# Complex structure on $S^6$ gets published in Journ. Math. Phys

A paper by Gabor Etesi was published that purports to solve a major outstanding problem:

Complex structure on the six dimensional sphere from a spontaneous symmetry breaking Journ. Math. Phys. 56, 043508-1-043508-21 (2015) journal version, current arXiv version.

Since this is obviously an important and groundbreaking result (if true), published in a physical journal, I am interested whether it is accepted by mathematicians.

• I do not think it is the wrong forum. This is not a preprint, it is a peer-reviewed and published paper claiming to solve a very famous and long-standing problem. It is completely natural to wonder whether it is actually correct or not. At any rate, I agree that the question could be improved. – Francesco Polizzi Jun 24 '15 at 22:38
• I honestly don't understand these comments. The person who posted the question is one of the leading experts in hyperkahler geometry worldwide. He is trying to understand whether the community has changed opinion regarding a paper he himself has already read and found incorrect a long time ago. It would be beneficial for the entire mathematical community to understand the status of the claimed result. (that being said, one might encourage the OP to rephrase the question so that it appears more polite and kind towards the author) – pro Jun 25 '15 at 1:04
• I've opened up a discussion at meta: meta.mathoverflow.net/questions/2328/… Please consider upvoting this comment for visibility. – Todd Trimble Jun 25 '15 at 12:49
• Professor Verbitsky: I don't know whether you've been following the meta discussion, but there seems to be growing consensus that the question might still be improved if you could perhaps relate, within the body of this question, some of the specific mathematical concerns you had which originally led to asking this question (concerns you alluded to before the edits). Many believe it is best practice for MathOverflow, and perhaps all the more so when asking questions about correctness of a paper, to pinpoint precise mathematical issues. Many thanks for your understanding. – Todd Trimble Jun 26 '15 at 2:07
• I have communicated with Etesi. He claims that the published paper is essentially all new material, and that it has very little in common with the previous (flawed) preprint. – André Henriques Jun 26 '15 at 10:42

I am Gabor Etesi, the author of the current paper "Complex structure on the six dimensional sphere from a spontaneous symmetry breaking", Journ. Math. Phys. 56, 043508-1-043508-21 (2015). First of all I would like to thank for the interest in my work on this classical problem. Because I have been asked by Andre Henriques, hereby I confirm that this published 2015 version is indeed completely independent of the wrong and withdrawn 2005 version available on arXiv. Therefore it is absolutely unnecessary to spend time with that version.

This new published 2015 version is self-contained and constructs a complex structure on $S^6$ using the following two simple observations:

(i) A complex structure on a complex manifold can always be re-interpreted as a spontaneously broken classical vacuum solution in a non-linear version of a Yang-Mills-Higgs theory formulated on the underlying manifold. This is because an almost complex tensor field $J$ is mathematically the same as a Higgs field $\Phi$. Although the terminology comes from theoretical physics, the corresponding mathematical structures are well-defined;

(ii) A complex structure on $S^6$ is then constructed as the Fourier expansion of the usual Samelson complex structure (regarded as spontaneously broken vacuum solution in the sense of item (i) above) on the exceptional Lie group $G_2$. The mathematical theory of this Fourier expansion is itself very useful and is contained in the text.

Please note that this is a freshly new approach to this old problem, apparently without a predecessor.

Finally, I would like to kindly ask everybody to read the paper first, before sending negative comments. I am ready to explain some details of the construction however please understand that I cannot take part in an infinitely long discussion. (Recently I have been working on different stuffs.)

Thanks again,

Gabor

Here is an answer to the question of YangMills (thank you!) regarding the subbundle $H \subset TG_2$:

I did not claim in the text that the subbundle $H \subset TG_2$ should be integrable in any sense. What I only need is the formal fact that the functional in eq. (15) vanishes on the constructed triple $(\nabla_H,J_H,g_H)$ as can be verified by a direct calculation. However I acknowledge that YangMills was right and the Levi--Civita connection does not preserve $H\subset TG_2$ as it was written in the text. For a corrected version please visit:

http://www.math.bme.hu/~etesi/s6-spontan.pdf

Neverthekess after this observation the proof proceeds as follows: $J_H$ is Fourier expanded and the corresponding ground mode, denoted by $J$ in the text after eq. (21), descends to $S^6$. The very important subtlety however is this (explained carefully in Section III): in our situation (i.e., Fourier expanding general sections of general vector bundles), there is NO canonical way to perform a Fourier expansion. Instead there is "moduli space" of possible Fourier expansions resulting in inequivalent ground modes. This is because doing fiberwise integration does NOT commute with gauge transformations hence Fourier expanding a gauge transformed (on $H$) section is NOT the same as gauge transforming (on $TS^6$) the ground mode of a Fourier expanded section. I construct in Lemma 5.1 a distinguished "$\alpha$-twisted" Fourier expansion of whose ground mode $J$ coincides with $J_H$ itself.

I think that most of the concerns and uncertainty about the published version is related with the historical remark that the "relationship" between Yang--Mills theory (mathematically invented in the 1980's) and classical complex manifold theory, more precisely the Kodaira--Spencer deformation theory (invented in the 1950-60's) is not fully clarified. By this I mean that apparently the action of the gauge group on an (almost) complex manifold $(M,J)$ is dubious: it can describe both just a symmetry transformation of $(M,J)$ or an effective deformation of $(M,J)$. But these certainly should be carefully distinguished. My suggestion is formulated in the "Principle" of Section II (but this point might require a more conceptional and less ad hoc work, I agree).