Witten and Knots

If you are at all interested in mathematical physics you will want to watch Ed Witten’s recent talk on his work in knot theory that he gave at the IAS. Witten gives a general overview of how he discovered that the Jones polynomial used to classify knots turns out to be “explained” as a path integral using Cherns-Simon theory in 3D. More recently the Jones Polynomial was generalised to Khovanov homology which describes a knotted membrane in 4D and Witten wanted to find a similar explanation. He was stuck until some work he did on Geometric Langlands gave him the tools to solve (or partially solve) the riddle.

Geometric Langlands was devised as a simpler variation on the original Langlands program that is a wide-ranging set of ideas trying to unify concepts in number theory. Witten makes some interesting comments during the question time. He says that one of the main reasons that physicists (such as himself) are able to use string theory to answer questions in mathematics is that string theory is not properly understood. If it was then the mathematicians would be able to use it in this way themselves, he says. Referring to the deeper relationship between string theory and Langlands he said.

“I had in mind something a little bit more ambitious like whether physics could affect number theory at a really serious structural level like shedding light on the Langlands program. I’m only going to give you a physicists answer but personally I think it is unlikely that it is an accident that Geometric Langlands has a natural description in terms of quantum physics, and I am confident that that description is natural even though I think it mught take a long time for the math world to properly understand it. So I think there is a very large gap between these fields of maths and physics. I think if anything the gap is larger than most people appreciate and therefore I think that the pieces we actually see are only fragments of a much bigger totality.”

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39 Responses to Witten and Knots

  1. Leo Vuyk says:

    For an alternative knot approach of the standard model see perhaps:

  2. In his last book, Helge Kragh has a whole chapter on the prehistory of knots for foundations of elementary particle physics.

  3. D R Lunsford says:

    What exactly is interesting about this? Why is physics so decadent? Does anyone work on anything substantial any more? This is going exactly nowhere, like everything Witten does. It’s not physics.


  4. Knot theory is not physics *yet* but probably is within few decades.

    I tend to believe that a proper generalization of braid diagrammatics will replace Feynman diagrammatics and resolve the well-known divergence problems relating to the use of local interaction vertices as a description of interactions.

    In the usual braid diagrams you do not have vertices – just braidings – and this is of course a problem. Could a proper generalization of braid diagrams give vertices for *entire braids* but *not for braid strands* coming from both future and past, which would just re-distribute in vertices? This would be nothing but a generalization of the ancient OZI rule from pre-QCD era. Interactions without interactions would Wheeler say. I almost dare argue that this is possible.

    For some concrete ideas in this direction see this.

    • Ervin Goldfain says:

      I cannot help thinking about the question raised by D R Lundsford. Are we attracted to Witten just because he is a brand name nowadays? Is there substantial evidence that these ideas would lead anywhere and will match observations in a compelling way?

      • D R Lunsford says:

        The answer is no. Knot theory has been around for quite a while, and so has the roster of physics posers who presume to see a royal road to quanta in it.


  5. Philip Gibbs says:

    I can’t provide any evidence that this will eventually be confirmed as real physics. Even if it is just maths it is very interesting and if Nature failed to include such beautiful ideas in its grand design even though they come so close to physical ideas then too bad.

    • D R Lunsford says:

      Well I can provide centuries of historical evidence that it will not be, and several decades worth of modern experience as punctuation to the lessons of history. How much damage can one or two generations do? Is not 30 years of abject failure embodied in the farce of string theory enough to call a halt to this decadent mental game that ignores phenomena in favor of the most wispy and dessicated of pointless abstractions? Ideas in themselves are not beautiful, and the multiplication of empty hypotheses is, to the contrary, explicitly ugly. I don’t know why this particular episode is so galling to me, but I think it must have something to do with being utterly fed up.


      • Dilaton says:

        @ D R Lunsford

        What about going somewhere else if You dont like this post and if You are “SO” fed up instead of writting trolling comments and trying to tell other people what to do?

        @ Phil

        Maybe here is some cleaning up to do now …;-)

  6. I am not fascinated by braids because Witten happens to study them. It just happens that knots and braids emerge in extremely natural manner in my own theory world from finite measurement resolution meaning replacement of space-time sheets with string world sheets with their ends defining braid strands. I am of course happy to see that real mathematical physicists end up to develop the mathematics which might apply also in TGD.

    Braids appear in topological QFTs and their generalizations in integrable 2-D QFTs, which as such are physically non-interesting as are also topological QFTs.

    The challenge is to generalize these theories so that they are not physically trivial anymore and TGD implies naturally this generalization. For instance, Kahler action- Maxwell action for Kahler form of CP_2 projected to space-time surface in M^4xCP_2, reduces to boundary terms for its preferred extremals since Coulomb term in the action vanishes for them and weak form of electric magnetic duality as a boundary condition reduces the resulting 3-D boundary terms to Chern-Simons terms.

    Therefore genuine dynamics making the theory almost topological and implies enormous calculational simplification and strong form of holography which is something very physical. This general picture is essentially what general coordinate invariance in strong form requires.

    • Ervin Goldfain says:


      But the question I posed before persists. Is there any compelling evidence at this point that your work invoking braids and knots (and anyone else’s for that matter) is real physics?



  7. Ervin and all,

    Compelling evidence this is real physics? Perhaps because a few of our bloggers have new similar ideas on this. Today I posted “New Thoughts for the New Physics” on my http://www.pesla.blogspot.com Now would you not say that Witten having similar intuitions is not evidence. (actually I did not praise his paper highly from a link on Kea’s blog I think, or maybe Motils for his is but to me a speculation that does not understand some things as well as his position people might wonder or not understand when string theory is real physics. He may want to unify the string ideas but is that needed in the new physics or maybe it all becomes something like symmetries of E8 generalized?

    Well, I do not know what possessed me to post this what amounts to a common sense old idea for the unitary field or toe like theories- more a question of logic and mathematics. So I looked up the Langlands link and find many things I have casually thought about and wrote about. This is the right track. It leads to what I have called a quasic generalization of which Witten is right- there is a great disconnect in the physics and math here still.

    I could have called the posting: “Beyond the New Physics” but that seemed a little too much or trivial as new theories might always arise. (I am not sure of some of the article as making decisive breakthroughs in certain methods mentioned but off hand I would have my doubts if the theory cannot decisive convince you of the concerns and possible solutions.

    So, thank Gibbs for this timely article and coincidence. It too makes me feel like our science is universally intelligible and needs not be in symbols and notions too hard for most of us if they care to understand.

    But a little bit of a more general view can lead to a lot of misunderstandings and partial representations- the price we pay I guess, and some pay with their careers.

    Forgive my informality, I had not planned many to read this anytime soon.

    The PeSla

  8. Kea says:

    So I think there is a very large gap between these fields of maths and physics.

    Yes, I believe Witten is one of only three people who understand the magnitude of this truth.

  9. Mark Thomas says:

    I while back I mentioned that there is a relation in ‘number theory’ and something to with the hyperbolic Leech lattice (pure maths) that is possibly directly related to physics. That relation is e^(2pi sqrt163) 70^2 = 337736875876935471466319632506024463200.0000008023 … which is also a ‘near integer’. This is an order magnitude ~10^38. This is ‘extremely near’ to the dimensionless physics relation hc/piGm^2 : Where h = Planck’s constant, c = speed of light, G = Newton’s constant and m = neutron mass

    Now that the NIST Codata 2010 has been published it is looking less a coincidence. Below are listed NIST 2006 Codata and the NIST 2010 Codata value of hc/piGm^2 and the reader can compare this to the math form above.

    Using 2006 Codata: hc/piGm^2 = 3.37700 x 10^38 (+- 0.00050) (where m = neutron mass).

    Using 2010 Codata: hc/piGm^2 = 3.37722 x 10^38 (+- 0.00040)

    Maybe this is a nexus.

    • Ulla says:

      There has been a longstanding question if G is fundamental or not. Can G be emergent, and from what then? Are there some views?

      As from Earth perspective g varies a little depending on the geoid structure.

      • Ulla says:

        Because their units cancel, ratios of like-dimensioned physical constants do not depend on unit systems in this way, so they are pure dimensionless numbers whose values a future theory of physics could conceivably hope to predict. Additionally, all equations describing laws of physics can be expressed without dimensional physical constants via a process known as nondimensionalisation, but the dimensionless constants will remain. Thus, theoretical physicists tend to regard these dimensionless quantities as fundamental physical constants.

        However, the term fundamental physical constant is also used in other ways. For example, the National Institute of Standards and Technology[1] uses the term to refer to any universal physical quantity believed to be constant, such as the speed of light, c, and the gravitational constant G.

        It is currently disputed whether any changes in dimensional physical constants such as G, c, ħ, or ε0 are operationally meaningful; however, a sufficient change in a dimensionless constant such as α is generally agreed to be something that would definitely be noticed.

        From Barrow 2002:

        [An] important lesson we learn from the way that pure numbers like α define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck’s constant, h. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature.

        Numbers are real?

      • Mark Thomas says:

        As the physics relation hc/piGm^2 is a dimensionless constant and is possibly equivalent to the ‘pure math form’ then the constant stays the same regardless of any running of the values (if the e^2{pi sqrt163} 70^2 form is true). It would not matter if you had the anthropic or natural values of the constants (or the fundamental values for that matter) as the relation would be invariant in (possibly) all cases. Here is something you will like Ulla as this gets close to the fundamentals of Nature and Platonism. If the Physics relation is equivalent to the ‘pure math’ relation then it can be said that “God is a mathematician.” in the Spinozan sense.

        Also, it should be noted that hc/piGm^2 = 2(M_pl/m)^2 (where M_pl = Planck mass and n = neutron mass) which again is ‘extremely close’ to e^2{pi sqrt163) 70^2. This strongly suggests the hierachy problem between the Planck scale and the lower proton, neutron energy scale could be related to the famous e^{pi sqrt163). All the relations above are quadratic and suggests along with the Leech lattice connection a highly symmetrical structure. The fact that this a very large number and that it is very close to a physics value is amazing. One last thing and I am gone. If you invert the value and multiply a factor 2 you get what looks to be the very very weak ‘gravitational coupling constant’ 5.92176970… *10^-39

      • Lawrence B. Crowell says:

        @Mark Thomas, These appear to indicate that the Leech lattice, the 24 dimensional subspace of the Jordan J^3(O), as the automorphism of the Fischer-Griess group, carry these numerical quantities. I have been looking some into these matter since we communicated last summer. The 163 is the top number in the Stark-Heegner theorem on algebraic integers. The algebraic integers determine the compactification of Calabi-Yau manifolds.


      • Kea says:

        Mark, they (163 and proton mass) are connected by the qubit/qutrit tetractys.

      • Kea says:

        Which is to say: the proton mass is 3 in natural QG units. The 24 dimensions of the Leech lattice sit in the hexagon at the centre of the tetractys, which are the leptons and quarks. The mirror neutrinos are the three outer points (the diagonal for J3(O)).

      • Ulla says:

        Cool. This also gives the Bohr orbits?

        Today I got another answer which meant this all is inherent in light, carried by em-force. You just have to change the spin. Yes, i like this.:D

      • Ulla says:

        In fact, all our pure numbers are defined from these fundamental constants.

  10. Ervin,

    braids emerge from TGD via two manners.

    a) At the level of principle braids emerge from the assumption that finite measurement resolution has as geometric correlate at the level of discretization at the level of partonic 2-surfaces at the ends of causal diamonds. The orbits of partonic 2-surfaces are light-like 3-surfaces and thus replaced by braids in finite measurement resolution.

    b) Concretely braids emerge from the study of the solutions of modified Dirac equation as a prediction of quantum TGD: dynamics itself represents the finite measurement resolution. What happens is a localization to braid strands using the analog of periodic boundary conditions. On braid strands with quantized length are allowed: the length is in the effective metric defined by modified gamma matrices defined as contractions of gamma matrices of M^4xCP_2 with canonical momentum densities of Chern-Simons action. See this.

    At quantum level discretization could be understood as following from cutoff to the modes of the induced spinor field. For finite number of modes the anti-commutations can hold true only for a finite number of points defining braid ends. Inclusions of hyper-finite factors of type II_1 provide the representation for finite measurement resolution at deeper level and quantum groups and braids relate to these very closely.

    • Ervin Goldfain says:


      Thanks for the reply, but your answer did not address my question. I asked you for specific experimental evidence in support of braids and knots.


  11. With all respect and deep admiration to Witten as a mathematical physicist, I dare to make some shy remarks. Take them as mumble of a crackpot. Maybe Witten’s vision is limited by the fact that he has worked in M-theory framework. Even Witten is a child of his time.

    a) D=10 and 11 are the dimensions of string and M-theories. Dimensions D=1,2,4,8 are extremely natural from number theory point of view and appear in TGD. The route from M^4xCP_2 to octonions is not long. Not surprisingly, one of the basic threads of TGD is “Physics as generalized number theory”.

    I would not be surprised if Witten had begun to realize it is 3- and 4-D surfaces instead strings and string world sheets which are the fundamental objects: of rather objects of dimension 1,2,3,4 together.

    b) p-Adic numbers is second thread of “Physics as generalized number theory” approach. I think that here again the lack of physical input is the problem. In my own case the success of p-adic mass calculations left no doubt that p-adic physics and real physics must be unified. Number theoretical universality would be the name of the principle and the challenge is to give it contents. The properties of p-adic number fields led also rapidly to the realization that p-adic physics would be excellent correlate for cognition and intentionality. But this requires that one tries to understand the puzzle of consciousness seriously. Even Witten would become unemployed if he showed public interest in this kind of topics.

    c) Finite measurement resolution is something very fundamental but for some reason it is not taken seriously by mathematical physicists. This leads to braids and string world sheets. And also to the realization of importance of hyper-finite factors and allows to interpret quantum groups physically without bringing in Planck length scale mystics. This also more or less forces the idea that the lines of Feynman graphs at deeper level are braids and that in vertices the lines only redistribute: this is the OZI rule of pre-QCD hadron physics. The ability to get rid of all n>2-vertices has obvious implications.

    d) M-theorists have never realized the enormous unifying power of sub-manifold geometry. Induced metric disappeared from string theory when Polyakov trick is accepted. String people never realized that the notion of induced spinor structure might be fundamental and solve basic problems due to the non-existence of spinor structure for a generic space-time. In lattice QCD the existence of 16 different spinor structures for periodic boundary conditions causes the problems with fermions. Induced spinor structure resolves all these problems. It also leads to a new view about baryons and leptons and predicts separate conservation for them. Proton does not decay and mathematical physicists should ask what stability could mean mathematically! Note also that the possibility of octonionic spinor structure makes the dimension D=8 unique and one can define the notion of quaternionic 4-surface.

  12. Kea says:

    Phil, why are you asleep? OPERA confirmed tachyons.

    • Philip Gibbs says:

      I know. Motl has the rumour a couple of days ago. They just ruled out one possible cause of error.

    • D R Lunsford says:

      He’s not asleep. He understands that there are no tachyons, no matter what this bad experiment says. Did you read Cohen and Glashow? Did you read Einstein for that matter? There are no tachyons, period.


      • Kea says:

        D R Lunsford, it is YOU that has not been following this conversation. I was one of the first people to read Cohen-Glashow, but I don’t believe in applying theories in an arbitrary manner to experiments that have the potential to rule out said theories. Especially given that I was working on neutrino gravity BEFORE the OPERA result. So, yes, verification is required. But are you, in your profound ignorance, correct, just because you say so? Me thinks not.

  13. D R Lunsford says:

    I just can’t believe it. People are reading about knot theory when they don’t even understand special relativity. Why bother even talking any more? Physics is finished.


    • Kea says:

      DRL, if you weren’t so ignorant you would know that knot theory is the very well established basis for 2d condensed matter systems in the laboratory, under the guise of topological field theory. It is also so deeply involved in the foundations of mathematics, that it cannot fail to be relevant to the mathematics of quantum gravity, however that turns out.

      • D R Lunsford says:

        Yes, Witten got his Fields mathematics medal for this nice work I think back in the 80s. I just don’t care. We are stuck with the same intractable problems as before. You can learn all you want about QFT from 100 different angles, and it’s still going to be impossible to do the simplest calculations with nucleons.


      • Kea says:

        So you think nothing has happened since the 1980s? Google Freedman’s and Kitaev’s work. That’s a start. And that’s just one of many, many topics …

      • Lawrence B. Crowell says:

        The Jones’ polynomial is a Skein relationship for a knot. The function W(C) = exp( i∫A*dx) is the Wilson line or loop integral for the valuation of a gauge connection. The expectation value is the path integral

        = ∫D[g,A]W(C)e^{-iS}

        Define the element α = 1 – 2πi/kN, for N = mode number and k = momentum vector, and z = -2πi/k. The Skein relationship is then

        α – α^{-1} = z.

        The + and – refer to crossing over or under. The Wilson line or loop is then a measure of the Ahronov-Bohm “quantum flux” due to a gauge potential. It then really does not make much sense to talk about knots existing as such. This is really nothing more than a technique for managing path integrals according to certain indices, which are first Chern classes. It actually makes little to talk about “knots existing” as such. This is basically a sort of mathematical method

        There are fermionic versions of this called the Skymrion (if I remember the spelling) which has a braid group interpretation. So electrons in condensed matter exist on or in these funny flux tubes.


    • Leo Vuyk says:

      There is indeed a frim reason to assume that Special relativity will have to be extended if we understand the internal structure of Quarks, Leptons and the oscillating massless Higgs-vacuum itself.


  14. Lawrence B. Crowell says:

    I hate these blog entries which kill off carrot signs. I repeat this from the statement of the path integral.

    \langle W(C)\rangle = ∫D[g,A]W(C)e^{-iS}

    Define the element α = 1 – 2πi/kN, for N = mode number and k = momentum vector, and z = -2πi/k. The Skein relationship is then

    α\langle W(L^+) \rangle – α^{-1}\langle W(L^-)\rangle = z\langle W(L^0)\rangle.

    The + and – refer to crossing over or under. The Wilson line or loop is then a measure of the Ahronov-Bohm “quantum flux” due to a gauge potential. It then really does not make much sense to talk about knots existing as such. This is really nothing more than a technique for managing path integrals according to certain indices, which are first Chern classes. It actually makes little to talk about “knots existing” as such. This is basically a sort of mathematical method

    There are fermionic versions of this called the Skymrion (if I remember the spelling) which has a braid group interpretation. So electrons in condensed matter exist on or in these funny flux tubes.


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