On AdS3 solutions of Type IIB

We study N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 supersymmetric AdS3× M7 backgrounds of Type IIB supergravity, with non-vanishing axio-dilaton, three-form and five-form fluxes, and a “strict” SU(3)-structure on M7. We derive the necessary and sufficient conditions for supersymmetry as a set of constraints on the torsion classes of the SU(3)-structure. Given an Ansatz for the three-form fluxes, the problem of also solving the equations of motion involves a “master equation”, which generalizes ones that have previously appeared in the literature.


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Sec. 5 [9] [10] [3] φ = 0 H = 0  to v foliates M 7 , and we find that the transverse six-dimensional space M 6 is conformally symplectic. On AdS 3 × M 7 , a solution to the supersymmetry equations also solves the equations of motion if and only if the Bianchi identities are satisfied by the fluxes (see for example [12]). By making an Ansatz for the three-form fluxes in our solution to the supersymmetry equations, we reduce the problem of finding a solution to the Bianchi identities, and hence the equations of motion, to two conditions: a "master equation" (5.10), which is a partial differential equation for the conformally Kähler metric on M 6 , and existence of a primitive (1,2)-form satisfying (5.7). Furthermore, supersymmetry is enhanced to N = 2. Similar master equations (and solutions thereof) associated with Bianchi identities appeared in [3,9,10], and the one presented here reduces to the ones of [3,9,10] in the appropriate limits. 3 The relation of these classes of solutions, and the corresponding master equations is depicted in figure 1. Solutions to the aforementioned conditions, as well as more general Ansätze will be reported in future work.
The rest of this note is organized as follows. In section 2, we present the supersymmetry equations as a set of equations involving a pair of polyforms on M 7 . In section 3, we introduce an SU(3)-structure in seven dimensions, and parameterize the polyforms in terms of it. In section 4, we derive a set of necessary and sufficient conditions for supersymmetry as restrictions on the torsion classes of the SU(3)-structure, and also give expressions for the fluxes in terms of the latter. A summary at the end of this section is included. section 5 presents a class of solutions to the equations of motion following an Ansatz, as described earlier. Our conventions and certain technical details are included in the appendix.

Supersymmetry equations
We start with a general bosonic background of Type IIB supergravity invariant under SO (2,2). The ten-dimensional metric is a warped product of a metric on AdS 3 and a metric on a seven-dimensional Riemannian manifold M 7 : (2.1)

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where A is a function on M 7 . 4 Conforming to the SO(2, 2) symmetry, the NSNS fieldstrength H 10d and the RR field-strengths F 10d , with F 10d denoting their sum in the democratic formulation, are decomposed as The magnetic fluxes H and F = p=1,3,5,7 F p , are forms on M 7 . The operator λ acts on a p-form F p as λ(F p ) = (−1) p/2 F p . The RR field-strengths are subject to d H 10d F 10d = 0, which decomposes as where d H ≡ d − H∧. We will refer to the first set of equations as equations of motion for F , and to the second one as the Bianchi identities.
In order to study the restrictions imposed by supersymmetry on the above bosonic background, we decompose the supersymmetry parameters of Type IIB supergravity, 1 and 2 under Spin(1, 2)× Spin(7) ⊂ Spin(1, 9): 5 Here, χ 1 and χ 2 are Majorana Spin(7) spinors; ζ is a Majorana Spin(1, 2) spinor satisfying the Killing equation: where the real constant parameter m is related to the AdS 3 radius L AdS 3 as L 2 AdS 3 = 1/m 2 . The above decomposition follows the requirement for N = 1 supersymmetry.
The necessary and sufficient conditions for preserving N = 1 supersymmetry can be derived following the derivation for Type IIA supergravity in the appendix of [15], with straightforward modifications. They are expressed in terms of bispinors ψ ± defined by Following the Fierz expansion of χ 1 ⊗ χ t 2 , and application of the Clifford map which maps anti-symmetric products of gamma matrices to forms, ψ + /ψ − become polyforms on M 7 , of even/odd degree.
The supersymmetry restrictions take the form of the following system of equations: (2.7d) 4 We work in string frame. 5 For the decomposition of the Clifford algebra see the appendix.

Supersymmetry and G-structures
This can be illustrated by the decomposition of χ 2 in terms of χ 1 (taking χ 1 , χ 2 to be of equal norm): where v is a real one-form with ||v|| = 1, and θ ∈ [0, π/2]. As θ varies from 0 to π/2, the G 2 × G 2 -structure varies from a "strict" SU(3)-structure, to an "intermediate" SU (3)structure, to a G 2 -structure. In this work we will consider the first case, i.e. θ = 0. An SU(3)-structure on M 7 is defined by a real one-form v, a real two-form J, and a complex decomposable three-form Ω, all nowhere-vanishing, These forms can be expressed as bilinears in terms of the spinors (χ 1 , χ 2 ); see appendix A for our conventions. The one-form v gives a foliation of M 7 with leaves M 6 ; accordingly, we define the volume form as vol 7 ≡ 1 3! v ∧ J ∧ J ∧ J and locally decompose the metric on M 7 as (3.3)

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Existence of an SU(3)-structure ensures that all forms on M 7 decompose into irreducible representations of SU(3). In particular, the local k-forms with no component along v can be decomposed into primitive (p, q)-forms. 8 We may also apply this decomposition to the exterior derivatives of the SU(3)-structure {v, J, Ω} itself. Doing so, we find a parameterization in terms of torsion classes. These constitute the components of the intrinsic torsion of the SU(3)-structure expressed in irreducible representations of SU(3). Specifically, we have (see for example [16]) In order to solve the supersymmetry equations, we parameterize the polyforms ψ ± as defined in (2.6) in terms of the SU(3)-structure data. Making use of (3.1), (A.7), (A.8) we find that in the general case, for ||χ 1 || 2 = ||χ 2 || 2 = e A . As stated earlier, we will study the case of a strict SU(3)-structure for which θ = 0 and hence (3.6) Substituting the above expressions in the supersymmetry equations (2.9), we will derive the restrictions on the intrinsic torsion of the SU(3)-structure imposed by supersymmetry. 8 A primitive k-form ω (k) satisfies J ω (k) = 0 for k = 2, 3, whereas k-forms with k = 0, 1 are primitive by definition. The (p, q) decomposition of k-form ω is defined by

JHEP05(2020)048 4 A class of solutions to the supersymmetry equations
In this section, we derive a class of solutions to the supersymmetry equations (2.9) by inserting the strict SU(3)-structure polyforms (3.6). The first constraint (2.9a) yields These in turn determine ReE v .
where H (2,1) and H (1,1) v are primitive, we also find expressions for several of the components in terms of torsion classes from (4.1). Using (A.12), we find:

(4.4)
The exterior derivatives of the the SU(3)-structure tensors now read We define a rescaled metric g M 7 = e −2A+φǧ M 7 and rescale the SU(3)-structure tensors accordingly as {v, J, JHEP05(2020)048 (4.8) We note that the condition dJ = 0 means that the six-dimensional leaves M 6 transverse tov admit a symplectic structure.
Turning to the second constraint (2.9b) we obtain: From these equations, employing (4.4) and (4.5) and the set of identities (A.10), (A.11), we can obtain expressions for the magnetic RR fluxes F p , p = 1, 3, 5, 7. We give these in (4.14) in the summary below. Finally, the third constraint (2.9c) reads 10) and plugging in the expressions for the RR fields we conclude that 3R + 6me −A + 4H R + 2ImE = 0 . (4.11)

Summary
Let us summarize our results. The differential constraints imposed on the SU(3)-structure by supersymmetry are: The expression for the NSNS field is:

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The expressions for the RR fields are: (4.15) The above solution to the supersymmetry equations also solves the equations of motion if and only if the Bianchi identities for the NSNS and RR fields are imposed in addition.

A new class of solutions
We make the following Ansatz: and recall that H (2,1) is primitive. This leads to v dA = 0 = v dφ and the following restrictions on the torsion classes: We thus have or in terms of the rescaled SU(3)-structure

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From the differential equations for {J ,Ω} we conclude thatM 6 is Kähler. In what follows we will introduce the exterior derivative d 6 onM 6 , Dolbeault operators ∂,∂ so that d 6 = ∂ +∂, and d c 6 = i(∂ − ∂). The remaining RR fields read Let us now examine the Bianchi identities. The first Bianchi identity, dF 1 = 0, enforces which is solved by setting φ = − log(ϕ + ϕ), with ϕ holomorphic. Next, the three-form Bianchi identities dH = 0 and dF 3 − H ∧ F 1 = 0 yield the constraints In analyzing the Bianchi identity for F 5 , we will invoke the results of [10]. The authors of [10] study supersymmetric solutions which descend from the solutions analyzed here, upon setting H (2,1) = 0. However, even when H (2,1) = 0, the SU(3)-structure of [10] and the expressions for F 1 and F 5 can be identified with the ones presented in this section. The map identifying the tensors there (left-hand side), with the tensors here (right-hand side) is 8) and in particular, (4.9b) is identified with (2.58) of [10]. 9 The authors of [10] showed that the Bianchi identity for F 5 , dF 5 = 0, amounts to which they refer to as the "master equation". In the above, R and R ij are respectively the Ricci scalar and the Ricci tensor ofǧ 6 , and contractions are also made usingǧ 6 . This master equation generalizes the one derived in [3] by including a varying axio-dilaton. For the case at hand the Bianchi identity of F 5 is dF 5 = H ∧F 3 , and the term on the right-hand side (a "transgression" term) modifies the master equation, which now becomes: (5.10) As noted above, in the limit H (2,1) = 0 the present class of solutions and master equation (5.10) reduce to the ones of [10]. Further setting the axio-dilaton to zero, they 9 One has to take into account that we work in the string frame whereas the Einstein frame is used in [10]. In addition, we use a different orientation on AdS3.

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reduce to the ones studied in [3]. Starting with the latter, the authors of [9] "turned on" a three-form flux G = G (1,2) , and taking the limit of vanishing axio-dilaton we recover their results. See figure 1.
Finally, the supersymmetry preserved by the class of solutions in this section enhances to N = 2, and the dual field theories are (0, 2) SCFTs [17]. The vector field dual to v generates a U(1) symmetry of the solutions, corresponding to the R-symmetry of the (0, 2) SCFTs. Thus we expect that a geometric dual of c-extremization exists for this class of solutions and would be very interesting to identify it.

Acknowledgments
We would like to thank N. Macpherson and A. Tomasiello for useful discussions. The work of A.P. is supported by is supported by Agence Nationale de la Recherche LabEx grant ENS-ICFP ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL. D.P. has been financially supported in part by INFN and by the ERC Starting Grant 637844-HBQFTNCER

A Conventions & identities
Clifford algebra decomposition. The ten-dimensional gamma matrices are decomposed as follows µ span Cliff(1, 2), γ m span Cliff(7) and the indices are spacetime indices. We take γ Spin (7). As noted above, we work with the Majorana representation of Cliff (7), for which the gamma matrices are imaginary and antisymmetric. The charge-conjugate of a Spin (7) spinor is the complex conjugate, and a Majorana spinor is real. The basis elements of Cliff (7)  As discussed in section 3, a pair of nowhere-vanishing Spin(7) Majorana spinors χ 1 and χ 2 define an SU(3)-structure {v, J, Ω} in seven dimensions. For the strict SU(3)-structure (θ = 0, or equivalently, χ t 1 χ 2 = 0), we introduce a Dirac spinor η as