382	9 Supergravity: An Anthology of Solutions

and the spectrum of fluctuations of the background arranges into Osp(N | 4) supermultiplets furthermore assigned to suitable representations of the bosonic flavor group.

9.3.4 The SO(8) Spinor Bundle and the Holonomy Tensor

We come next to discuss a very important property of 7-manifolds with a spin structure which plays a crucial role in understanding the supergauge completion. This is the existence of an SO(8) vector bundle whose non-trivial connection is defined by the Riemannian structure of the manifold. To introduce this point and in order to illustrate its relevance to our problem we begin by considering a basis of D = 11 gamma matrices well adapted to the compactification on AdS4 × M7.

9.3.5 The Well Adapted Basis of Gamma Matrices

According to the tensor product representation well adapted to the compactification, the D = 11 gamma matrices can be written as follows:

Γa = γa	18×8	(a = 0, 1, 2, 3)	(9.3.62)
Γ3+α = γ5 τα		(α = 1, . . . , 7)

where, following the old Kaluza Klein supergravity literature [26, 30, 41] the matrices τα are the real antisymmetric realization of the SO(7) Clifford algebra with negative metric:

{τα , τβ } = −2δαβ ;	τα = −(τα )T	(9.3.63)
In this basis the charge conjugation matrix is given by:
C = C 18×8		(9.3.64)
where C is the charge conjugation matrix in d = 4:
C γa C −1 = −γaT ;	C T = −C	(9.3.65)

9.3.6 The so(8)-Connection and the Holonomy Tensor

Next we observe that using these matrices the covariant derivative introduced in (9.3.48) defines a universal so(8)-connection on the spinor bundle which is given once the Riemannian structure is given, namely once the vielbein and the spin con-

9.3 Flux Vacua of M-Theory and Manifolds of Restricted Holonomy						383
nection {Bα , Bαβ } are given:
1
Uso(8) ≡ −				Bαβ ταβ − eBα τα		(9.3.66)
			4
More precisely, let ζA7 be an orthonormal basis:
				ζB	= δAB	(9.3.67)
	ζ A
7				7	77

of sections of the spinor bundle over the Einstein manifold M7. Any spinor can be written as a linear combination of these sections that are real. Furthermore the bar operation in this case is simply the transposition. Hence, if we consider the so(8) covariant derivative of any of these sections, this is a spinor and, as such, it can be expressed as a linear combinations of the same:

so(8) ζA	≡ d + Uso(8) ζA	= UAB ζB	(9.3.68)
7	7	77 7

According to standard lore the one-form valued, antisymmetric 8 × 8 matrix UAB

77

defined by (9.3.68) is the so(8)-connection in the chosen basis of sections. If the manifold M7 admits N Killing spinors, then it follows that we can choose an orthonormal basis where the first N sections are Killing spinors:

ζA = ηA; so(8) ηA = 0, A = 1, . . . , N

(9.3.69)

and the remaining 8 − N elements of the basis, whose covariant derivative does not vanish are orthogonal to the Killing spinors:

ζ

= ξ

;

so(8) ξ

= 0,

A

= 1, . . . , 8 − N

A

A

A

ηA = 0

(9.3.70)

ξ

B

ξ

= δ

ξ

B

C

BC

It is then evident from (9.3.69) and (9.3.70) that the so(8)-connection UAB takes

77

values only in a subalgebra so(8−N ) so(8) and has the following block diagonal form:

UAB =	0	U				(9.3.71)
	0	0
77
			AB

Squaring the SO(8)-covariant derivative, we find

2ζA	= (dUAB					− UAC				UCB ) ζB
7				77		77				77		7
						FAB [ ]
						77		U
			1
	= −			Rγ δ αβ − 4e2δγ δ αβ									τγ δ ζA	(9.3.72)
			4
													7
							C		γ δ
									αβ

where C γ δ αβ is the so called holonomy tensor, essentially identical with the Weyl tensor of the considered Einstein 7-manifold.

384	9 Supergravity: An Anthology of Solutions

9.3.7 The Holonomy Tensor and Superspace

As a further preparation to our subsequent discussion of the gauge completion let us now consider the form taken on the AdS4 × G /H backgrounds by the operator Kab introduced in (9.3.9) and governing the mechanism of supersymmetry breaking. We will see that it is just simply related to the holonomy tensor discussed in the previous section, namely to the field strength of the SO(8)-connection on the spinor bundle. To begin with, we calculate the operator Fa introduced in (9.3.3), (9.3.5). Explicitly using the well adapted basis (9.3.62) for gamma matrices we find:

Fa = Fa = −2eγa γ5 18

Fα = −e14 τα

Using this input we obtain:

Kab = 0

K

Kaβ

=

0

1

γ δ

2

γ δ

ab

(

δ

) τ

K

αβ = −

4

R

αβ −

4e

αβ

γ δ

=

Cγ δ αβ

(9.3.73)

(9.3.74)

Where the tensor Cγ δ αβ defined by the above equation is named the holonomy tensor and it is an intrinsic geometric property of the compact internal manifold M7. As we see the holonomy tensor vanishes only in the case of M7 = S7 when the Riemann tensor is proportional to an antisymmetrized Kronecker delta, namely, when the internal Einstein 7-manifold is maximally symmetric. The holonomy tensor is a 21 × 21 matrix which projects the SO(7) Lie algebra to a subalgebra:

Hhol SO(7)

(9.3.75)

with respect to which the 8-component spinor representation should contain singlets in order for unbroken supersymmetries to survive. Indeed the holonomy tensor appears in the integrability condition for Killing spinors. Indeed squaring the defining equation of Killing spinors with m = 1 we get the consistency condition:

Cγ δ αβ τγ δ η = 0

(9.3.76)

which states that the Killing spinor directions are in the kernel of the operators Cγ δ αβ τγ δ , namely are singlets of the subalgebra Hhol generated by them.

In view of this we conclude that the gravitino field strength has the following structure:

	ab		ρab = 0
				= 0
ρ		=	ρaβ		(9.3.77)
			ραβ	0	zero at θ = 0
				= ;
					depends only on the broken θ s

9.3 Flux Vacua of M-Theory and Manifolds of Restricted Holonomy

385

Table 9.1 The homogeneous 7-manifolds that admit at least 2 Killing spinors are all Sasakian or tri-Sasakian. This is evident from the fibration structure of the 7-manifold, which is either a fibra-

tion in circles S1 for the N = 2 cases or a fibration in S3 for the unique N = 3 case corresponding to the N 010 manifold

N Name Coset	Holon. so(8) Fibration
	bundle

8	S7	SO(8)
		SO(7)

2 M111 SU(3)×SU(2)×U(1)

SU(2)×U(1)×U(1)

2 Q111 SU(2)×SU(2)×SU(2)×U(1)

U(1)×U(1)×U(1)

2	V 5,2	SO(5)
		SO(2)

3 N 010 SU(3)×SU(2) SU(2)×U(1)

π

1

S7 =3P3

1

(p) S

1

p P ; π −

π

; π −1(p) S1

SU(3)

p P2 × P1

M

111

= P2

× P1

π

× P1; π −1

(p) S1

SU(3)

p P1 × P1

Q111

= P1 × P1

× P1

SU(3)

V

5,2

π

1

1

= Ma

quadric in P4

p Ma ; π − (p) S

π

SU(2)

p P2; π −1(p) S3

N 010

= P2

Table 9.2 The homogeneous 7-manifolds that admit just one Killing spinors are the squashed 7-sphere and the infinite family of N pqr manifolds for pqr = 010.

N	Name	Coset	Holon. so(8)
			bundle
1	S7	SO(5)×SO(3)	SO(7)+
	squashed	SO(3)×SO(3)
1	N pqr	SU(3)×U(1)	SO(7)+
		U(1)×U(1)

As a preparation for our next coming discussion it is now useful to remind the reader that the list of homogeneous 7-manifolds G /H of Englert type which preserve at least two supersymmetries (N ≥ 2) is extremely short. It consists of the Sasakian or tri-Sasakian homogeneous manifolds7 which are displayed in Table 9.1. For these cases our strategy in order to obtain the supergauge completion will be based on a superextension of the Sasakian fibration. The cases with N = 1 are somewhat more involved since such a weapon is not in our stoke. These cases are also ultra-few and they are displayed in Table 9.2.

7The theory of Sasakian manifolds, as applied to supergravity compactifications was discussed in [39]. In short an odd dimensional manifold is named Sasakian if the even dimensional cone constructed over it has vanishing first Chern class. After several manipulations this implies that the Sasakian manifold is an S1-fibre bundle over a suitable complex base manifold.