400	9 Supergravity: An Anthology of Solutions

the dilatino equation (9.4.47) is satisfied on the eigenspace of eigenvalue 2, which is indeed the case:

Pη = 2η		2 W + τ7	η = 0	(9.4.49)
		1

9.4.5 Gauge Completion in Mini Superspace

As a necessary ingredient of our construction let ηA (A = 1, . . . , 6) denote a complete and orthonormal basis of solutions of the internal Killing spinor equation, namely:

PηA = 2ηA
(D[6] + eQ)ηA = 0	(9.4.50)
ηAT ηB = δAB ; A, B = A = 1, . . . , 6

On the other hand let χx denote a basis of solutions of the Killing spinor equation on AdS4-space, namely (9.3.84), normalized as in (9.3.85). Furthermore let us recall the matrix K defining the intrinsic components of the Kähler 2-form.

In terms of these objects we can satisfy the rheonomic parameterizations of the one-forms spanning the d = 10 super-Poincaré subalgebra of the FDA with the following position:8

V a = Ba −

1

x γ a χy

xy

χ

8e

Ψ = χx

1

ηAΦx|A

V α = Bα −

ηAT τ α ηB A AB

8

ωab = Bab +

1

x γ abγ5χy

xy

χ

2

e

e

ωαβ = Bαβ +

ηAT τ

αβ ηB A AB −

K αβ KAB A AB

4

4

(9.4.51)

(9.4.52)

(9.4.53)

(9.4.54)

(9.4.55)

The proof that the above ansatz satisfies the rheonomic parameterizations is by direct evaluation upon use of the following crucial spinor identities.

Let us define

U =	3	1	+	1	P	(9.4.56)
	2			4

8With respect to the results obtained for the mini superspace extension of M-theory configuration everything is identical in (9.4.51)–(9.4.54) except the obvious reduction of the index range of (α, β, . . . ) from 7 to 6-values. The only difference is in (9.4.55) where the last contribution proportional to the Kähler form is an essential novelty of this new type of compactification.

9.4 Flux Compactifications of Type IIA Supergravity on AdS4 × P3

401

We can verify that:

ηAτ α U τ α ηB − ηAτ αβ ηB A AB = K αβ KAB A AB									(9.4.57)
Furthermore, naming:
		1
Bα = −				ηAT τ α ηB A AB					(9.4.58)
			8
	e						e
Δωαβ =		ηAT τ αβ ηB A AB −						K αβ KAB A AB	(9.4.59)
	4						4
we obtain:
−Δωαβ				Bβ =	e	ηAT τ α ηB A AC A CB			(9.4.60)
					8

These identities together with the d = 4 spinor identities (C.3.11), (C.3.12) suffice to verify that the above ansatz satisfies the required equations.

9.4.6 Gauge Completion of the B[2] Form

The next task is the derivation of the explicit expression for the B[2] form. Differently from the case of the 3-form this is possible and has a great value since it allows an explicit expression for the Green Schwarz σ -model describing string propagation in this background.

There is an ansatz for B[2] which is the following one:

B[2] = αχ	x χy		Aτ7ηB ΦAx ΦBy			(9.4.61)
		η
By explicit evaluation we verify that with
	α =			1		(9.4.62)
				4e

The rheonomic parameterization of the H-field strength is satisfied, namely:

dB[2] = −i		Γa Γ11ψ V a	(9.4.63)
	ψ

9.4.7Rewriting the Mini-Superspace Gauge Completion as Maurer Cartan Forms on the Complete Supercoset

Next we can rewrite the mini-superspace extension of the bosonic solution solely in terms of Maurer Cartan forms on the supercoset (9.4.2). Let the graded matrix

(9.4.68)

402					9		Supergravity: An Anthology of Solutions
		|
L		Osp(6 4) be the coset representative of the coset M 10|24, such that the Maurer
Cartan form Σ can be identified as:
			Σ = L−1 dL							(9.4.64)
Let us now factorize L as follows:
				L = LF LB						(9.4.65)
where LF is a coset representative for the coset:
			Osp(6|4)				#	L	F	(9.4.66)
			SO(6)	×	Sp(4, R)

while LB rather than being the Osp(6|4) embedding of a coset representative of just AdS4, is the embedding of a coset representative of AdS4 × P3, namely:

LB =	LAdS	4	0	;	Sp(4, R)		# LAdS4 ;	SO(6)		# LP3 (9.4.67)
	0		LP3		SO(1, 3)			U(3)

In this way we find:

Σ = L−B 1ΣF LB + L−B 1 dLB

Let us now write the explicit form of ΣF :

ΣF

=

2

F

ΦA

3

(9.4.69)

−eAAB

4ieΦAγ5

where ΦA is a Majorana-spinor valued fermionic one-form and where

F is an

sp(4, R) Lie algebra valued one-form presented as a 4 × 4 matrix. Both ΦA as

F

and AAB depend only on the fermionic θ coordinates and differentials.

On the other hand we have:

LB−1 dLB =

AdS

4

0

(9.4.70)

0

AP3

where the

AdS4

is also an

sp

(4, R) Lie algebra valued one-form presented as a 4

×

4

μ

matrix, but it depends only on the bosonic coordinates x

of the anti de Sitter space

AdS4. In the same way AP3 is an su(4) Lie algebra element presented as an so(6) antisymmetric matrix in 6-dimensions. It depends only on the bosonic coordinates yα of the internal P3 manifold. According to (C.1.5) we can write:

AdS4 = −	1
	4 Babγab − 2eγa γ5Ba	(9.4.71)

where {Bab, Ba } are respectively the spin-connection and the vielbein of AdS4.