String Theory and M-Theory

434 String geometry
folds, constrains M7 to have G2 holonomy. In such a compactification to
flat D = 4 Minkowski space-time, there should exist one spinor (with four
independent components) satisfying
δψM = ∇Mε = 0. (9.211)
The background geometry is then ¡ 3,1 × M7, where M7 has G2 holonomy,
and ε is the covariantly constant spinor of the G2 manifold tensored with a
constant spinor of ¡ 3,1 . As in the case of Calabi–Yau three-folds, Eq. (9.211)
implies that M7 is Ricci flat. Of course, it cannot be Kähler, or even complex,
since it has an odd dimension. Let us now examine why Eq. (9.211) implies
that M7 has G2 holonomy.
The exceptional group G2
G2 can be defined as the subgroup of the SO(7) rotation group that preserves
the form
where
ϕ = dy 123 + dy 145 + dy 167 + dy 246 − dy 257 − dy 347 − dy 356 , (9.212)
dy ijk = dy i ∧ dy j ∧ dy k , (9.213)
and y i are the coordinates of ¡ 7 . G2 is the smallest of the five exceptional
simple Lie groups (G2, F4, E6, E7, E8), and it has dimension 14 and rank 2.
Its Dynkin diagram is given in Fig. 9.11. Let us describe its embedding in
Spin(7), the covering group of SO(7), by giving the decomposition of three
representations of Spin(7), the vector 7, the spinor 8 and the adjoint 21:
• Adjoint representation: decomposes under G2 as 21 = 14 + 7.
• The vector representation is irreducible 7 = 7.
• The spinor representation decomposes as 8 = 7 + 1.
G2
Fig. 9.11. The G2 Dynkin diagram.
The singlet in the spinor representation precisely corresponds to the covariantly
constant spinor in Eq. (9.211) and this decomposition is the reason
why G2 compactifications preserve 1/8 of the original supersymmetry, leading
to an N = 1 theory in four dimensions in the case of M-theory. While