流形(英语:G2 manifold)和 S p i n ( 7 ) {\displaystyle \mathrm {Spin} (7)} 流形(英语:Spin(7) manifold)。 注意 S p ( n ) ⊂ S U ( 2 n ) ⊂ U ( 2 n ) ⊂ S O ( 4 n ) {\displaystyle
discovered by John H. Schwarz in 1972) need to be compactified on a Calabi–Yau manifold. (In string theory, compactification is a generalization of Kaluza–Klein