I was up in Boston for a few days, and managed to attend a few of the talks at the conference in honor of George Lusztig’s 60th birthday. Lusztig started out his career in geometry and topology; his thesis was in the area of index theory, working with Michael Atiyah and using the families version of the index theorem. He soon turned his attention to representation theory, which is the field that he has worked in for most of his career, often from a quite algebraic point of view. His papers are dense and can be difficult to read, especially for someone like me who is not so algebraically inclined, but many speakers at the conference remarked on how their work had drawn important inspiration from one or another of these papers.
Among the things he is famous for are his work on quantum groups, on the representation theory of reductive groups over finite fields (called Deligne-Lusztig theory, for an introduction, see here), on a whole new field in Lie theory known as Kazhdan-Lusztig theory (for an introduction, see the article by Deodhar in the proceedings of the 1991 AMS summer institute on algebraic groups), and many other things.
Of the few talks I heard at the conference, two were really exceptional. One of these was by Michael Atiyah, with the title “Quaternions in Geometry, Analysis and Physics”. He began by explaining that not only was Lusztig 60, but, if he were alive, the Irish mathematician Hamilton would be 200. There’s a famous story about Hamilton’s discovery of the quaternions: this took place in a flash of insight on October 16, 1843, after which he supposedly engraved the defining relations of the quaternion algebra into a Dublin bridge. Atiyah described a piece of history I didn’t know, showing an extract from a 1846 paper of Hamilton’s in which he takes a square root of the Laplacian and essentially writes down the Dirac equation (in Euclidean signature, this was long before special relativity…).
Hamilton was very taken with quaternions as a generalization of complex numbers, and wanted to develop a “quaternionic analysis” that would be a generalization of complex analysis, a project he thought would take him at least ten years. It turns out that you can’t simply generalize the beautiful subject of complex analysis and algebraic geometry over the complex numbers to the quaternionic case. Because of non-commutativity, polynomials behave very differently. Atiyah explained that in his view the correct generalization of complex analysis to the quaternionic case was Penrose’s twistor theory. Here one considers all possible ways of identifying R4 with C2, forming a 3 complex dimensional “twistor space”. Complex analysis on this twistor space is what Atiyah claimed should be thought of as the quaternionic analog of complex analysis (on the complex plane).
He reviewed the story of how solutions to various linear equations are related to sheaf cohomology groups on the twistor space, then went on to the non-linear case, where solutions of the anti-self-dual Yang-Mills equations correspond to holomorphic bundles on the twistor space. One can generalize twistor theory to what Atiyah claimed should be thought of as quaternionic analogs of Riemann surfaces: 4d Riemannian manifolds with holonomy in Sp(1)=SU(2), these are self-dual Einstein manifolds, what Penrose would call a “non-linear graviton” (although this is the Riemannian, not pseudo-Riemannian case). The twistor space of these 4d manifolds is a 3d complex manifold, and Atiyah considers complex analysis on this to be the quaternionic analog of complex analysis on a Riemann surface.
The quaternionic analog of higher dimensional complex manifolds are manifolds of dimension 4k, with holonomy Sp(k). Unlike in the complex case, there are few compact examples. Atiyah went on to discuss how examples (mostly non-compact) could be generated as quotients using the quaternionic analog of symplectic reduction. He described several different classes of examples, noting that this construction first appeared in work with physicists studying supersymmetric non-linear sigma models. While I was a post-doc at Stony Brook, Nigel Hitchin was visiting there and working with Martin Rocek and others on this, leading to the 1987 paper in CMP by Hitchin, Karlhede, Lindstrom and Rocek. Atiyah said that he wouldn’t try and describe the relation to supersymmetry, since “I don’t know much about supersymmetry, and if I tried to explain it, you would understand even less”. That Atiyah, after many years of working in this area, still finds supersymmetry to be something he can’t quite understand, is an interesting comment, reflecting the way the subject is still very imperfectly integrated into mathematician’s traditional ways of thinking about geometry and algebra.
Atiyah also commented that off and on over the years he had pursued the idea that quantum groups (which aren’t quite groups), are in some sense the quaternionification of a Lie group (which doesn’t quite exist). He said he hadn’t been successful with this idea, but still thought there was something to it, and hoped that someone else would take up the challenge of trying to make sense of it.
The second wonderful talk I heard was that of Igor Frenkel, from Yale, with the title “Quantum deformation, geometrization, categorification: What is next?”. Unlike Atiyah’s talk, which I pretty much completely understood, Frenkel’s covered much too quickly a lot of material I had never understood, but putting it into an intriguing perspective close to the unsolved problems that seem to me the most important ones for mathematicians and physicists to be looking at. Frenkel began by saying that for many years he had been trying to solve the problem of how to generalize the constructions of representations of loop groups that are related to 2d CFT to representations of 3d gauge groups that should be related to 4d QFT. Some of his thoughts about this are in the write up of his talk at the 1986 ICM. He described himself as having for a long time given up on this problem, moving on to simpler things that he could do: quantum groups which are deformations of the affine Lie algebra story. He went on to talk about “Geometrization”, by which he meant the principle that “all structure constants are Euler characteristics of some variety, all vector spaces are cohomologies”, then “Categorification”, to him the principle that “all structure constants are dimensions of vector spaces, all vector spaces the Grothendieck groups of an Abelian category”. Many of the examples he was using to flesh this out are not well-known to me, I need to do some serious work learning about them before I can say that I clearly understand exactly what he has in mind here.
The last part of his talk, the “What Next?”, went by way too fast but sounded fascinating. He claimed to have some new ways of thinking about the problem of what a representation of these higher dimensional analogs of loop groups should me. I hope to learn more from him in the future to get a better idea of what he has in mind here. He and collaborators at Yale have papers forthcoming, which I look forward to reading. When and if I ever better understand this stuff, I may try and write about it again here.
Not sure why he thinks that the complex analysis on twistor space generalizes complex analysis to the case of quaternions. What is quaternionic about the twistor space? I don’t see any noncommutative algebra here. Is it just about counting dimensions?
The interpretation of anti-self-dual Yang-Mills solutions in terms of twistor space you described seem to miss all the new developments since the 2003 Witten paper – which is an expansion around the self-dual Yang-Mills to the full Yang-Mills. The research since that paper has at least doubled the amount of interesting insights in this realm.
Supersymmetry has not yet been seen in the colliders but it is an extremely physical framework, and it is not too surprising that pure mathematicians may find it counterintuitive. Surely mathematicians in the 21st century will need SUSY because string theory will dominate mathematics of the 21st century. That’s why mathematicians will have to be learning many things that are currently taught in physics departments only.
Strictly speaking, manifolds of Sp(k) holonomy don’t define quaternionic manifolds. The holonomy of quaternionic Kahler manifolds is Sp(k) times Sp(1) because it can allow a multiplication by quaternions (Sp(1)) – or quaternionic matrices (Sp(k)) – from both sides. And they don’t commute with each other. The manifolds of the more special, Sp(k) holonomy are called hyperKahler manifolds, not quaternionic manifolds.
Lubos,
Atiyah’s point was that what you get by naively replacing complex variables by quaternionic ones is not the really interesting generalization of complex analysis that you get in the twistor space picture. He was using quaternions to motivate this, but the twistor space of R^4 is something different than just using quaternions to put a multiplication on R^4.
Related to this, the 4k dim manifolds that Atiyah was discussing were the hyperkahler ones, not the quaternionic kahler ones.
Dear Peter, I understand that the twistor is something “different” than using the quaternion multiplication table. The word “different” is clear. What is not clear is why you or he uses the “same” word for “different” things. đ
Your text also made it clear that he was using Sp(k) holonomy manifolds. But I was trying to convince you that you incorrectly called them the quaternionic analogue of Kahler complex manifolds.
The quaternionic analogue of Kahler (U(k) holonomy) manifolds are the quaternionic Kahler manifolds whose holonomy is Sp(k).Sp(1).
The manifolds with Sp(k) holonomy are called hyperKahler manifolds, and they are, in some sense, the quaternionic generalization of (SU(k) holonomy) Ricci-flat Kahler or Calabi-Yau manifolds. Your sentence about the quaternionic generalization is not quite right.
Lubos,
I know very well what the difference is between a hyperkahler and a quaternionic kahler manifold. So does Atiyah. I was reporting on his talk, and quoting him; he was the one claiming that hyperkahler (not quaternionic kahler) manifolds are the interesting quaternionic analog of kahler manifolds. If you don’t like this, please go write to Sir Michael and explain to him why he’s wrong.
I agree that the hyperKahler manifolds are more interesting than quaternionic Kahler manifolds, but I disagree that they are the generalization of Kahler manifolds to the case of quaternions. Instead, hyperKahler manifold is the quaternionic generalization of Ricci-flat Kahler manifolds whose holonomy is SU(n), not U(n).
What you wrote is, in fact, completely incorrect. You did not even use the word “Kahler”. You wrote that Sp(k) manifolds generalize “complex manifolds”. That’s nonsense because “complex manifolds” can have virtually any holonomy you want while Sp(k) is highly constrained. What you wanted to say was “Ricci-flat Kahler manifolds”, not “complex manifolds”.
If Sir Atiyah used your sentence literally and meant every word in that sentence to be treated seriously, I will happily explain the error to Sir Atiyah, too. Unlike you, I don’t think that certain people above some level of dignity are infallible and uncriticizable.
Lubos,
If you want to explain to Sir Michael your views on why he’s wrong to think of hyperkahler manifolds as the quaternionic generalization of Kahler ones and why his analogy is “nonsense”, definitely go ahead and e-mail him. I’m sure he’ll be fascinated by your insights into this and glad to be set straight.
Then again, before you hit send on that e-mail, you might take a moment to reflect and realize that there are some people in this world who know about a hundred times more than you do about certain subjects, and geometry might be one of them.
Dear Peter, I don’t have any trustworthy data that would indicate that Sir Atiyah makes incorrect analogies between different structures that would resemble your incorrect sentence.
If the talk is available online (proceedings or arXiv) that would settle the issue. Why argue the details of what is essentially second-hand information?
Sunderpeeche,
The talk is not available online. You’re quite right, this is second-hand information, produced by me wasting much of my evening trying to write out the clearest explanation I could of an analogy (“analogy”, as in not precisely the same, check your dictionary) described by one of the world’s greatest mathematicians in a talk I attended, based on my written notes and what I remember.
Jeez, why waste time trying to explain something interesting here? I should just go back to string theory bashing.
Among those who practice David Hestenes’ Geometric Algebra, it is said that this is the natural generalization of complex analysis. An introduction is here:
http://faculty.luther.edu/~macdonal/GA&GC.pdf
Carl
Peter,
As far as I’m concerned, you haven’t wasted your evening and I want to thank you very much for your efforts to keep us all informed.
We all benefit from your reporting, including the emotionally unstable kid who reads your blog so frequently and so thoroughly that he’s typically the first to write a response to it.
It’s nice to hear that Frenkel continues to explain the importance of categorification. I got excited about this concept around 1994 when I read his paper with Louis Crane on Hopf categories and the canonical bases for quantum groups; you can see my excitement in “week38” of This Week’s Finds. It took a while for this idea of Frenkel’s – categorifying quantum groups and their resulting tangle invariants – to bear fruit, but his student Khovanov has turned it into a hot topic. Since you’re presumably right next door to Khovanov, you could probably get him to explain anything about categorification that interests you! I’m very excited that my student Aaron Lauda is going to Columbia next year to work with Khovanov on this stuff. I hope he looks you up sometime.
I was in Cambridge this weekend too, but too busy to attend any talks. I’ll probably be free on Monday night. If any cool people want to talk, email me. I’d like to have dinner in Harvard Square. I haven’t had time to go there this visit, and I haven’t been there for over 5 years. I hear it’s been much gentrified since I was a grad student here back in ’82-’86.
Atiyah’s statement RE quaternion analysis is absolutely fascinating and obviously correct. It’s the conformality that makes complex analysis so rich and interesting.
-drl
Carl Brannen mentions “David Hestenesâ Geometric Algebra” as “the natural generalization of complex analysis”.
Geometric Algebra is basically Clifford algebra.
The real Clifford algebra Cl(0,1) has dimension 2^1 = 2 and is the Complex numbers.
The real Clifford algebra Cl(0,2) has dimension 2^2 = 4 and is the Quaternions.
Peter mentions with respect to Atiyah’s talk “… a âquaternionic analysisâ that would be a generalization of complex analysis …”.
Since both Complex numbers and Quaternions are Clifford algebras, a natural way to look at “… a âquaternionic analysisâ that would be a generalization of complex analysis …” is to look at Clifford analysis.
John Ryan, in math.CV/0303339 , Introductory Clifford Analysis, said:
“… one can extend basic results of one complex variable analysis on holomorphic function theory to four dimensions using quaternions. … This was developed by the Swiss mathematician Rudolph Fueter in the 1930âs and 1940âs … So it seems reasonable to ask if all that is known in the quaternionic setting extends to the Clifford algebra setting … the answer is yes …
…[the] type of diffeomorphisms acting on subdomains of Rn [that] preserve Clifford holomorphic functions … is a conformal transformation …
… for dimensions 3 and greater the only conformal transformations … are Moebius transformations …”.
Lars Ahlfors, in his paper Clifford Numbers and Moebius Transformations in R^n, published on pages 167-175 of the book Clifford Algebras and Their Applications in Mathematical Physics, Proceedings of NATO and SERC Workshop, Canterbury, Kent, 1985, edited by Chisholm and Common, NATO ASI Series (Reidel 1986), said:
“… Moebius transformations in any dimension can be expressed through 2×2 matrices with Clifford numbers as entries. This technique is relatively unknown in spite of having been introduced as early as 1902 … by K. Th. Vahlen …”.
Pertti Lounesto, in his book Clifford Algebras and Spinors (Cambridge, 2nd edition, 2001), said ( denoting nxn matrices of X by Mat(n,X) and denoting the quaternions by H ):
“… Cl( p+1 , q+1 ) = Mat(2, Cl(p,q) ) …
… Cl(1,3) = Mat(2, H ) which implies Cl(2,4) = Mat(4, H) …”.
Since the bivector Lie algebra of the Clifford algebra Cl(2,4) gives the Lie group Spin(2,4) = SU(2,2) that is the 15-dimensional Conformal group over Minkowski spacetime R(1,3),
the quaternionic structure of that Conformal group is made clear by Cl(2,4) = Mat(4, H ).
Roger Penrose, in The Road to Reality (Knopf 2005) said at page 972, using notation O(2,4) for the 15-dimensional Conformal group:
“… The shortest … way to describe a (Minkowski-space) twistor is to say that it is a reduced spinor ( or half spinor ) for O(2,4). …”.
The above outline shows:
1 – a useful quaternionic “generalization of complex analysis”, based on Clifford algebra (as indicated by Carl Brannen’s comment)
2 – the answer to Lubos Motl’s question “What is quaternionic about the twistor space?”
3 – more details about the view (expressed by Atiyah) that “the correct generalization of complex analysis to the quaternionic case was Penroseâs twistor theory”.
All the above is consistent with D. R. Lunsford’s comment that what makes such stuff “so rich and interesting” is “the conformality”.
Tony Smith
http://www.valdostamuseum.org/hamsmith/
We used quaternions a lot in GNC (Guidance, Navigation & Control) software for spacecraft at JPL, yet I wrote ground test software that used Euler angles and matrices to validate & verify the flight software of Gaileo, to avoid repeating any conceptual errors of the flight software. I got deep into qaternionic analysis then and since. I’ve published a paper deadling with whether certain physical quantities assumed to be real (such as momentum and acceleration) for elementary particles, might actually be complex, or quaternions, or Cayley algebra…
As to manifolds, what’s the deal on the latest episode of this saga: Joe Lau writes on Slashdot to mention a story running on the Xinhua News Agency site, reporting a proof for the Poincare Conjecture in an upcoming edition of the Asian Journal of Mathematics.
http://news.xinhuanet.com/english/2006-06/04/content_4644754.htm
From the article:
“A Columbia professor Richard Hamilton and a Russian mathematician Grigori Perelman have laid foundation on the latest endeavors made by the two Chinese. Prof. Hamilton completed the majority of the program and the geometrization conjecture. Yang, member of the Chinese Academy of Sciences, said in an interview with Xinhua, ‘All the American, Russian and Chinese mathematicians have made indispensable contribution to the complete proof.'”
Frenkel began by saying that for many years he had been trying to solve the problem of how to generalize the constructions of representations of loop groups that are related to 2d CFT to representations of 3d gauge groups that should be related to 4d QFT. Some of his thoughts about this are in the write up of his talk at the 1986 ICM.
Another place where this thoughts as of 1993 can be found is hep-th/9303047. Ref 10 of that paper contains the first interesting representations of multi-dimensional current algebras. The same construction applied to multi-dimensional diffeomorphism algebras led Rao and Moody to the first interesting representations of the multi-dimensional Virasoro algebra.
There is also another extension of current algebras in 3D, the Mickelsson-Faddeev extension. However, it was proved by Pickrell that this algebra lacks unitary reps on a separable Hilbert space; for a simple plausibility argument, see math-ph/0501023, where the two types (KM and MF) of current algebra extensions are contrasted in a Fourier basis.
Re: Poincare: I’m guessing they’re just claiming to have finished filling in the details of Perelman’s program. A fair number of people have been working on that as I understand it.
Re: Poincare
Yes, Cao and Zhu are claiming to have come up with a full proof based on Perelman’s outline. It has been vetted by Yau and is supposed to appear in this month’s Asian Journal of Mathematics, which is on-line now, minus the pdf file of the article. I was waiting for the article to appear before writing about this, if anyone knows where a copy is available, let me know.
Peter quoted
âAtiyah said that he wouldnât try and describe the relation to supersymmetry, since âI donât know much about supersymmetry, and if I tried to explain it, you would understand even lessâ. That Atiyah, after many years of working in this area, still finds supersymmetry to be something he canât quite understand, is an interesting comment, reflecting the way the subject is still very imperfectly integrated into mathematicianâs traditional ways of thinking about geometry and algebra.â
I would like to add that also modular localization (the adaptation of Tomita-Takesaki modular theory to local quantum physics) which is presently the best sniffer dog in the conceptual arsenal of theoretical particle physics (a gift of AQFTists, who were deeply involved in its independent discovery within a more limited context) is also unable to attribute any significance to supersymmetry. If one starts from a given observable local net structure (pure bosonic in the sense of spacelike commutativity), and hence leaves its previously mentioned very radical and yet insuffiently understood monade approach (leading to spacetime symmetries and observable nets from more basic assumptions of the kind favored by Lee Smolin) outside, there is the DHR theory (which may be viewed as part of modular theory) which is capable to uniquely extend the observable algebra to the charge-carrying field algebra from which one can then read off the compact group symmetry (in low dimensional spacetimes the picture is more complicated). As mentioned on a previous occasion it does so in the inverse problem spirit of Marc Kacâs âhow to hear the shape of a drumâ. It leads to compact groups, but not to graded groups. Of course it is not forbidden to extend to supersymmetries by hand, even though modular theory which gives spacetime- as well as inner symmetries does not lead to susy in any natural way.
In addition there is the curious observation that putting a supersymmetric QFT into a heat bath, the resulting situation is characterized by a collape and not a spontaneously broken symmetry as it would occur with Lorentz symmetry
http://br.arxiv.org/abs/math-ph/0604044 see in particular ref. [3] and [4]
The situation throws doubt whether the spontaneous breaking mechanism for supersymmetry permits any consistent formulation (i.e. even outside the heat bath setting) at all. Supersymmetry looks like a man-made thing which is despite its unattractive rigid appearance apparently extremely fragil.
Today seems to be a really bad day for supersymmetry
http://br.arxiv.org/abs/hep-th/0606029
… but a good one for satanists. Maybe Lubos foresaw all of this in a dream, where the String God appeared to him and intoned the words: “Lubos, my good and faithful servant. I can protect you only until the satanists rise again. Then nameless evils will happen, and I will no longer be able to prevent the publication of Woit’s book.”
I had only a second-hand commentary on Frenkel’s conception of categorification by Dror Bar-Natan about half way down this post. From the perspective presented in the first half of the post, which might be phrased “If it moves, salute it; if it doesn’t, categorify it”, this is somewhat limited.
Well 6’s are representative of “hybrid strings”, that is, those that close on themselves but leave a tell-tale serif flapping in the vacuum. So 666 is really the Mark of the Proton, forseen in the Book of Zweig, and detailed in Gell-Mann’s Letters to the Experimentalists.
-drl
The Coming of the Experimentalists, as foretold in the Book of Zweig, is feared above all things by those who worship the String God. If I may quote a passage (Book 5, v. 12-15):
666 means read/write for owner, group and others. But no execute permisions. Better 777.
Chris Oakley said “… But the people were confounded. For they knew not the meaning of “666”. …”.
In the 6×6 magic square
01 35 34 03 32 06
30 08 28 27 11 07
24 23 15 16 14 19
13 17 21 22 20 18
12 26 09 10 29 25
31 02 04 33 05 36
the columns, rows, diagonals all add to 111 and the total number is 666.
Cornelius Agrippa (1486-1535) identified that 666 magic square with the sun
( see http://www.geocities.com/CapeCanaveral/Lab/3469/examples.html ).
Maybe because some religions considered by Christians to be pagan
were sun-god religions, Christians demonized the sun-number 666.
Maybe, in Chris Oakley’s Book of Zweig, verse 16. may have said:
16. What the people did not know was that 666 = Sun = Light of Experimental Evidence that is to come from the Temple of Experiment in Geneva, capable of destroying even that most fervently worshipped of Theoretical Gods, SuperSymmetry.
Tony Smith
http://www.valdostamuseum.org/hamsmith/
Tony,
What you say is interesting, but is not what I have. My version reads:
06/06 is the Swedish national day, which we celebrate, perhaps, because some obscure coup d’etat in 1809 replaced one king by another. 2006 is the first year it is a holiday.
I am intrigued: Why did you choose Zweig?
Alejandro – no idea. It seems to be lost in the mists of time. But to continue: