Generalized Calabi-Yau manifolds(19)

A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphis

Nowsupposethatρisreal.Proposition2saysthattherearecomplexpurespinorsα,βwithρ=α+β.Realityo erstwopossibilities:αandβarebothreal,orβ=α¯.Ifα,βarerealthensois α,β andsofrom(10)q(ρ)>0.Ifβ=α¯then α,α¯ isimaginaryandq(ρ)<0.FromProposition2,wededuce:

Proposition3Letρ∈Sbearealspinorwithq(ρ)<0.Thenρistherealpartofapurespinor with , ¯ =0.

Thesearepreciselythepurespinorsweneedinthede nitionofageneralizedCalabi-Yaumanifold.

WhenthevectorspaceVisreal,theopenset

U={ρ∈S:q(ρ)<0}

isactedontransitivelybytherealgroupR ×Spin(6,6).Weshallstudynextthegeometryofthisspace,followingcloselytheparalleldiscussionofthreeformsinsixdimensions,asin[10].Infact,whatwearedoinghereisadirectgeneralizationofthatwork.

5.2Thesymplecticgeometryofthespinrepresentation

De nition4OntheopensetU Sforwhichq(ρ)<0,de nethefunctionφ,homogeneousofdegree2,by φ(ρ)=

搜索“diyifanwen.net”或“第一范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，第一范文网，提供最新工程科技Generalized Calabi-Yau manifolds(19)全文阅读和word下载服务。