On supersymmetric AdS$_3$ solutions of Type II

We classify supersymmetric warped AdS$_3 \times_w M_7$ backgrounds of Type IIA and Type IIB supergravity with non-constant dilaton, generic RR fluxes and magnetic NSNS flux, in terms of a dynamic $SU(3)$-structure on $M_7$. We illustrate our results by recovering several solutions with various amounts of supersymmetry. The dynamic $SU(3)$-structure includes a $G_2$-structure as a limiting case, and we show that in Type IIB this is integrable.


Introduction
Technically more approachable, but not less physically interesting than their higherdimensional counterparts, AdS 3 /CFT 2 dualities provide a hospitable environment for finding answers to questions on both sides of the holographic correspondence. Conformal field theories in two dimensions, which underlie string theory and are key tools in the description of critical phenomena, feature a highly-constraining infinite-dimensional algebra of conformal transformations that often allows for their exact solution. Gravity in three-dimensional asymptotically anti-de Sitter spacetime provides a toy model for quantum gravity. There is thus a clear motivation for the study of AdS 3 backgrounds of string theory.
Owing to the high dimensionality of the internal space, the problem of exploring the space of AdS 3 backgrounds is challenging. A way forward is to impose a symmetry on the background, in the form of supersymmetry or isometry, at the expense of the size of the subspace of backgrounds one can access, depending on the degree of the symmetry. In the present work we impose the minimal amount of supersymmetry, aiming for a more comprehensive scan of supersymmetric AdS 3 backgrounds of Type II supergravity. We classify warped AdS 3 × w M 7 backgrounds with non-constant dilaton, generic RR fluxes and magnetic NSNS flux. Minimal supersymmetry equips the internal manifold M 7 with a dynamic SU (3)-structure, due to the existence of two Majorana spinors on M 7 . In the limiting case where the spinors are parallel to each other, the dynamic SU (3)-structure corresponds to a G 2 -structure. We translate the necessary and sufficient conditions for supersymmetry to restrictions on the torsion classes of the SU (3)-structure, and obtain expressions for the supergravity fields in terms of the geometric data. We illustrate our results by recovering several AdS 3 solutions with various amounts of supersymmetry. In Type IIB supergravity we take a closer look at the G 2 -structure limiting case, show that it is integrable, and reduce the problem of finding a solution to not only the supersymmetry equations but also the equations of motion, to a single geometric equation. Generically, the dual superconformal field theories in two dimensions preserve N = (0, 1) supersymmetry, and include well-studied families of theories with higher supersymmetry such as those arising from D3-branes wrapped on Riemann surfaces [1][2][3], whose duals appear in section 5.1.
The rest of this paper is organized as follows. In section 2, we present the supersymmetry equations as a set of equations for a pair of polyforms on M 7 . In section 3, we review G 2 -and SU (3)-structures in seven dimensions, and parameterize the polyforms in terms of the latter. Sections 4 and 5 contain our results for Type IIA and Type IIB supergravity respectively.

Supersymmetry equations
We consider bosonic backgrounds of Type II supergravity whose spacetime is a warped product AdS 3 × w M 7 , where M 7 is a seven-dimensional Riemannian manifold. The ten-dimensional metric reads: where A is a function on M 7 . Preserving the symmetries of AdS 3 , the NSNS fieldstrength H 10d , and the sum of the RR field-strengths F 10d in the democratic formulation of Type II supergravity [4], are decomposed as The magnetic fluxes H, and where d H ≡ d − H∧. The first set of equations act as equations of motion for F , and the second one as Bianchi identities.
We also decompose the ten-dimensional supersymmetry parameters, ǫ 1 and ǫ 2 , under Spin(1, 2)× Spin(7) ⊂ Spin (1,9): The upper sign in ǫ 2 corresponds to Type IIA, and the lower sign to Type IIB. χ 1 and χ 2 are Majorana Spin (7) spinors; ζ is a Majorana Spin(1, 2) spinor that solves the Killing equation where the real constant parameter m is related to the AdS 3 radius L AdS 3 as L 2 The Cliff (1,9) gamma matrices are decomposed as where γ µ are Cliff(1, 2) gamma matrices, γ m are Cliff(7) gamma matrices and µ, m are spacetime indices. We choose γ (3) µ to be real, and γ m imaginary and antisymmetric. For more details see the appendix of [5].
Necessary and sufficient conditions for supersymmetry are generally given in terms of a set of Killing spinor equations. For AdS 3 backgrounds, these can be rewritten in terms of a pair of bispinors ψ ± defined by Taking into account the Fierz expansion of χ 1 ⊗ χ t 2 , and by mapping anti-symmetric products of gamma matrices to forms, ψ + and ψ − can be treated as polyforms on M 7 , of even and odd degree respectively. The necessary and sufficient conditions for supersymmetry in terms of differential constraints on these polyforms were derived in [6] for Type IIA, and in [5] for Type IIB.

Supersymmetry imposes
where c ± are constants defined by In what follows we will consider backgrounds with zero electric component for H 10d , κ = 0, and thus ||χ 1 || 2 = ||χ 2 || 2 . In Type IIB, κ = 0 can be set to zero by applying an SL(2, R) duality transformation. 1 In Type IIA, as shown in [6], κ = 0 leads to zero Romans mass; such AdS 3 backgrounds can thus be studied in M-theory, see [7][8][9]. Without loss of generality, we set c + = 2, that is (2.11) Given the above choices, the system of supersymmetry equations then reads: Here, an upper sign applies to Type IIA and a lower one to Type IIB; (ψ ∓ , F ) 7 ≡ (ψ ∓ ∧ λ(F )) 7 , with (·) 7 denoting the projection to the seven-form component.
We can decompose χ 2 in terms of χ 1 as follows: where without loss of generality, we take v to be a real one-form of unit norm and restrict θ ∈ [0, π/2]. At the boundary value θ = 0, χ 1 and χ 2 are orthogonal and define a "strict" SU (3)-structure on T M 7 . At the other boundary value θ = π/2, χ 1 and χ 2 are parallel and define a G 2 -structure. At intermediate values of θ, the pair (χ 1 , χ 2 ) define a "dynamic" SU (3)-structure on T M 7 , or alternatively a G 2 × G 2 -structure on the generalized tangent bundle T M 7 ⊕ T * M 7 .
In the next section we review G 2 -and SU (3)-structures in seven dimensions, and parameterize ψ ± in terms of the latter.

G 2 -and SU (3)-structures in seven dimensions
We briefly summarize the mathematical formalism for G 2 -and SU (3)-structures on seven-dimensional Riemannian manifolds that we will use in analyzing the supersymmetry equations (2.12).
A G 2 -structure on a seven-dimensional Riemannian manifold M 7 is defined by a nowhere-vanishing, globally defined three-form ϕ. Equivalently, a G 2 -structure is defined by a nowhere-vanishing, globally defined Majorana spinor. The three-form ϕ is constructed as a bilinear of the Majorana spinor as where χ is taken to have unit norm. The three-form ϕ is normalized so that In the presence of a G 2 -structure, the differential forms on M 7 can be decomposed into irreducible representations of G 2 . In particular, this may be applied to the exterior derivative of the three-form ϕ and its Hodge dual ⋆ 7 ϕ: The k-forms τ k are the torsion classes of the G 2 -structure. τ 0 transforms in the 1 representation of G 2 , τ 1 in the 7, τ 2 in the 14, and τ 3 in the 27.
An SU (3)-structure on a seven-dimensional Riemannian manifold M 7 is defined by a nowhere-vanishing, globally defined triplet comprising a real one-form v, a real twoform J, and a complex decomposable three-form Ω, subject to the following defining Equivalently, an SU (3)-structure is defined by a pair of non-parallel Majorana spinors; see the appendix of [5]. The one-form v foliates M 7 with leaves M 6 . The metric on M 7 is then locally decomposed as with an accompanying volume form vol where d 6 is the exterior derivative on M 6 . k-forms orthogonal to v can be decomposed into primitive (p, q)-forms with respect to J.
The intrinsic torsion of an SU (3)-structure splits into torsion classes, which transform in irreducible representations of SU (3). They parameterize the exterior derivatives of (v, J, Ω) as: R is a real scalar, E and W 1 are complex scalars, V 1 , V 2 and W 5 are complex (1, 0)forms, W 0 and W 4 are real one-forms, T 1 and T 2 are real primitive (1, 1)-forms, W 2 is a complex primitive (1, 1)-form, W 3 is a real primitive (2, 1) + (1, 2)-form, and S is a complex primitive (2, 1)-form. R, E and W 1 transform in the 1 representation of SU (3), V 1 , V 2 and W 5 in the 3, W 0 and W 4 in the 3 + 3, T 1 , T 2 and W 2 in the 8, W 3 in the 6 + 6, and S in the 6.
As detailed in [5], the polyforms ψ ± are parameterized in terms of the SU (3)structure as where θ is the angle appearing in (2.13). When θ = π/2, the one-form v drops out of the decomposition (2.13) of χ 2 , the spinors (χ 1 , χ 2 ) become parallel, and thus define merely a G 2 -structure rather than an SU (3)-structure. Nevertheless, the above decomposition is still valid: it can be shown that for compact M 7 , existence of a Spin (7) structure implies existence of an SU (3)-structure [10]. Hence, we may decompose the three-form ϕ defined by the spinor χ 1 = χ 2 in terms of an SU (3)-structure (v, J, Ω), leading to the above result even in this limiting case. The phase e iθ multiplying Ω can be "absorbed" by a redefinition e iθ Ω → Ω, and we will apply this redefinition in the following sections.
2 In terms of local coordinates,

Type IIA
In this section we analyze the Type IIA supersymmetry equations (2.12) (upper sign): by substituting (3.9) in (2.12) (redefining e iθ Ω → Ω), the necessary and sufficient conditions for supersymmetry translate to restrictions on the torsion classes of the SU (3)structure defined by (χ 1 , χ 2 ) on M 7 . Furthermore, the RR and NSNS field-strengths are expressed in terms of the SU (3)-structure data.
The geometry of M 7 and the NSNS field-strength are constrained by equation (2.12a), which yields: 3 The RR field-strengths are derived from (2.12b), corresponding to From (4.1), using (3.8), we obtain the following relations for the torsion classes of the SU (3)-structure:  Here we have introduced the notationḟ ≡ v df for any function f . Using (4.3) and (4.5), as well as the identities in the appendix of [5] to Hodge dualize, we derive the following expressions for the RR field-strengths:

Solutions
We now look at solutions of the supersymmetry conditions we have derived. In particular we will recover the N = 8 supersymmetric AdS 3 × S 6 solution of [6] (realizing the F (4) superalgebra), and the N = (4, 0) supersymmetric AdS 3 × S 3 × S 3 × S 1 solution of [11]. In addition to the supersymmetry equations, the equations of motion are solved provided that the fluxes satisfy the Bianchi identities (see for example [12]), and this is the case for the solutions below.
The angle θ, the warp factor A, and the dilaton φ satisfy The one-form v is closed -see (4.1a) given (4.9) -and locally a coordinate z can be introduced such that v = ξ(z)dz , (4.10) for a function ξ(z) which can be fixed by a change of coordinate. Following [6], it is fixed to Accordingly, the metric on M 7 (3.6) reads and the metric on M 6 is taken to be conformal to the round metric on the six-sphere S 6 : ds 2 (M 6 ) = e 2Q(z) ds 2 (S 6 ) . (4.14) It follows thatĴ ≡ e −2Q J andΩ ≡ e −3Q Ω define a nearly-Kähler structure on S 6 : dĴ = 3ImΩ , dΩ = 2Ĵ ∧Ĵ .  The only non-zero fluxes are F 0 and F 6 = 5qvol(S 6 ).
This solution arises as a near-horizon limit of a D2-O8 configuration. The internal space is non-compact, with z ∈ [0, ∞]. Near z = 0, there is an O8-plane singularity, and at z → ∞ a type of D2-brane singularity; see [6] for more details. Flux quantization imposes 2πF 0 ∈ Z, and q/(6π 2 ) ∈ Z. [11, sec. 4.1] belongs to the class of solutions with strict SU (3)-structure, i.e. θ = 0. The warp factor A, and the dilaton φ satisfy The one-form v is closed and locally is set to v = e φ 0 e 3A(ρ) dρ, where φ 0 is a constant. The metric on M 7 reads Furthermore, the torsion classes of the SU (3)-structure are restricted so that The dilaton φ and the warp factor A are given by: where ℓ is a constant. The only non-zero fluxes are F 0 and The Bianchi identity to be satisfied is that of F 4 , dF 4 = 0, which yields Thus, what remains to be done in order to solve the equations of motion is to find a suitable manifold M 6 admitting an SU (3)-structure with the right torsion classes. On S 3 ≃ SU (2), a set of left-invariant forms (σ 1 , σ 2 , σ 3 ) can be found such that where j, k, l ∈ {1, 2, 3}, and ǫ jkl is the Levi-Civita symbol. Let σ j a be the left-invariant forms on S 3 a , a ∈ {1, 2}, and let where ds 2 (S 3 a ) = 1 4 j (σ j a ) 2 . Making use of (4.24), the non-vanishing torsion classes are determined to bê (4.28) as desired. The fluxes now read: The coordinate ρ here is related to the coordinate r in [11] via and also e −φ 0 = qL 5 , where (L, ν = ±1, c, q) are constant parameters in [11].

Type IIB
In this section we analyze the Type IIB supersymmetry equations (2.12) (lower sign) in a way similar to that of the analysis of the Type IIA supersymmetry equations in the previous section.
Here, we will examine the other limiting case: G 2 -structure solutions, i.e., solutions with θ = π/2. Although equivalent, it turns out to be more convenient to work directly with the G 2 -structure rather than to use the θ = π/2 limit of the supersymmetry conditions derived above.
The polyforms ψ ± are parameterized in terms of the G 2 -structure, defined by the three-form ϕ, as Plugging these expressions into the supersymmetry equations (2.12), and making use of (3.3) leads to the following constraints for the torsion classes Vanishing of the τ 2 torsion class means that the G 2 -structure is integrable, meaning one can introduce a G 2 Dolbeault cohomology [15]. Furthermore, we obtain and Next, we examine the Bianchi identities, which reduce to dF 3 = 0. Imposing the Bianchi identities in addition to the supersymmetry conditions yields a solution to the equations of motion. We will work with a rescaled G 2 -structureφ = e −3A ϕ and corresponding metric ds 2 (M 7 ) = e −2A ds 2 (M 7 ). The rescaled torsion classes are given bŷ (5.11) Using these, the Bianchi identities read Thus, the problem of finding a solution to the equations of motion is reduced to this purely geometric condition. Note that (up to constant prefactors), the same condition appears for heterotic backgrounds on G 2 -structure spaces [16, eq. (2.13)].

Examples
Let us now give several examples of solutions to the Bianchi identities, which have been reduced to the constraint (5.12).
The next two examples are solutions in the presence of spacetime-filling O5-plane and D5-brane sources, which wrap calibrated three-cycles inside M 7 . The presence of these lead to a source term in the Bianchi identity, dF 3 = J 4 . This thus modifies the right-hand side of (5.12) such that the sourced Bianchi identities instead reduce to The first sourced example is given by the twisted toroidal orbifold M 7 = T 7 /(Z 2 × Z 2 × Z 2 ): we refer the reader to [19] which we follow closely, as well as [20] for details. Given a set of coordinates y m on M 7 , we may introduce a twisted frame {e m (y)}. In terms of this frame, the three-form determining the G 2 -structure can then be defined as This twisting breaks the G 2 -holonomy of the toroidal orbifold by introducing nonvanishing torsion classes τ 0 , τ 3 , such that ϕ is co-closed, but no longer closed. The representation of the Z 2 -involutions on the coordinates y m of M 7 , as well as the consistency constraint d 2 e m = 0, restrict the possible values τ m np can take. We will restrict our attention to τ m np being the structure constants of SO(p, q) × U (1) with p + q = 4: this comes down to setting    with a i constant and all other τ m np vanishing. Neither τ 0 nor τ 3 vanishes generically, with τ 0 = − 2 7 −a 1 − a 2 + a 3 + a 4 + a 5 − a 6 + a 1 a 4 a 2 − a 1 a 6 a 2 − a 3 a 6 a 2 + a 1 a 5 a 3 − a 2 a 4 a 3 + a 2 a 5 a 3 .

(5.20)
As discussed in [20], setting A = 0 leads to solutions with source term J 4 given by with ψ m ∈ {e 3456 , e 1256 , e 1234 , e 2457 , e 1467 , e 2367 , e 1357 } and k m (a i ) dependent on the twisting parameters. Note that the AdS 3 radius is proportional to the torsion class τ 0 (a i ), and hence the twisting parameters a i are restricted such that τ 0 = 0.
The second example of a sourced solution can be obtained by taking M 7 = H(3, 1), the generalized Heisenberg group, as discussed in [21] and recently investigated in the context of three-dimensional heterotic Minkowski backgrounds [22]. Geometrically, H(3, 1) is a nilmanifold, for which a frame can be found satisfying with k 1 = (a + b) 2 + (ab + bc + ca) and (a, b, c) cyclically permuted for k 2 , k 3 .