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

Abstract

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 S 6 . 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 G 2 .

Hodge numbers of a hypothetical complex structure on S6

“…Third, Gabor Etesi [Ete05v8] (see Remark 5.1 (3) in the arXiv version 8, arXiv:math/0505634v8) expects that h 0,1 (X) = h 1,1 (X) = 1 and h 0,2 (X) = h 1,2 (X) = h 1,0 (X) = h 2,0 (X) = 0. Bott-Chern cohomology in this case is considered in [McH1, Section 3.2].…”

Explicit construction of the complex structure on the six dimensional sphere

“…The proof is based on identifying S 6 with an exceptional conjugate orbit inside the exceptional compact Lie group G 2 and then restricting the Samelson complex structure of G 2 to this orbit. The proof to be presented here is elementary and self-contained hence is independent of our former Yang-Mills-Higgs theoretic approach [4].…”

“…The proof is based on identifying S 6 with an exceptional conjugate orbit [3] inside the exceptional compact Lie group G 2 and then restricting a Samelson complex structure on G 2 to this orbit. The proof to be presented here is elementary and self-contained hence is independent of our former Yang-Mills-Higgs theoretic approach [4]; nevertheless those considerations definitely have been used here as a source of ideas. We just note that meanwhile the treatment in [4] is based on the well-known SU(3)-fibration: the projection (i.e., a surjective mapping) π : G 2 → S 6 , our present proof rests on a less-known but very remarkable injection (i.e., an injective mapping) f : S 6 → G 2 .…”

“…The proof to be presented here is elementary and self-contained hence is independent of our former Yang-Mills-Higgs theoretic approach [4]; nevertheless those considerations definitely have been used here as a source of ideas. We just note that meanwhile the treatment in [4] is based on the well-known SU(3)-fibration: the projection (i.e., a surjective mapping) π : G 2 → S 6 , our present proof rests on a less-known but very remarkable injection (i.e., an injective mapping) f : S 6 → G 2 . Therefore the two approaches are dual to each other in this sense.…”

“…In this paper we shall construct explicitly the recent complex structure on the six dimensional sphere found in [4]. The proof is based on identifying S 6 with an exceptional conjugate orbit [3] inside the exceptional compact Lie group G 2 and then restricting a Samelson complex structure on G 2 to this orbit.…”