Complex Structures on the Six-sphere

A preprint by Andrei Marshakov and Antti Niemi appeared on hep-th this evening making a remarkable claim. According to this preprint, a few weeks before passing away recently at the age of 93, Shiing-Shen Chern completed a preprint entitled “On the Non-existence of a Complex Structure on the Six Sphere”.

Whether or not a given manifold defined using real coordinates can be given the structure of a complex manifold is often a difficult problem. For the case of a d-dimensional sphere, clearly you can’t do this in odd dimensions, but for even dimensions, you certainly can for the case d=2. For the cases d=4 and d=8 or more, there is a topological obstruction to even finding an “almost complex structure”. In other words, you can’t find a continous choice for each point on the sphere of what it means to multiply elements of the tangent space by the square root of minus one. The case d=6 is special: you can use the octonions to construct an almost complex structure, but this complex structure is not “integrable”, it doesn’t come from any local choice of complex coordinates. One of the most famous open problems in geometry has long been the following: is there another almost complex structure on the six-sphere that is actually integrable?

It has long been conjectured that there is no such integrable almost complex structure, but no one has ever been able to prove this. Chern’s preprint contains a purported proof, but Marshakov and Niemi devote only a paragraph to the non-trivial part of his argument. From their preprint you can’t tell whether Chern has a valid argument.

I’ve heard via e-mail from a knowledgeable authority on the subject who points out that there are serious flaws in the manuscript that was privately circulated. His opinion is that Chern’s argument actually does prove something interesting, but not the full result Chern claims, so the conjecture about the non-existence of a complex structure on the six-sphere remains open.

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11 Responses to Complex Structures on the Six-sphere

  1. Peter says:

    The section of the paper on “Chern’s Last Theorem” is almost entirely devoted to explaining well-known facts about G_2/SU(3)=S^6, including the well-known fact that the almost complex structure you get from the octonions is not integrable. I’m pretty sure all of this material goes back to Elie Cartan in the early part of the last century. The new argument due to Chern is only dealt with in the last paragraph of the section (I said it was only one paragraph, not one sentence. String theorists have a weird inability to quote me correctly…). It’s the argument in this last paragraph that is new, but also has a hole in it.

  2. DMS says:


    You have captured what I tried to say. And in the process you have given me a glimpse of the complexities and subtleties of (dis-)proving statements in mathematics. Thanks!

  3. Lubos Motl says:

    It is not true that there is just one sentence dedicated to Chern’s proof in Marshakov et al. Look at their conclusions – and you will see that they say that “we have explained his proof in detail”. I also think that there’s a lot of it over there.

  4. Peter says:

    I think what you mean by “physicist’s proof” is something along the lines of what a mathematician would call an outline of a proof, with some steps in the outline possibly justified by appeal to a physical argument that such a step has to somehow work out correctly.

    There certainly are some reasonably good examples of this kind of thing. One would be Witten’s work on supersymmetry and Morse theory. There he outlined an argument relating Morse theory, Hodge theory and index theory, using at crucial points expectations of what should happen based on semi-classical approximation techniques in quantum mechanics.

    But the Ricci-flow case is very different. Based on personal experience, I can tell you that claims by some physicists that the RG provides a “physicist’s proof” of geometrization have convinced some mathematicians that said physicists are arrogant idiots. In this subject you can’t assume that metrics are well-behaved, since they aren’t. Singularities develop when you follow the Ricci flow, and understanding what happens then is what the whole subject is about. There is no “physical” argument for ignoring this phenemenon. If there were it would imply an infinite number of untrue mathematical theorems.

  5. DMS says:

    Someone wrote:
    “If it’s wrong for a mathematician, it must also be wrong for the physicist.”

    Of course that is obvious: don’t need a Harvard education to see that.

    The question was: is there a strong physical reason for one to believe a particular statement to be true. Of course, such physical reasons may turn out to be wrong. Alternatively, the statement may be correct, but may require newer mathematics to demonstrate that rigorously (like Dirac delta functions —> distribution theory).

    “A difference between a physicist’s proof and a mathematician’s proof can only exist in physics where the physicists know what they mean and mathematicians are slower.”

    Well, the recent proof of Thurston geometrization conjecture relied on Hamilton’s Ricci flows, reminiscent of RG.
    A physicist would be happy if he could show it for “well-behaved metrics”, not a (“slow”) mathematician.

  6. plato says:

    It’s important that we can see where you are both focused instead of having the ramble of content that deters from appropriate discussion.

    So we now have you both fixed in the appropriate direction, so that you can philosphically:) discuss the idea being expounded at the forefront of our knowledge base.

    Is one, going to reject the mathematics of another? On what grounds? Physics?:)

  7. Luboš Motl says:

    Impressive guy (Chern), even it it’s not right.

    Someone asked whether the proof is valid as a “physicist’s proof”. That’s a ludicrous question. A difference between a physicist’s proof and a mathematician’s proof can only exist in physics where the physicists know what they mean and mathematicians are slower.

    In mathematics, for example in discussions about the existence of a complex structure, the physicist’s proof and a mathematician’s proof is the same thing. If it’s wrong for a mathematician, it must also be wrong for the physicist.

  8. plato says:

    At planck scale with topological considerations, no determination can ever be foolproof, as it is conjectured math already based, unless, you willingly admit and move into abstract spaces, detached from reality:)

    But its still fun to conjecture?

  9. Peter says:

    The Marshakov-Niemi paper just gives an outline of Chern’s argument, it doesn’t in any sense provide a “physicist’s proof”.

  10. DMS says:

    What about Marshakov-Niemmi paper itself? Is it valid as a “physicist’s proof”?

    That is, even if Chern was wrong, it may be that someone is going to prove it rigorously, so it is basically correct (like a lot of QFT proofs).

  11. The claim that S^6 does not possess complex structure is very interesting from TGD point of view. Since I cannot expect that anyone sees the trouble of getting bored of what follows I demonstrate my deep ignorance by representing my question immediately:

    Does the possible existence of complex structure in S^6 imply the existence of Kaehler structure?

    In the following I try to the physical meaning of Chern’s last theorem emerges in TGD Universe.

    1. Number theoretical dynamics for space-time surfaces

    The motivation for my question comes from what I believe but cannot yet quite prove(;-). I have been working hardly to concretize my belief that the classical dynamics of space-times identified as 4-dimensional surfaces in 8-D imbedding space M^4xCP_2 can be expressed purely number theoretically. I have gone through several variants of the hypothesis and thought to describe one option since it seems most promising at this moment and also relates very closely to the Chern’s last theorem.

    The first guess was that tangent spaces of H and M^4 could be given octonionic resp. quaternionic structures. The Minkowskian signature however forces to modify the approach by replacing these structures by their hyper counter parts obtained by multiplying imaginary units by commuting sqrt(-1).

    [This structure is not identical with the commutative algebra structure that I called hyper-complex structure and talked about in an earlier email containing several mistakes.]

    Clearly a number theoretic analog for the transition from Riemannian to pseudo-Riemannian geometry is in question. M^4 resp. M^8 could be seen as a sub-space of complexified quaternions resp. octonions. This space does not form field nor algebra but this can be tolerated: for instance, the overall important notion of hyper-prime makes sense. For instance, in the quaternionic case the interpretation would be as four-momenta satisfying automatically the stringy mass formula M^4=p. Light-like hyper numbers having no inverse correspond to massless 4- or 8-momenta and hyper-units to Lorentz boosts in analogy with pslash/m in the case of Dirac equation. The notion of pole of analytic function is replaced with lightlike 3-surface, etc…

    2. Number theoretical variant of spontaneous compactification as M^8M^4xCP_2 duality?

    Since HO/HQ power series with real coefficients give end result in HO/HQ, the notion of HO/HQ manifold with hyper transition functions between coordinate charts makes sense. HO analyticity is not plagued by the complications due to non-commutativity and non-associativity. The reason is that this notion results also if product is Abelianized by assuming that different HO imaginary units multiply to zero.

    The problem is that M^4xCP_2 very probably does not allow HO structure.

    Here comes in rescue the old idea is that four-surfaces in M^8 define four-surfaces in M^4xCP_2 in a natural manner and vice versa assuming that field equations are satisfied. This duality would be a number-theoretical counterpart of spontaneous compactificatition. It would have nothing to do with dynamics but would be one item in the list of dualities relevant to TGD.

    3. Justification for the number theoretical spontaneous compactification

    The justification for the hypothesis comes from following observation. The space of quaternionic sub-spaces of octonions with a priori fixed complex structure (containing a fixed octonionic imaginary unit) is CP_2. Same applies in the hyper case.

    This means that if one has a four-surface X^4 in M^8==HO with a hyper-quaternionic tangent space and a fixed complex structure, it defines a surface in M^4xCP_2. The M^4 coordinates of a given point are obtained by a canonical projection from M^8 to M^4 and CP_2 coordinates as parameters characterizing the HQ tangent space at given point.

    One can assume that the local complex structure depends on space-time point and thus characterized by a map

    f: X^4–>S^6.

    It is here where S^6 pops up in TGD framework.

    4. Foliations of HO=M^8 and M^4xCP_2 by space time surfaces maps g: OH–>SU(3) satisfying integrability conditions.

    For a given map f: HO –> S^6 defining a local preferred imaginary unit M^8-M^4xCP_2 duality allows to construct a foliation of HO/M^4xCP_2 by HQ space-time surfaces in terms of maps

    g: HO–> SU(3)

    satisfying certain integrability conditions guaranteing that the distribution of hyper-quaternionic planes integrates to a foliation by 4-surfaces.

    The reason is that the bundle projection SU(3)–>CP_2 defines the local tangent plane at each point of SU(3). The foliation defines a four-parameter family of 4-surfaces in M^4xCP_2. The dual of this foliation defines a 4-parameter family HQ space-time surfaces.

    5. Hyper-octonion analytic functions OH–>OH as a solution to the integrability conditions?

    HO analytic functions HO–> HO with real Taylor coefficients provide a physically motivated ansatz, which might satisfy the integrability conditions.

    a) The basic observation is that the complexified octonions decompose as 1+1+3+3bar under SU(3) automorphisms leaving a preferred imaginary unit fixed (easy to remember if you have heard about leptons and quarks!).

    [Notice that SU(3) has interpretation as color group and isometry group of CP_2 whereas U(2) would act as vector fields in the tangent space of X^4 as well as the holonomy group of CP_2 identifiable as electro-weak gauge group. Standard model gauge structure would have purely number theoretical origin.]

    b) If you have a map HO–>HO you can form tensor product 3x3bar of the states of 3 and 3bar states defined by the map and identify it as a Lie algebra element of SU(3) and exponentiate it to an element g of HO-local SU(3). The resulting map g defines the foliation of M^4xCP_2 by hyper-quaternionic space-time surfaces as desired.

    6. How Chern’s last theorem could be relevant for TGD?

    If the conjecture holds it would mean that the foliations giving families of solutions of field equations are characterized by two functions.

    a) The function f: HO–>S^6 characterizing the selection of preferred imaginary unit in OH. Physically the function would characterize the choice of the ground state. This function has also an interpretation as SO(7) local group element with SO(6) gauge invariance. The interpretation would be in terms of zero modes.

    It is here, where Chern’s last theorem becomes relevant. The interpretation as zero modes would conform with the claim that S^6 does not allow complex structure. Indeed, Kahler structure is absolutely essential for the identification of these degrees of freedom as quantum fluctuating degrees of freedom in infinite-dimensional context. As a believer on TGD based world view I would dare to believe also the non-existence conjecture.

    b) The hyper-octonion analytic function g: HO–>SU(3) or effectively g: HO–>CP_2 guaranteing the integrability conditions would be second function involved with the general solution ansatz. Here complex structure is present and quantum fluctuating degrees of freedom are in question.

    7. Is number theoretical dynamics equivalent with absolute minimization of Kaehler action?

    The basic conjecture is that the absolute minima of Kaehler action correspond to the hyper-quaternionic surfaces in the proposed sense. The enormous vacuum degeneracy of Kaehler action would relate to the local selection of octonionic imaginary unit characterized by the map f: HO–>S^6. The known facts about the solution spectrum of Kahler action conform with the proposed general picture.

    This conjecture has several variants. It could be that only the asymptotic behavior of absolute minima corresponds to a hyper-octonion analytic function. It could also be that maxima of Kaehler
    function K of configuration space of 3-surfaces of H, with K being determined by absolute minimum of Kahler action, correspond to this kind of 4-surfaces. Etc…

    With Best Regards,

    Matti Pitkanen

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