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by Andre Henriques, hereby I confirm that this published 2015 version is indeed completely independent of the wrong and withdrawn 2005 version available on arXiv.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.)