396	9 Supergravity: An Anthology of Solutions
since we have:
	D Q ≡ dQ + e2Ω Q + Q Ω = 0	(9.4.29)

as it can be explicitly checked. The above result follows because the matrix Kαβ commutes with all the generators of u(3). In view of (9.4.29) the integrability of the Killing spinor equation (9.4.26) becomes the following one:

Hol η = 0

(9.4.30)

where we have defined the holonomy 2-form:

Hol ≡ D 2 + e2Q Q =	−	4 Rαβ ταβ + e2Q Q		(9.4.31)
		1

and Rαβ denotes the curvature 2-form (9.4.12). Explicit evaluation of the holonomy 2-form yields the following result.

					0	0	0	0	0	0	8[B2							B6					B3				B5				8B5						B6						8B2					B3
					0	0	0	0	0	0	8B3					B4					−8B1						B6]				8	[	B1					B3−						B4			B6
																		B5−								B4										B5					−8B1							B2]
					0	0	0	0	0	0	8	[	B1										B2							]	8B4
			2		0	0	0	0	0	0	8				2					3	−		B	5			B	6			8	B			2					6−				B	3				5
Hol		e										B			B						−				1				3]						B						−					1		B
	=				0	0	0	0	0	0		[		4					6													[		3					4											6]
											8B				1		B			2−		8B				B				8B					1		B			5−			8B			B
					0	0	0	0	0	0	8		B					B					B	4				5			8		B					B						B	2				4
												[									−						B			]		[									−							B
					0	0	0	0	0	0																																									]
																					0																				8	K
																																						−
																																									0 =
				0		0 0		0	0 0										8K
																					=

(9.4.32) It is evident by inspection that the holonomy 2-form vanishes on the subspace of spinors that belong to the eigenspace of eigenvalue 2 of the operator P . In the chosen basis this eigenspace is spanned by all those spinors whose last two components are zero and on such spinors the operator Hol vanishes.

Let us now connect these geometric structures to the compactification ansatz.

9.4.3 The Compactification Ansatz

As usual we denote with Latin indices those in the direction of 4-space and with Greek indices those in the direction of the internal 6-space. Let us also adopt the notation: Ba for the AdS4 vielbein just as Bα is the vielbein of the Kähler threefold described in the previous section. With these notations the Kaluza-Klein ansatz is the following one:

G = 2e exp[−ϕ0]Kαβ ab 0 otherwise

9.4 Flux Compactifications of Type IIA Supergravity on AdS4 × P3				397
Ga a a a			−e exp[−ϕ0]εa1a2a3a4
	1 2 3 4		= 0 otherwise
Ha1a2a3			= 0
		ϕ = ϕ0 = const
		V a = Ba		(9.4.33)
		V α = Bα

ωab = Bab

ωαβ = Bαβ

where Ba , Bab respectively denote the vielbein and the spin connection of AdS4, satisfying the following structural equations:

0 = dBa − Bab Bcηbc

dBab − Bac Bdbηcd = −16e2Ba Bb

(9.4.34)

Ricab = −24e2ηab

while Bα and Bαβ are the analogous data for the internal P3 manifold:

0 = dBα − Bαβ Bγ ηβγ

dBαβ − Bαγ Bδβ ηγ δ = −Rαβγ δ Bγ Bδ

(9.4.35)

Ricαβ = 16e2ηαβ

whose geometry we described in the previous section.

With these normalizations we can check that the dilaton equation (6.7.44) and the Einstein equation (6.7.39), are satisfied upon insertion of the above Kaluza Klein ansatz. All the other equations are satisfied thanks to the fact that the Kähler form K=is closed and coclosed.

9.4.4 Killing Spinors on P3

The next task we are faced with is to determine the equation for the Killing spinors on the chosen background, which by construction is a solution of supergravity equations.

Following a standard procedure we recall that the vacuum has been defined by choosing certain values for the bosonic fields and setting all the fermionic ones equal to zero:

398	9			Supergravity: An Anthology of Solutions
	ψL/R|μ = 0
	χL/R = 0			(9.4.36)
	ρL/R|ab = 0

The equation for the Killing spinors will be obtained by imposing that the parameter of supersymmetry preserves the vanishing values of the fermionic fields once the specific values of the bosonic ones is substituted into the expression for the susy rules, namely into the rheonomic parameterizations.

To implement these conditions we begin by choosing a well adapted basis for the d = 11 gamma matrices. This is done by setting:

Γ a = γ a 1

Γ a = Γ α = γ 5 τ α (9.4.37)

Γ 11 = iγ 5 τ 7

Next we consider the tensors and the matrices introduced in (6.7.20), (6.7.22) and (6.7.23), (6.7.24). In the chosen background we find:

Mαβ =	1		eKαβ ; Mabcd =				1
							16 eεabcd
		4						(9.4.38)
					1
N0 =	0;			Nαβ =		eKαβ ;	Nabcd = −	1	eεabcd
					2			24

all the other components of the above matrices being zero. Hence in terms of the operators introduced in the previous section we find:

M± = ie 4 1 W −					2 iγ5 1
			1		3
N±(even) = e	2	1 W iγ5 1					(9.4.39)
	1

N±(odd) = 0

It is now convenient to rewrite the Killing spinor condition in a non-chiral basis introducing a supersymmetry parameter of the following form:

						ε = εL + εR								(9.4.40)
In this basis the matrices M and N (even) read
		1							1
M = M+						1 + Γ	11 + M−					1	− Γ	11
					2					2
			i						i
= −				eϕ GabΓ abΓ 11 −							eϕ Gabcd Γ abcd =
			8					16
=	e	γ5 (W τ7					+ 61)							(9.4.41)
	4

9.4 Flux Compactifications of Type IIA Supergravity on AdS4 × P3

399

			1						1
N (even) = N+(even)					1 +	Γ 11 + N−(even)				1	− Γ	11
				2					2
=	1	eϕ GabΓ ab +				1		eϕ Gabcd Γ abcd
	4						24
=	e	1	(W		+ 2τ7)							(9.4.42)
	2

Upon use of this parameter the Killing spinor equation coming from the gravitino rheonomic parameterization (6.7.32) takes the following form:

D ε = −M Γa V a ε

(9.4.43)

while the Killing spinor equation coming from the dilatino rheonomic parameterization is as follows:

0 = N (even)ε

(9.4.44)

Let us now insert these results into the Killing spinor equations and let us take a tensor product representation for the Killing spinor:

ε = ε η

(9.4.45)

where ε is a 4-component d = 4 spinor and η is an 8-component d = 6 spinor. With these inputs (9.4.43) becomes:

0 = D[4]ε η − eγa γ5Ba ε						2 +		4 P η
						3		1
+ε	D[6] + e	2	+	4 P τα Bα η						(9.4.46)
		3		1

while (9.4.44) takes the form:

0 = ε	1
	2 W + τ7 η	(9.4.47)

Let us now recall that (9.4.26) is integrable on the eigenspace of eigenvalue 2 of the P -operator. Then (9.4.46) is satisfied if:

D[4] − 2eγa γ5Ba ε = 0
Pη = 2η	(9.4.48)
(D[6] + eQ)η = 0

The first of the above equation is the correct equation for Killing spinors in AdS4. It emerges if the eigenvalue of P is 2. The second and the third are the already studied integrable equation for six Killing spinors out of eight. It should now be checked that