Luscious Langlands II
Since domains on the complex plane might really represent moduli, and we know everything should be about categories at the end of the day, it would be better to replace the eigenfunction $f$ with a more sheaf theoretic concept. As Kapustin and Witten say in their abstract: The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions.
Yes, N=4 SUSY Yang-Mills turned up when we were worrying about twistor string theory and calculating gluon amplitudes. Actually, these days the geometric Langlands conjecture is about an equivalence of categories, namely derived categories of sheaves. Schreiber has been blogging about such things. But where did the number theory go in all this String geometry? Isn't that what this is really about?
Yes, N=4 SUSY Yang-Mills turned up when we were worrying about twistor string theory and calculating gluon amplitudes. Actually, these days the geometric Langlands conjecture is about an equivalence of categories, namely derived categories of sheaves. Schreiber has been blogging about such things. But where did the number theory go in all this String geometry? Isn't that what this is really about?
posted by Kea | 10:21 AM
11 Comments:
That Kapustin and Witten paper http://arxiv.org/PS_cache/hep-th/pdf/0604/0604151.pdf, 10-d supersymmetry research, doesn't seem to get anywhere.
There might be a problem with these hundreds-of-pages long maths papers which go through rigorous derivations and thousands of lines of equations, and lead nowhere.
I've always wondered how mathematicians can justify a lot of maths which doesn't get solid results. How do you judge what is useful pure/applied mathematics if has no real world applications? It's just going to be down to the subjective judgement of the examiner or audience. That's the danger of speculating mathematically about physics. The maths of the the Kelvin-Helmholtz "vortex atom" and Maxwell's "aether" disappeared.
How does the mathematics community judge results in number theory or anything astract that can't be checked in the real world? Consensus? Presumably, it is like the art world's problems in deciding what exactly is a work of art, and how beautiful different abstract things are? It does seem a little bit of a trick for people to work mathematically on 10-d supersymmetry ideas without it being clear whether what they are doing is just maths or whether it is claimed to be physics.
The problem is that the work passes for physics (and is published as such) yet risks nothing, and if/when found to not be/contain any physics, it can then retreat behind the mathematics banner.
This must be how the vortex atom became a success. Notice that Kelvin, even long after the discovery of radioactivity (his vortex atom is completely stable), never admitted defeat. You can't disprove a mathematical theorem, and that is the problem of this sort of work. Someone writes down a load of mathematics of 10 dimension compactified to give 4 observable dimensions, and those theorems can't be disproved. The vortex atom of Kelvin is purely mathematical and it's theorems were never disproved; they were found simply to be of no use to modelling the atom.
Above where I said "The maths of the the Kelvin-Helmholtz "vortex atom" and Maxwell's "aether" disappeared", I mean with regard to the original subject (atomic theory, and mechanical modelling of an elastic solid vacuum).
Maxwell's equations (mainly the equations of others, like Gauss, Ampere, Faraday's inductance law, and first written in vector calculus by Heaviside) survived in a sense. Some of the perfect fluid ideas of the vortex atom proved useful in hydrodynamics.
But this "usefulness elsewhere" analogy breaks down for 10-dimensions, where nothing physically observed is being modelled to begin with, just speculations like supersymmetry (supersymmetry is not an observed physical property of the universe like say the speed of light).
Witten himself was sighted at the AAS (American Astronomical Society) meeting. He is quite interested in astronomer's confirmation of theory. Thank you for your latest spin on neutron stars too.
02 22 07
"How do you judge what is useful pure/applied mathematics if has no real world applications? It's just going to be down to the subjective judgement of the examiner or audience. That's the danger of speculating mathematically about physics"
Nigel I think your view is somewhat myopic with respect to mathematics and physics. Mathematics is applied and gives us physics. I don't care what other people may say but they are really two halves of the same whole.
One could argue, given your logic that all theoretical physics is speculative, but that would presuppose that the math isn't useful.
I refuse to believe that. I went through a few lines to derive an alternating series rep in binary for 1/3. However, it turns out that those lines relating to a theoretical problem have broad applicability in computer science, biology etc.
Perhaps you have to think broader about the utility of formalism and rigor. One day it may save you.
Where did the numbers go! This question bothers also me when I try to read these papers. I am also frustrated about the more or less complete disappearance of Galois group for the closure of rationals in number theoretic Langlands.
For a simple mind like me the first thing come in mind is that this monstrous Galois group must correspond to infinite symmetric group permuting roots of an irreducible polynomial with infinite degree whose roots lead to algebraic closure or rationals, idealization of course but might make sense as some kind of limiting case.
Combined with the observation that the group algebra of S_infty, which by the way extends naturally to infinite braid group, is hyper-finite factor of type II_1 leads to a fresh approach to the understanding of Langlands program.
The close contact with physics inspires also new conjectures (there seems to be quite many on the market already;-)). Even spontaneous symmetry breaking of S_infty to finite Galois group represented diagonally and corresponding to finite group defining inclusions of hyperfinite factor of type II_1 emerges naturally. Galois groups would be also symmetries of physical systems and make even elementary particles carriers of information.
The extension of S_infty to braid group in turn gives connection with homotopies so that geometric and number theoretic Langlands programs might be put under same umbrella provided by conformal field theories involving the notion of number theoretic braid.
For an effort to understand Langlands along these lines see the link here, where one can also find also a link to a chapter Langlands Program and TGD.
Best Regards, Matti Pitkanen
Hi Mahndisa,
You may be missing my point.
Most of the maths of physics consists of applications of equations of motion which ultimately go back to empirical observations formulated into laws by Newton, supplemented by Maxwell, Fitzgerald-Lorentz, et al.
The mathematical model follows experience. It is only speculative in that it makes predictions as well as summarizing empirical observations. Where the predictions fall well outside the sphere of validity of the empirical observations which suggested the law or equation, then you have a prediction which is worth testing. (However, it may not be falsifiable even then, the error may be due to some missing factor or mechanism in the theory, not to the theory being totally wrong.)
Regarding supersymmetry, which is the example of a theory which makes no contact with the real world, Professor Jacques Distler gives an example of the problem in his review of Dine's book Supersymmetry and String Theory: Beyond the Standard Model:
http://golem.ph.utexas.edu/~distler/blog/
"Another more minor example is his discussion of Grand Unification. He correctly notes that unification works better with supersymmetry than without it. To drive home the point, he presents non-supersymmetric Grand Unification in the maximally unflattering light (run α 1 ,α 2 up to the point where they unify, then run α 3 down to the Z mass, where it is 7 orders of magnitude off). The naïve reader might be forgiven for wondering why anyone ever thought of non-supersymmetric Grand Unification in the first place."
The idea of supersymmetry is the issue of getting electromagnetic, weak, and strong forces to unify at 10^16 GeV or whatever, near the Planck scale. Dine assumes that unification is a fact (it isn't) and then shows that in the absense of supersymmetry, unification is incompatible with the Standard Model.
The problem is that the physical mechanism behind unification is closely related to the vacuum polarization phenomena which shield charges.
Polarization of pairs of virtual charges around a real charge partly shields the real charge, because the radial electric field of the polarized pair is pointed the opposite way. (I.e., the electric field lines point inwards towards an electron. The electric field likes between virtual electron-positron pairs, which are polarized with virtual positrons closer to the real electron core than virtual electrons, produces an outwards radial electric field which cancels out part of the real electron's field.)
So the variation in coupling constant (effective charge) for electric forces is due to this polarization phenomena.
Now, what is happening to the energy of the field when it is shielded like this by polarization?
Energy is conserved! Why is the bare core charge of an electron or quark higher than the shielded value seen outside the polarized region (i.e., beyond 1 fm, the range corresponding to the IR cutoff energy)?
Clearly, the polarized vacuum shielding of the electric field is removing energy from charge field.
That energy is being used to make the loops of virtual particles, some of which are responsible for other forces like the weak force.
This provides a physical mechanism for unification which deviates from the Standard Model (which does not include energy sharing between the different fields), but which does not require supersymmetry.
Unification appears to occur because, as you go to higher energy (distances nearer a particle), the electromagnetic force increases in strength (because there is less polarized vacuum intervening in the smaller distance to the particle core).
This increase in strength, in turn, means that there is less energy in the smaller distance of vacuum which has been absorbed from the electromagnetic field to produce loops.
As a result, there are fewer pions in the vacuum, and the strong force coupling constant/charge (at extremely high energies) starts to fall. When the fall in charge with decreasing distance is balanced by the increase in force due to the geometric inverse square law, you have asymptotic freedom effects (obviously this involves gluon and other particles and is complex) for quarks.
Just to summarise: the electromagnetic energy absorbed by the polarized vacuum at short distances around a charge (out to IR cutoff at about 1 fm distance) is used to form virtual particle loops.
These short ranged loops consist of many different types of particles and produce strong and weak nuclear forces.
As you get close to the bare core charge, there is less polarized vacuum intervening between it and your approaching particle, so the electric charge increases. For example, the observable electric charge of an electron is 7% higher at 90 GeV as found experimentally.
The reduction in shielding means that less energy is being absorbed by the vacuum loops. Therefore, the strength of the nuclear forces starts to decline. At extremely high energy, there is - as in Wilson's argument - no room physically for any loops (there are no loops beyond the upper energy cutoff, i.e. UV cutoff!), so there is no nuclear force beyond the UV cutoff.
What is missing from the Standard Model is therefore an energy accountancy for the shielded charge of the electron.
It is easy to calculate this, the electromagnetic field energy for example being used in creating loops up to the 90 GeV scale is the energy of a charge which is 7% of the energy of the electric field of an electron (because 7% of the electron's charge is lost by vacuumn loop creation and polarization below 90 GeV, as observed experimentally; I. Levine, D. Koltick, et al., Physical Review Letters, v.78, 1997, no.3, p.424).
So this physical understanding should be investigated. Instead, the mainstream censors physics out and concentrates on a mathematical (non-mechanism) idea, supersymmetry.
Supersymmetry shows how all forces would have the same strength at 10^16 GeV.
This can't be tested, but maybe it can be disproved theoretically as follows.
The energy of the loops of particles which are causing nuclear forces comes from the energy absorbed by the vacuum polalarization phenomena.
As you get to higher energies, you get to smaller distances. Hence you end up at some UV cutoff, where there are no vacuum loops. Within this range, there is no attenuation of the electromagnetic field by vacuum loop polarization. Hence within the UV cutoff range, there is no vacuum energy available to create short ranged particle loops which mediate nuclear forces.
Thus, energy conservation predicts a lack of nuclear forces at what is traditionally considered to be "unification" energy.
So there would seem to discredit supersymmetry, whereby at "unification" energy, you get all forces having the same strength. The problem is that the mechanism-based physics is ignored in favour of massive quantities of speculation about supersymmetry to "explain" unification, which are not observed.
Figure 4, page 27 of the Cachazo and Svrcek paper is likely missing one possible representation:
the catenoid equivalent of hyperbolas and cones.
Combined with the observation that the group algebra of S_infty, which by the way extends naturally to infinite braid group, is hyper-finite factor of type II_1 leads to a fresh approach to the understanding of Langlands program.
Thanks for your comment, Matti. I appreciate that you have done a lot of interesting work on understanding how Langlands (and Riemann hyp for that matter) fits into new physics. I also have a big problem with the Galois groups - but I'm trying to understand it using a pure category theory approach.
Nigel, if you don't mind me saying so, Mahndisa had a very valid point. Witten is a great physicist, who uses real physical intuition in attacking problems. His version of Langlands is not mathematical, although it is great mathematics, too. The fact that there are many problems with Stringy physics does not mean that one can dismiss these papers.
02 24 07
"For a simple mind like me the first thing come in mind is that this monstrous Galois group must correspond to infinite symmetric group permuting roots of an irreducible polynomial with infinite degree whose roots lead to algebraic closure or rationals, idealization of course but might make sense as some kind of limiting case."
Hello Matti and everyone:
There is nothing simple about your mind. Regardig the infinite heirarchy of irrational roots of rational numbers, I see that more and more. The Galois group seems to pop up more and more I look! I also know that Wiles messed around with quite a bit of the Galois group when solving Fermat's last theorem.
Nigel: I think we respectfully have a different philsophical approach to what math and physics is! I really don't see a separation between the two disciplines despite what one might observe in the academy.
You are operating under a false premise, which is that number theoretic answers cannot be checked by real applications. This simply means that you have not seen the right applications of number theory. Computing languages are ONE good example. Cryptography is another. Underlying symmetries of genetic code can also be described with number theoretic influences. Matti has developed a whole RIGOROUS theory dedicated to using number theory as a FUNDAMENTAL in physics.
So I suppose I don't understand your aversion to math. It doesn't mean that I don't like you though:)
Hi Kea,
Yes, Witten is certainly a great physicist and mathematician. I'm only worried about papers which build a vast amount of maths on speculation, when papers which try to connect to physics and take risks have been censored off arxiv.
Hi Mahndisa,
"... I don't understand your aversion to math." - Mahndisa.
Mathematical speculation that leads to no physical results useful to experimentalists is what is objected to, not mathematics.
I do like maths which is useful and spend a lot of my spare time studying mathematical papers, which is not an aversion to maths; the papers I study are those which give physical results that have been checked or can be.
Your statement that I must hate mathematics generally because I want physics papers to do physics, reminds me a little of Professor Hardy's autobiography, A Mathematician's Apology.
Hardy, a pure mathematician, wrote that Hogben (who had written a popular book on applied mathematics and its history) was hated maths. What he meant was that Hogben hated the kind of maths which doesn't give results which apply to this universe. ;-)
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