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Existence of a complex structure on the 6 dimensional sphere is proved in this paper. The proof is based on re-interpreting a hypothetical complex structure as a classical ground state of a Yang–Mills–Higgs-like theory on S6. This classical vacuum solution is then constructed by Fourier expansion (dimensional reduction) from the obvious one of a similar theory on the 14 dimensional exceptional compact Lie group G2. AMS Classification: Primary: 32J17, 32J25; Secondary: 43A30, 53C56, 81T13

Complex structure on the six dimensional sphere from a spontaneous symmetry breaking