390	9 Supergravity: An Anthology of Solutions

9.3.11 The 3-Form

We have found an explicit expression for the supervielbein V a , the gravitino 1-form Ψ and an the spin-connection ωab . In order to complete the description of the superextension we need also to provide an expression for the 3-form A[3]. According to the general definitions of the FDA curvatures (6.4.2) and the rheonomic parameterization we find that:

dA[3] = F[4] −	1	Ψ ΓabΨ V a V b	(9.3.96)
	2

dA[3] = eεabcd Ea Eb Ec Ed + 1 χ x γabχy Φx

2 A

+ 1 χ x χy ζAταβ ζB Φx Φy Bα Bβ 2 A B

+ χ x γa γ5χy ζAτβ ζB ΦAx ΦBy Ea Bβ

Φy Ea Eb

(9.3.97)

The expression of dA[3] as a 4-form is completely explicit in (9.3.97) and by construction it is integrable in the sense that d2A[3] = 0. One might desire to solve this equation by finding a suitable expression for A[3] such that (9.3.97) is satisfied. This is not possible in general terms, namely by using only invariant constraints. In order to find explicit solutions, one needs to use some explicit coordinate system and some explicit solution of the constraints. This analysis is not in the spirit we have adopted. Here it is just the constraints what matters, not their explicit solutions.

In the main application one might consider, namely while localizing the action of the supermembrane M2 on such backgrounds, we can easily avoid all such problems. We simply substitute the world volume integral of A[3] with:

A[3] →	dA[3]	(9.3.98)
W V3	W V4

where the 4-dimensional integration volume W V4 is such that its boundary is the original supermembrane world-volume:

∂W V4 = W V3

(9.3.99)

and we circumvent the problem of solving (9.3.97).

With this observation we have concluded our proof that any AdS4 × G /H bosonic solution of M-theory field equations can be gauge completed to a solution in the mini-superspace containing all the theta variables associated with unbroken supersymmetries.

9.4 Flux Compactifications of Type IIA Supergravity on AdS4 × P3

391

9.4Flux Compactifications of Type IIA Supergravity on AdS4 × P3

As a further example of supergravity exact solution we consider the compactification of type IIA supergravity on the following direct product manifold:

M10 = AdS4 × P3

(9.4.1)

which was constructed in [45].

In this case not only we are able to write an exact solution of the bosonic field equations but we are also able to construct an explicit expression for all the bosonic and fermionic p-forms that close the type IIA differential algebra in such a way that the rheonomic solution of the Bianchi identities is matched. In other words, for this background we possess an explicit and simple integration of the rheonomic conditions just as it is the case for the compactification of M-theory on AdS4 × S7. This is the main pedagogical reason to present the solution we are going to consider. From the string theory point of view the interest in such backgrounds streams from the recently discovered duality between N = 6 superconformal Chern-Simons theory in three dimensions and superstrings moving on AdS4 × P3 backgrounds [46–49, 51–55] which has prompted the study of superstrings on Osp(N |4) backgrounds [43, 56–58]. Indeed the explicit integration of the rheonomic conditions is obtained through an appropriate use of the isometry supergroup of the background (9.4.1) which is identified with the following supergroup OSp(6|4). The Maurer Cartan forms of this supergroup on the following super-coset manifold

M 10|24		OSp(6|4)			(9.4.2)
	= SO(1, 3)		×	U(3)

allow for an explicit supergauge completion of the solution of the bosonic field equations provided by the geometry of the Riemannian space AdS4 × P3 in a very similar way to the M-theory cases discussed in the previous section.

To be precise what we will present is not a full gauge completion involving 10 bosonic coordinates and 32 fermionic ones, but only a partial gauge-completion in the mini-superspace given by the coset (9.4.2) which contains all the 10 bosonic coordinates but only 24 of the fermionic ones, those associated with the supersymmetries preserved by the background. The gauge completion in the remaining eight θ s associated with the broken supersymmetries is a rather difficult problem that, however, has been solved by the authors of [59].

9.4.1 Maurer Cartan Forms of OSp(6|4)

The bosonic subgroup of OSp(6|4)	is Sp(4, R)		×	SO(6). The Maurer-Cartan one-
	xy
forms of sp(4, R) are denoted by		(x, y = 1, . . . , 4), the so(6) one-forms are

Sp(4,R) , P3 ≡ SO(6) SO(1,3) U(3)

392	9 Supergravity: An Anthology of Solutions

denoted by AAB (A, B = 1, . . . , 6) while the (real) fermionic one-forms are denoted by ΦAx and transform in the fundamental representation of Sp(4, R) and in the fundamental representation of SO(6). These forms satisfy the following OSp(6|4) Maurer-Cartan equations:

	dΔxy +			xz			ty εzt = −4ieΦAx ΦAy
	dAAB − eAAC ACB = 4iΦAx ΦBy εxy									(9.4.3)
dΦAx + xy εyzΦAz − eAAB ΦBx = 0
where
					0		0	0	1
εxy	= −	εyx			0		0	−1	0	(9.4.4)
			=		0		1	0	0
						1	0	0	0
				−

The Maurer-Cartan equations are solved in terms of the super-coset representative of (9.4.2). We rely for this analysis on the general discussion in Appendix C. It is convenient to express this solution in terms of the one-forms describing the bosonic submanifolds AdS4 ≡ of (9.4.2) and of the one-forms on the

fermionic subspace of (9.4.2). Let us denote by Bab, Ba and by Bαβ , Bα the connections and vielbein on the two bosonic subspaces respectively. The supergauge completion is accomplished by expressing the p-forms satisfying the rheonomic parameterization of the type IIA Free Differential Algebra in terms of the minisuperspace one-forms. The final expression of the D = 10 fields will involve not only the bosonic one-forms Bab, Ba , Bαβ , Bα , but also the Killing spinors on the background. The latter play indeed a special role in this analysis since they can be identified with the fundamental harmonics of the cosets SO(2, 3)/SO(1, 3) and SO(6)/U(3), respectively. The Killing spinors on the AdS4 were already discussed. We wills study those on P3.

9.4.2 Explicit Construction of the P3 Geometry

The complex three-fold P3 is Kähler. Indeed the existence of the Kähler 2-form is one of the essential items in constructing the solution ansatz.

Let us begin by discussing all the relevant geometric structures of P3. We have to construct the explicit form of the internal manifold geometry, in particular the spin connection, the vielbein and the Kähler 2-form. This is fairly easy, since P3 is a coset manifold:

P3	=	SU(4)			(9.4.5)
		SU(3)	×	U(1)

so that everything is defined in terms of the structure constants of the su(4) Lie algebra. The quickest way to introduce these structure constants and their chosen

9.4 Flux Compactifications of Type IIA Supergravity on AdS4 × P3

393

normalization is by writing the Maurer-Cartan equations. We do this introducing already the splitting:

su(4) = H K

(9.4.6)

between the subalgebra H ≡ su(3) × u(1) and the complementary orthogonal subspace K which is tangent to the coset manifold. Hence we name H i (i = 1, . . . , 9) a basis of one-form generators of H and Kα (α = 1, . . . , 6) a basis of one-form generators of K. With these notation the Maurer-Cartan equations defining the structure constants of su(4) have the following form:

dKα + Bαβ Kγ δβγ = 0

(9.4.7)

dBαβ + Bαγ Bδβ δγ δ − X αβγ δ Kγ Kδ = 0

where:

1.the antisymmetric one-form valued matrix Bαβ is parameterized by the 9 generators of the u(3) subalgebra of so(6) in the following way:

0

H 9

−H 8

H 1 + H 2

H 6

−H 9

0

H 7

H 6

H 1 + H 3

Bαβ

H 8

H 2

H 7

0

H 5

H 4

H 1

−

−H 6

H 5

−0

H 9

=

−

H 4

H 9

0

−

H 6

H 1 H 3

−

5

− −4

−2

− H

3

− 8

−H

7

H

−H

−H

H

2. the symbol X	αβ	denotes the following constant, 4-index tensor:
	γ δ
			αβ	αβ	+ K αβ K γ δ + K αγ K	β
		X	γ δ	≡ δγ δ		δ

−H 5

H 4

H 2 + H 3

−H 8

H 7

0

(9.4.8)

(9.4.9)

3. the symbol K αβ denotes the entries of the following antisymmetric matrix:

		0	0	0	0	−1	0
		0	0	0	−1	0	0
K		0	0	0	0	0	−1		(9.4.10)
		1	0	0	0	0	0
	=	0	1	0	0	0	0
		0	0	1	0	0	0

The Maurer Cartan equations (9.4.7) can be reinterpreted as the structural equations of the P3 6-dimensional manifold. It suffices to identify the antisymmetric one-form valued matrix Bαβ with the spin connection and identify the vielbein Bα with the coset generators Kα , modulo a scale factor λ

Bα =	1	Kα	(9.4.11)

λ

394	9 Supergravity: An Anthology of Solutions

With these identifications the first of (9.4.7) becomes the vanishing torsion equation,

αβ

while the second singles out the Riemann tensor as proportional to the tensor X γ δ of (9.4.9). Indeed we can write:

Rαβ = dBαβ + Bαγ Bδβ δγ δ
= R		αβ		(9.4.12)
			γ δ Bγ Bδ
where:
	αβ		αβ	(9.4.13)
R			γ δ = λ2X γ δ

Using the above Riemann tensor we immediately retrieve the explicit form of the Ricci tensor:

Ricαβ = 4λ2ηαβ

(9.4.14)

For later convenience in discussing the compactification ansatz it is convenient to rename the scale factor as follows:

λ = 2e	(9.4.15)
In this way we obtain:
Ricαβ = 16e2ηαβ	(9.4.16)

which will be recognized as one of the field equations of type IIA supergravity. Let us now come to the interpretation of the matrix K . This matrix is im-

mediately identified as encoding the intrinsic components of the Kähler 2-form. Indeed K is the unique antisymmetric matrix which, within the fundamental 6- dimensional representation of the so(6) su(4) Lie algebra, commutes with the entire subalgebra u(3) su(4). Hence K generates the U(1) subgroup of U(3) and this guarantees that the Kähler 2-form will be closed and coclosed as it should be. Indeed it is sufficient to set:

K== Kαβ Bα Bβ			(9.4.17)
namely:	B4 + B2 B5 + B3	B6
K = −2 B1			(9.4.18)
=

and we obtain that the 2-form K=is closed and coclosed:

dK== 0, d K== 0

(9.4.19)

Let us also note that the antisymmetric matrix K satisfies the following identities:

K 2 = −16×6

(9.4.20)

8Kαβ = εαβγ δτ σ K γ δ K τ σ

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395

Using the so(6) Clifford Algebra defined in Appendix C.3.1 we define the following spinorial operators:

W = Kαβ τ αβ ; P = W τ7

(9.4.21)

and we can verify that the matrix P satisfies the following algebraic equations:

P2 + 4P − 12 × 1 = 0

(9.4.22)

whose roots are 2 and −6. Indeed in the chosen τ -matrix basis the matrix P is diagonal with the following explicit form:

		0	2	0	0	0	0	0		0
		2	0	0	0	0	0	0		0
		0	0	2	0	0	0	0		0
		0	0	0	2	0	0	0		0
		0	0	0	0	2	0	0		0
P	=											(9.4.23)
		0	0	0	0	0	2	0		0
		0	0	0	0	0	0		6	0
		0	0	0	0	0	0	−
								0		6
										−

Let us also introduce the following matrix valued one-form:

Q ≡	3	1	+	1	P τα Bα	(9.4.24)
	2			4

whose explicit form in the chosen basis is the following one:

				0			2B3				−2B2								0				2B6				2B5						2B4			2B1
		−	2B3				0				2B1						2B6						− 0				−	2B4					−2B5			2B2
							−2B1										−2B5																−
		2B2									2		0		5								2B4			3	2		0		2		−2B6			2B3
Q				0			2	B		6			B						0				2	B					B				2	B	1	2	B	4
		2			6											4	2				3											1	2		2	2		5
				B			− 0					2		B					B				− 0					2		B				B			B
	=										−																−
			2B			5	2B		4				0					2B				2	2B		1				0				2B		3	2B		6
		−															−
				0			0						0						0				0						0					0			0
				0			0						0						0				0						0					0			0
																																				(9.4.25)
and let us consider the following Killing spinor equation:
															D η + eQη = 0																					(9.4.26)
where, by definition:
														D = d −						1		Bαβ ταβ														(9.4.27)
																				4

denotes the so(6) covariant differential of spinors defined over the P3 manifold. The connection Q is closed with respect to the spin connection

Ω = −	1
	4 Bαβ ταβ	(9.4.28)