea on my own back in "week158", but it's an obvious guess after you learn that the 2form field called B in 10d superstring theory really is a connection on a U(1) gerbe:
4) Alan L. Carey, Stuart Johnson and Michael K. Murray, Holonomy on Dbranes, available as arXiv:hepth/0204199.
Unfortunately there are some problems with naively pushing this idea up a dimension. For example, a crucial factor of 1/6 in the Lagrangian for 11d supergravity is not explained by thinking of A this way.
Another possible explanation was that this 3form is the ChernSimons form of an E_{8} bundle over spacetime:
5) Emanuel Diaconescu, Gregory Moore and Edward Witten, E_{8} gauge theory, and a derivation of Ktheory from Mtheory, Adv. Theor. Math. Phys. 6 (2003) 10311134. Also available as arXiv:hepth/0005090.
6) Emanuel Diaconescu, Daniel S. Freed and Gregory Moore, The Mtheory 3form and E_{8} gauge theory, available as arXiv:hepth/0312069.
This idea explains that factor of 1/6. And, it might move towards an explanation of how the octonions get into the act, because the group E_{8} is deeply related to the octonions. But as the authors of the above paper say, "the E_{8} gauge field plays a purely topological role and appears, in some sense, to be a 'fake'." In particular, you don't see any E_{8} connection staring you in the face in the Lagrangian for 11d supergravity that I wrote down in "week158".
Later, it started becoming clear that both ideas  the twice categorified U(1) connection and the E_{8} gauge theory  fit together in some way:
7) Paolo Aschieri and Branislav Jurco, Gerbes, M5brane anomalies and E_{8} gauge theory, JHEP 0410 (2004), 068. Also available as arXiv:hepth/0409200.
It all became a lot clearer to me when Urs Schreiber read these papers and translated them into a language I like:
8) Leonardo Castellani, Lie derivatives along antisymmetric tensors, and the Mtheory superalgebra, available as arXiv:hepth/0508213.
9) Pietro Fré and Pietro Antonio Grassi, Pure spinors, free differential algebras, and the supermembrane, available as arXiv:hepth/0606171.
The idea is to think of 11d supergravity as a twice categorifed gauge theory  not just the 3form field in 11d supergravity, but all the fields, in a unified way!
For this, we need to do something much more clever than taking 11d spacetime and slapping a U(1) 2gerbe on top of it. We need to combine the graviton, the gravitino and the 2form field into a connection on a nonabelian 2gerbe.
Here things get a bit technical, but Urs has covered the technical points quite nicely in his blog, so right now I'll just try to give you some handwavy intuitions.
Very roughly speaking, an connection on a bundle takes any path in spacetime
f x>yand gives you an element of some group, which says how a particle would transform if you moved it along this path. This group could be U(1)  the group of phases  or it could be something more fun, like a nonabelian group.
If we categorify this concept, we get the concept of a connection on a "2bundle" (which is more or less the same as a gerbe). Such a connection takes any path and gives you an object in some 2group, but it also takes any surface like this:
f > /  \ x F y \ \/ / > gand gives you a morphism in this 2group. You see, 2group is a kind of category that acts like a group, and a category has "objects" and "morphisms". The morphisms go between objects. For more on 2groups, try:
10) Higherdimensional algebra V: 2Groups, with Aaron D. Lauda, Theory and Applications of Categories 12 (2004), available at http://www.tac.mta.ca/tac/volumes/12/14/1214abs.html Also available as math.QA/0307200.
If we categorify once more, we get connections on a "3bundle", which is more or less the same thing as a "2gerbe"  unfortunately the numbering systems are off by one. This gives us objects, morphisms and 2morphisms in a 3group, which describe what happens when we move particles, strings and 2branes.
And so on:
group point particles 2group point particles and strings 3group point particles, strings and 2branes 4group point particles, strings, 2branes and 3branesetc.
So, if 11d supergravity is a twice categorified gauge theory, we need to know its symmetry 3group.
But actually, since we're doing geometry, this 3group should be a "Lie 3group". In other words, very roughly speaking, a 3group that has a manifold of objects, a manifold of morphisms, and a manifold of 2morphisms, where all the operations are smooth.
But actually, since we're doing supersymmetric geometry, we need a "Lie 3supergroup"! In other words, very roughly speaking, a 3group that has a supermanifold of objects, a supermanifold of morphisms, and a supermanifold of 2morphisms, where all the operations are smooth. (Maybe I should say "supersmooth", just to be consistent.)
If you don't know what a supermanifold is, now is probably not the time to learn. I mean, not right this second. The point is just this: supersymmetry infests everything once you let it in the door, just like ncategories, and just like manifolds  and now we're doing all three.
In fact, nobody has even written down a rigorous definition of a Lie 3supergroup yet! But, Lie algebras are in some ways simpler than Lie groups, and they're a good start, so we can be glad that people do know what a Lie 3superalgebra is!
And Urs describes, in his blog, the relevant Lie 3superalgebra for 11d supergravity!
I would like to say more about this, but it's getting a bit tough trying to talk about this stuff in a fun, easily accessible style, and I have the feeling I'm no longer succeeding. In fact, I don't think I can give a "fun, easily accessible" description of this specific Lie 3superalgebra  at least not yet. So, now I'll completely give up trying to be comprehensible, and simply state some facts.
As shown here:
11) Higherdimensional algebra VI: Lie 2Algebras, with Alissa Crans, Theory and Applications of Categories 12 (2004), available at http://www.tac.mta.ca/tac/volumes/12/15/1215abs.html Also available as math.QA/0307200.
the category of Lie nalgebras is equivalent to the category of L_{∞} algebras which as chain complexes have only n nonvanishing terms, the 0th to the (n1)st. L_{∞} algebras are just algebras of Stasheff's L_{∞} operad in the category of chain complexes of vector spaces  see "week191" and especially these:
12) Martin Markl, Steve Schnider and Jim Stasheff, Operads in Algebra, Topology and Physics, AMS, Providence, Rhode Island, 2002.
James Stasheff, Hartford/Luminy talks on operads, available at http://www.math.unc.edu/Faculty/jds/operadchik.ps
But, we can replace vector spaces by Z/2graded vector spaces and everything still works. Physicists call Z/2graded vector spaces "super vector spaces". So, a "Lie nsuperalgebra" is an algebra of the L_{∞} operad in the category of chain complexes of super vector spaces.
Given this, to specify a Lie 3superalgebra we first need to specify the 0chains, then the 1chains, then the 2chains.
For the particular one Urs mentions, we have
{0chains} = 11d Poincaré Lie superalgebra {1chains} = {0} {2chains} = RHere R is the real numbers, and this 1dimensional thing is what ultimately gives the 3form field A in 11d supergravity. As a vector space, the 11d Poincaré Lie superalgebra is the direct sum of an even part, which is the usual Poincaré Lie algebra iso(11,1), and an odd part, which is the 32dimensional real spinor rep of so(11,1). These give the graviton (or more precisely the LeviCivita connection) and the gravitino in 11d supergravity.
Next we need to make this stuff into a chain complex. That's easy: the differential has to be zero.
Next, we need to specify the L_{∞} structure on this chain complex. First, we need a binary bracket operation, like in an ordinary Lie superalgebra. The bracket of 0chains is the usual bracket in the 11d Poincaré Lie superalgebra. All the other binary brackets are zero.
Then, we need a ternary bracket operation, which expresses how the Jacobi identity holds only up to chain homotopy. This is zero.
Then, we need a quaternary bracket operation (since that chain homotopy satisfies its own identity only up to chain homotopy). This is nonzero: when we take the quaternary bracket of four 0chains we get a 2chain, and there's a nontrivial way to define this! This is the interesting bit, since ultimately it relates the graviton/gravitino to the 3form field.
How do we get that quaternary bracket? Well, here's where things get funky: D'Auria and Fré dreamt up a formula that gives a number from 2 spinors and 2 vectors:
(ψ, φ, v, w) → ψ* Γ^{ab} φ v_{a} w_{b}
where that "ψ*" should really be "ψbar". And, magically, in 11 dimensions this gives a 4cocycle on the Poincaré Lie superalgebra! The proof of this uses some Fierz identity in 11 dimensions:
12) R. D'Auria and Pietro Fré, Geometric supergravity in D = 11 and its hidden supergroup, Nucl. Phys. B201 (1982), 101140. Also available at http://www.math.unihamburg.de/home/schreiber/sdarticle.pdf
And, from HDA6 we know that the 4cocycle condition is just what's needed to make the quaternary bracket satisfy the identity we need for a Lie 3superalgebra. (Alissa and I just did the calculation for Lie nalgebras, but the "super" stuff should work too with a few signs thrown in.)
So, this is all very cool, but I need to understand Fierz identities in different dimensions to see what if anything is special to 11d here  or, alternatively, work out the cohomology of Poincaré Lie superalgebras, to see when they can be deformed to Lie nsuperalgebras. Sounds like a lot of work  maybe someone already did it. Actually D'Auria and Fre make it look like a matter of understanding tensor products of irreps of so(n,1), which is not bad. A worthwhile project in any event.
I also need to understand what all this has to do with E_{8}. For that the paper by Diaconescu, Freed and Moore should help.
Well, this is just the beginning, but Urs explains the rest.
Addenda: I thank Noam Elkies for a correction. Aaron Bergman has this to say about E_{8} and Mtheory:
John Baez wrote:Urs Schreiber replied:> I also need to understand what all this has to do with E_{8}.E_{8} might not have much of anything to do with this. As mentioned in Diaconescu, Freed and Moore, E_{8} appears to function solely as a convenient standin for K(Z,3).On the other hand, the split form of the Eseries (up to E_{11} if you're feeling particularly speculative) is known to show up in describing the fields of 11D SUGRA, but the I don't think anyone knows of a connection between the two E_{8}s. Just to add to the fun, E_{8} gauge fields also show up on the fixed points of Mtheory on S^{1}/Z_{2} giving the E_{8} × E_{8} heterotic string.
Aaron
Aaron Bergman wrote:Aaron Bergman replied:> E_{8} gauge fields also show up on the fixed points of > Mtheory on S^{1}/Z_{2} giving the E_{8} × E_{8} heterotic string.The topological part of the membrane action involves the integral of the sugra 3form over the worldvolume. By DFW, part of that 3form can be thought of as an E_{8} CS 3form.So part of the membrane action looks similar to an E_{8} CStheory over the worldvolume.
Now let the membrane have a boundary. A bulk E_{8} CStheory is well known to induce an E_{8} WZW theory on the boundary.
Could this be the connection between the DWF E_{8} and the HoravaWitten E_{8}?
I have asked this question before: http://golem.ph.utexas.edu/string/archives/000791.html. Jarah then agreed that this must be about right. But it is not completely clear to me yet.
One problem is that in CStheory we vary the connection, while in the topological membrane the E_{8} connection on the background is fixed and we vary the embedding by which we pull it back to the worldvolume. Under suitable assumptions that might be equivalent to varying an E_{8} connection on the worldvolume itself?
Urs Schreiber wrote:> Now let the membrane have a boundary. A bulk E_{8} CStheory is > well known to induce an E_{8} WZW theory on the boundary. > > Could this be the connection between the DWF E_{8} and the > HoravaWitten E_{8}?A similar observation was made by Horava way back in arXiv:hepth/9712130 (in the final section).Aaron
© 2006 John Baez
baez@math.removethis.ucr.andthis.edu
