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Published for SISSA by Springer

Received: September 4, 2013

Accepted: September 26, 2013

Published: October 15, 2013

Parinya Karndumria,b and Eoin Colginc

aDepartment of Physics, Faculty of Science, Chulalongkorn University, 254 Phayathai Road, Pathumwan, Bangkok 10330, Thailand

bThailand Center of Excellence in Physics, CHE, Ministry of Education, Bangkok 10400, Thailand

cDepartamento de Fsica, Universidad de Oviedo,

Avda de Calvo Sotelo s/n, 33007 Oviedo, Espaa

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: AdS3 solutions dual to N = (0, 2) SCFTs arise when D3-branes wrap Kahler

two-cycles in manifolds with SU(4) holonomy. Here we review known AdS3 solutions and identify the corresponding three-dimensional gauged supergravities, solutions of which uplift to type IIB supergravity. In particular, we discuss gauged supergravities dual to twisted compactications on Riemann surfaces of both N = 4 SYM and N = 1 SCFTs

with Sasaki-Einstein duals. We check in each case that c-extremization gives the exact central charge and R symmetry. For completeness, we also study AdS3 solutions from intersecting D3-branes, generalise recent KK reductions of Detournay & Guica and identify the underlying gauged supergravities. Finally, we discuss examples of null-warped AdS3 solutions to three-dimensional gauged supergravity, all of which embed in string theory.

Keywords: Supergravity Models, Supersymmetric E ective Theories, Gauge-gravity correspondence, AdS-CFT Correspondence

ArXiv ePrint: 1307.2086

3D supergravity from wrapped D3-branes

JHEP10(2013)094

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP10(2013)094

Web End =10.1007/JHEP10(2013)094

Contents

1 Introduction 1

2 AdS3 from wrapped D3-branes 42.1 Review of wrapped D3-branes 42.2 D3-branes with N = (0, 2) SCFTs duals 6

3 An example of a generic reduction 10

4 Less generic reductions 144.1 Twists of SCFTs with Sasaki-Einstein duals 164.2 Intersecting D3-branes 22

5 Null-warped AdS3 solutions 27

6 Outlook 30

A Type IIB supergravity conventions 32

B Connection between [11] and [38] 33

C Details of reduction of D = 5 U(1)3 gauged supergravity 34C.1 Killing spinor equations 36

D Curvature for Kahler-Einstein space-times 37

E Details of reduction on H2 KE4 38

F Details of reduction on S2 T 4 40

1 Introduction

Gauged supergravity is a very useful tool in many areas of string theory such as ux compactications and the AdS/CFT correspondence (see [1] for a review). Due to these applications, gauged supergravities in various dimensions as well as their Kaluza-Klein (KK) dimensional reductions have been extensively explored. It is well known that lower-dimensional gauged supergravities can be obtained from dimensional reductions of higher-dimensional theories. Up to now, many examples have appeared and amongst them, [2, 3] and [58] are recognizable primary examples. In this paper, we are interested in gauged supergravities in three dimensions in order to incorporate both the principle of c-extremization and null-warped AdS3 solutions.

JHEP10(2013)094

The complete classication of Chern-Simons gauged supergravities in three dimensions has been given in [9]. Most theories constructed in this formulation have no known higher-dimensional origin. The three-dimensional gauged supergravities obtainable from dimensional reductions form a small part, with non-semisimple gauge groups, in this classication [10]. Unlike in higher-dimensional analogues, only a few examples of three-dimensional gauged supergravities, which play an important role in AdS3/CFT2 correspondence, have been obtained by dimensional reductions [1113]. In this paper, we will extend this list with more examples of gauged supergravities in three dimensions arising from wrapped D3-branes in type IIB supergravity.

Recently, c-extremization for N = (0, 2) two-dimensional SCFTs has been proposed

and various examples of gravity duals in ve- and seven-dimensional gauged supergravities exhibited [14, 15]. Recall that c-extremization is a procedure that allows one to single out the correct U(1)R symmetry of the CFT from the mixing with other U(1) symmetries. Soon after, c-extremization was formulated purely in the context of the AdS3/CFT2 correspondence by explicitly showing that, in the presence of a gauged SO(2)R U(1)R

R symmetry, the so-called T tensor of the three-dimensional gauged supergravity can be extremized leading to the exact central charge and R symmetry [16]. This realization is similar to how a-maximization of four-dimensional SCFTs [17] can be encoded in ve-dimensional gauged supergravity [18] in the context of the AdS5/CFT4 correspondence.1

Interestingly, in three dimensions, not only is the central charge reproduced, but the moment maps comprising the T tensor give information about the exact R symmetry. In this work we will provide more details of the results quoted in [16] and also exhibit another (related) example by considering twists of generic SCFTs with Sasaki-Einstein duals.

In three dimensions, where a vector is dual to a scalar, the matter coupled super-gravity theory can be formulated purely in terms of scalar elds resulting in a non-linear sigma model coupled to supergravity. N = 2 supersymmetry in three dimensions requires

the scalar target manifold to be Kahler. Gaugings of the theory are implemented by the embedding tensor specifying the way in which the gauge group is embedded in the global symmetry group. In general, the moment map of the embedding tensor, given by scalar matrices V, determines the T tensor which plays an important role in computing the scalar

potential and supersymmetry transformations. As a general result, N = 2 supersymmetry

allows any proper subgroup of the symmetry to be gauged. Furthermore, there is a possibility of other deformations through a holomorphic superpotential W . The scalar potential generally gets contributions from both the T tensor and the superpotential. However, any gauging of the R symmetry requires vanishing W .

The particular higher-dimensional theories we choose to reduce can all be motivated from the perspective of ten dimensions. From either an analysis of the Killing spinor equations [20], or by following wrapped D-brane intuition [21], it is known that supersym-metric AdS3 solutions supported by the ve-form RR ux of type IIB supergravity, or in other words, those corresponding to wrapped D3-branes, have seven-dimensional internal manifolds Y7 and bear some resemblance to Sasaki-Einstein metrics. More precisely, Y7

1A concrete realization is presented in [19].

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can be expressed locally in terms of a natural U(1) bration (the R symmetry) over a six-dimensional Kahler base that is subject to a single di erential condition

R = 12R2 RijRij, (1.1)

where R and Rij are, respectively, the Ricci scalar and Ricci tensor of the metric of the Kahler manifold. Through the supersymmetry conditions [20, 21], the Ricci scalar R is related to an overall warp factor for the ten-dimensional space-time.

Of course the above equation can be simplied considerably by assuming that the Kahler manifold is also Einstein, but in general, solutions with non-trivial warp factors can be di cult to nd. A search for IIB solutions tailored to this context can be found in [22], where a solution originally found in [23] was recovered. The challenges here are reminiscent of generalisations of direct-product AdS4 and AdS5 solutions to warped products. To support this observation, we recall that, for an Ansatz covering the most general supersym-metric warped AdS5 solutions of type IIB supergravity [24], the only warped geometry2 noted by the authors beyond the special case of Sasaki-Einstein was the Pilch-Warner solution [26]. On a more recent note, warped AdS4 solutions of eleven-dimensional supergravity generalising Sasaki-Einstein have been found [27, 28]. In the face of these di culties, it is a pleasant surprise to witness the ease at which supersymmetric solutions with warp factors can be constructed in ve-dimensional supergravity through twisted compactications on a constant curvature Riemann surface g of genus g and how the principle of c-extremization accounts for the central charge and exact R symmetry of the dual N = (0, 2) SCFT [14, 15].

c-extremization aside, we can further motivate the study of three-dimensional gauged supergravities through the continued interest in null warped AdS3 space-times. Over the last few years, we have witnessed a hive of activity surrounding warped AdS3 space-times and their eld theory duals [29], primarily in Topologically Massive Gravity (TMG) [30, 31]. Indeed, the mere existence of these solutions and the fact that they are deformations of AdS3 with SL(2, R) U(1) isometry, raises very natural questions about the putative

dual CFT. Since relatively little is known about these theories, the common approach is to extract information holographically from warped AdS3 solutions. To date, in three dimensions, warped AdS3 solutions have cropped up in a host of diverse settings, including of course, solutions [29, 32, 33] to TMG, solutions [34] to New Massive Gravity [35], Higher-Spin Gravity [36], topologically gauged CFTs [37] and three-dimensional gravity with a Chern-Simons (CS) Maxwell term [38], where the latter is embeddable in string theory. As we shall see, within the last class of three-dimensional theories, one also nds gauged supergravities.

Indeed, null warped AdS3 are central to e orts to generalise AdS/CFT to a non-relativistic setting, where holography may be applicable to condensed matter theory via a class of Schrdinger space-times. Taking the catalyst from [39, 40], through edgling embeddings in string theory [4144], various attempts have been made to provide a working description of non-relativistic holography. On one hand, one may wish to start with a

2A class of solutions can be generated via TsT transformations [25] starting from AdS5 S5, but as the transformation only acts on the internal S5, the nal solution is not warped.

JHEP10(2013)094

recognisable theory with Schrdinger symmetry, such as a non-relativistic limit [45, 46] of ABJM [47], but holographic studies [4850] fail to capture the required high degree of supersymmetry. On the other hand, if one starts from gravity solutions with Schrdinger symmetry, one may be more pragmatic and obtain an e ective description of the dual non-relativistic CFT, valid at large N and strong coupling [51].3 Similar points of view were also advocated in [5557]. Whether the dual theory is a genuine CFT as proposed in [29], or some warped CFT, is an open question drawing considerable attention.4

The structure of the rest of this paper is as follows. In section 2, we present an overview of our knowledge of supersymmetric AdS3 geometries arising from wrapped D3-branes. In section 2.2, we focus on geometries with a U(1) R symmetry dual to N = (0, 2) SCFTs and

present known examples preserving at least four supersymmetries, all of which will correspond to the vacua of the gauged supergravities we discuss later. In section 3, we provide more details of the KK reduction reported in [16]. In section 4.1, we present the three-dimensional gauged supergravity corresponding to a twisted compactication of an N = 1

SCFT with a generic Sasaki-Einstein dual. In section 4.2, we generalise the KK reductions discussed in [38] and identify the corresponding gauged supergravities. In section 5 we present some simple constructions of null-warped AdS3, or alternatively Schrdinger geometries with dynamical exponent z = 2, before discussing some open avenues for future study in section 6.

2 AdS3 from wrapped D3-branes

2.1 Review of wrapped D3-branes

In this section we review supersymmetric AdS3 geometries arising from D3-branes wrapping calibrated two-cycles in manifolds with SU(2), SU(3) and SU(4) holonomy. To this end, we follow the general ten-dimensional classication presented in [21] and later indicate where particular explicit solutions t into the bigger picture. The approach of [21] builds on earlier work concerning wrapped M5-branes [59, 60] and M2-branes [61].

We recall that the general wrapped-brane strategy [59] involves rst assuming that AdS3 geometries start o as warped products of the form

ds210 = L1ds2 R1,1

+ ds2 (M8) , (2.1) where both the warp factor L and the metric on M8 are independent of the Minkowski

factor. Here the Minkowski space-time should be regarded as the unwrapped part of the D3-brane, and as expected, the D3-branes source a self-dual RR ve-form ux F5 = + 10

invariant under the symmetries of the Minkowski factor.

For the particular geometries of interest to us, the metric and the ux for the geometry may be expressed as [21]

ds210 = L1ds2 R1,1

d L1J2d

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+ ds2 (M2d) + Lds2

, (2.2)

3Separately it has been argued [52, 53] that generic non-relativistic quantum eld theories have a holo-graphic description in terms of Hoava gravity [54].

4See [58] for a recent discussion.

R82d

= vol R1,1

wrapped brane manifold supersymmetry R symmetry Kahler 2-cycle CY2 N = (4, 4) SO(4) U(1)

Kahler 2-cycle CY3 N = (2, 2) U(1) U(1)

Kahler 2-cycle CY4 N = (0, 2) U(1)

Table 1. Wrapped D3-brane geometries and their supersymmetry.

where d = 2, 3, 4. In each case we require the existence of globally dened SU(d) structures, specied by everywhere non-zero forms J2d, 2d on M2d. The accompanying torsion

conditions follow from the SU(4)R8 case of [62], with the conditions for smaller structure groups being determined through decompositions of the form

J2d+2 = J2d e2d+1 e2d+2, 2d+2 = 2d

e2d+1 ie2d+2 . (2.3)

As explained in detail in [21], the supersymmetry conditions for AdS3 space-times may then be derived by introducing an AdS3 radial coordinate r, writing the (unit radius) AdS3 metric in the formds2 (AdS3) = e2rds2 R1,1

+ dr2, (2.4)

redening the warp factor, L = e2r, and performing a frame rotation of the form

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dr = sin + cos v, (2.5)

where parametrises the frame-rotation, which is further assumed to be independent of the AdS3 radial coordinate, and, v are respectively unit one-forms on M2d and the overall

transverse space.5 Omitting various technicalities associated to this frame-rotation one arrives at a simple but e ective derivation of the supersymmetry conditions for various AdS3 space-times of type IIB supergravity. A summary of the outcome may be encapsulated in table 1 which we reproduce from [21].

As can be seen from the above table, in each case the cycle being wrapped is the same, but as the dimensionality of the Calabi-Yau n-fold (CYn) increases, the preserved super-symmetry decreases. For D3-branes wrapping Kahler two-cycles in CY2 manifolds, one can generically have SO(4) U(1) R symmetry provided the radial direction (2.5) involves a

rotation. Upon analytic continuation, one recovers the half-BPS LLM solutions [63] with isometry RSO(4)SO(4)U(1), however there appear to be no known AdS3 space-times

in this class. On the contrary, when = 0, i.e. when the radial direction is purely trans-verse, one recovers the well known AdS3 S3 CY2 solution6 with R symmetry SO(4). In

either case the supersymmetry is N = (4, 4).

5For SU(4) structure manifolds there is no transverse space so there = /2.

6Specialising to CY2 = T 4 and performing T-dualities we arrive at the usual form of the D1-D5 near-horizon sourced by three-form RR ux. We also remark that the geometry sourced by ve-form ux and three-form ux are also related via fermionic T-duality [64] as explained in [65].

For D3-branes wrapping Kahler two-cycles in CY3, supersymmetry is reduced to N =

(2, 2), while the associated R symmetry group is U(1) U(1). Examples of these space-

times can be found in the literature [66, 67]. Finally, for D3-branes wrapping Kahler two-cycles in CY4 the dual SCFTs preserve N = (0, 2) supersymmetry and the U(1) Killing

direction is dual to the R symmetry. A rich set of examples of these geometries exist in the literature [14, 15, 22, 23, 67, 68]. In the notation of [21], the metric and ux may be expressed as

ds210 = 1ds2 (AdS3) + ds2 (M6) + 1 (d + B)2 , (2.6)

= vol (AdS3)

32J3, (2.9)

d = 2i (d + B) . (2.10) The rst condition implies that M6 is a Kahler manifold, while the last condition simply

identies the Ricci form R = 2dB.2.2 D3-branes with N = (0, 2) SCFTs duals

Now that we have covered AdS3 space-times arising from D3-branes wrapping Kahler two-cycles in Calabi-Yau manifolds in a general manner, here we focus on the particular case where the manifold is CY4. Since this case preserves the least amount of supersymmetry, it includes geometries dual to two-dimensional SCFTs with N = (2, 2) and N = (4, 4)

supersymmetry as special cases.

While the characterisation of wrapped D3-branes [21] presented in the previous section o ers a welcome sense of overview, henceforth we switch to the notation of [22], which is itself based on the work of [20]. The generic AdS3 solutions corresponding to wrapped

D3-branes are then of the form [22],

ds2 = L2

e2Ads2 (AdS3) + 14e2A (dz + P )2 + e2Ads2 (M6) ,

F5 = L4 vol(AdS3)

18d e4A (dz + P )

2 6 dR

2(d + B) 21J , (2.7)

where is the Killing vector dual to the R symmetry. The SU(3) structure manifold M6

is subject to the the conditions [21]:

dJ = 0, (2.8)

J2 dB =

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+ 1

16L4

J R (dz + P ) +

, (2.11)

where L is an overall scale factor, 6 refers to Hodge duality with respect to the metric of the

Kahler space, dP = R, with R being the Ricci form on M6.7 The warp factor is related to

7The Ricci form is dened by Rij = 12 RijklJkl, where Rijkl is the Riemann tensor. Recall also that the Ricci scalar R and the Ricci tensor Rij may be expressed in terms of the Ricci form as R = JijRij and

Rij = J kiRkj.

the Ricci scalar through 8e4A = R, a relation that can be inferred from (2.9). The closure of F5 leads to the di erential condition on the curvature (1.1). Finally, to make direct comparison with the previous incarnation of this solution (2.6), one can simply redene

= e2A, z = 2, P = 2B, = eiz ~

, (2.12)

where we have added a tilde to di erentiate between complex forms. The ve-form uxes (2.7) and (2.11) are related up to a factor of 4 and follow from the choice of nor

malisation adopted in [20]. This point should be borne in mind when making comparisons.

Examples. To get better acquainted with the form of the general soution, we can consider some supersymmetric solutions that will correspond later to the vacua of our gauged supergravities. We begin with the well-known AdS3 S3 T 4 solution corresponding to

the near-horizon geometry of two intersecting D3-branes. Via T-duality it is related to the D1-D5 near-horizon where the geometry is supported by a RR three-form.

To rewrite the solution in terms of the general description (2.11), we take

A = 0,dz + P = (d3 cos 1d2) , ds2 (M6) = ds2 T 4

+ 14 d21 + sin2 1d22

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, (2.13)

where i parametrise the coordinates on the S3 normalised to unit radius, the same radius as the AdS3 factor. Despite this solution tting into the general ten-dimensional framework, it preserves sixteen supercharges and is dual to a SCFT with N = (4, 4) supersymmetry.

Before illustrating the most general solution of [14, 15] in its ten-dimensional guise, we can satisfy the required supersymmetry condition

a1 + a2 + a3 = , (2.14)

where is the curvature of the Riemann surface g, more simply through setting all the ai equal, ai = 13, and taking the Riemann surface to be a unit radius Hyperbolic space, = 1.

This solution originally featured in [67]. With these simplications the solution reads

ds2 = 49ds2 (AdS3) +

13ds2 H2

Xi=1d2i + 2i

di +

2, (2.15)

F5 = (1 + )

"3281 vol (AdS3) vol H2

427 vol (AdS3)

Xi=1d 2i

di + #,

P3i=1 2i = 1. Note now that all Ai are equal, Ai =, and d = 13 vol(H2). It is easy to determine the one-form K = 12e2A(dz + P )

corresponding to the R symmetry direction

K = 2

Xi=12i

where the i are constrained so that

di + #, (2.16)

and check that it has the correct norm K2 = e2A = 49 [20]. Taking into account the factor of 4 in the denitions of the ux, and also setting L = 1, we then learn from

comparing (2.11) with (2.15) that

3281 vol(H2)

4 27

Xi=1d(2i) (di +) = 2J + d(e2AK). (2.17)

One can then determine J

J = 4

27 vol H2

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+ 2

Xi=1d 2i

di + , (2.18)

which comes with the correct factor of vol(H2),

ds2 (M6) =

4 27ds2(H2) +

d21 + d22 + d23 + 2122 (d1 d2)2

+ 2123 (d1 d3)2 + 2223 (d2 d3)2

, (2.19)

4 9

13! J3. Observe also that J is independent of K since idi = 0 follows from the fact that the i are constrained. In addition, the nal di erence in angular coordinates 2 3 can be written as a linear combination of the other two, so we only

have four directions separate from those along the H2. As a further consistency check, we have conrmed that the Ricci scalar for M6 is R = 8e4A, in line with our expectations.

We can now repeat for general ai subject to the single constraint (2.14). This also comprises the only example we discuss where the warp factor A is not a constant. In the notation of [14, 15], the ten-dimensional solution is

ds2 =

so that vol(M6) =

he2fds2 (AdS3) + e2gds2 ( g)i+ 123

Xi=1 X1i

d2i + 2i di + Ai

2 , (2.20)

F5 = (1+) vol (AdS3)

Xi=1 e3f+2g

2Xi X2i2i

vol ( g)

ai2 e4gX2i

d 2i

(d+Ai)

where

Xi=1Xi2i, X1X2X3 = 1, (2.21)

and as before the i are constrained. The constrained scalars Xi can be expressed in terms of two scalars i in the following way

X1 = e

[parenleftBig]

2p6 1+22[parenrightBig], X2 = e

[parenleftBig]

2p6 12

2 [parenrightBig], X3 = e

2p6 1. (2.22)

To give the full form of the solution one also needs to specify the values of the various warp factors ef, eg and scalars Xi [14]:8

ef = 2X1 + X2 + X3 , e2g =

a1X2 + a2X1

2 ,

X1X13 = a1

(a2 + a3 a1) (a1 + a2 a3)

, X2X13 = a2

(a1 + a3 a2) (a1 + a2 a3)

. (2.23)

From the higher-dimensional perspective a orded to us here, the canonical R symmetry corresponds with the Killing vector [15]

= 2

Xi=1XiX1 + X2 + X3 i . (2.24)

Again, one is in a position to determine the dual one-form

K = ef

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Xi=12i (di + Ai) , (2.25)

and conrm that it squares correctly K2 = e2A =

e2f. Proceeding in the same fashion

as above, one can then determine J

J =

Xi=11 4

ai (2ai + )e3fd 2i

(di + Ai) + 2ai (2ai + )

2ie3f vol ( g)

, (2.26)

where we have adopted the notation of [15], namely

= a21 + a22 + a23 2 (a1a2 + a1a3 + a2a3) , = (a1 + a2 + a3) (a1 a2 + a3) (a1 + a2 a3) . (2.27)

The accompanying expression for the manifold M6 is

ds2 (M6) = e2g+2fds2 ( g) + e2f

X11d21 + X12d22 + X13d23

+ X3

2122 (X2D1 X1D2)2 +

2123 (X3D1 X1D3)2

+X1

2223 (X3D2 X2D3)2

, (2.28)

where we have further dened Di = di + Ai. One can check it is consistent with the expression for J and furthermore that one recovers the previous expressions upon simplication, i.e. setting ai = 13, = 1.

These solutions will all be utilised later when we come to discuss three-dimensional gauged supergravities with vacua corresponding to the above supersymmetric solutions. In the next section, we begin by discussing an example of a generic reduction, in other words one where the warp factor is not a constant, by providing further details of the reduction and resulting three-dimensional N = 2 supergravity initially reported in [16].

8The solutions with g = 1 were studied in [69], while for g = 0, g > 1, modulo issues related to the range of the parameters, the solutions can be mapped to (4.6) of [70] through interchanging the scalars 1 $ 2 and redening the parameters accordingly ai = mi/(m1 + m2 + m3), where = 1 for g = S2 and = 1 for g = H2.

3 An example of a generic reduction

In this section we illustrate an example of a generic reduction, where we use the word generic to draw a line between dimensional reductions with non-trivial warp factors from the ten-dimensional perspective, and those that are direct products. Recall that, in addition to the famous KK reductions based on spheres [2, 3, 58], which give rise to maximal gauged supergravities in lower dimensions, generic KK reductions based on gaugings of R symmetry groups, notably gaugings of U(1) R symmetry [71, 72] and SU(2) R symmetry [73, 74] exist despite the internal space not being a sphere. This observation leads to the natural conjecture [72] that gaugings of R symmetry groups are intimately connected to the existence of consistent KK dimensional reductions. Here should be no exception, so we expect that one can gauge the existing U(1) R symmetry present in (2.11) and reduce to three dimensions.

However, in contrast to similar reductions to four and ve dimensions, for instance [71, 72], here in addition to retaining the gauge eld from the R symmetry gauging, we also require an additional scalar so that the three-dimensional gauged supergravity ts into the structure of N = 2 gauged supergravity as laid out in [9]. More concretely, we require

an even number of scalars to constitute a Kahler scalar manifold. While the reduction we discuss presently assumes additional structure for the M6, i.e. the existence of a Riemann

surface, it would be interesting to identify truly generic reductions without having to specify

the internal six-dimensional Kahler manifold.

Here we will present further details of the dimensional reduction from ve-dimensional U(1)3 gauged supergravity to three-dimensional N = 2 gauged supergravity reported

in [16]. While not being the most general reduction, from the ten-dimensional vantage point it provides a neat example of a reduction where the warp factor, and the associated Ricci scalar of the internal M6, is not a constant. We also do not need to address the full

embedding of the three-dimensional theory in ten dimensions, since we can work with the U(1)3 gauged supergravity in ve dimensions.

The bosonic sector of the action for ve-dimensional U(1)3 gauged supergravity can be found in [75]. It arises as a consistent reduction from type IIB on S5, so it is directly connected to ten dimensions9 via the equations of motion, and corresponds to the special case where only the SO(2)3 Cartan subgroup of SO(6) is gauged. The action reads

L5 = R 1

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Xi=1di di 1 2

Xi=1

X2iF i F i

+ 4g2

Xi=1

X1i vol5 +F 1 F 2 A3, (3.1)

9The bosonic sector also appears as a reduction from D = 11 supergravity [76] where it is based on the existence of near-horizon black holes [77]. Interestingly, one can start from D = 11 and reduce to D = 4 U(1)4 gauged supergravity, which, for consistency, requires F i ^ F j = 0. Taking a near-horizon limit prescribed in [77] one nds the bosonic sector of D = 5 U(1)3 gauged supergravity, without such a condition.

where g is the gauge coupling and the constrained scalars Xi we have dened earlier (2.22). From varying the potential with respect to the scalars it is easy to see that there is only a single supersymmetric AdS5 vacuum at Xi = 1.

As commented in [75], or by inspection from the equations of motion in appendix C, one can consistently truncate the theory by setting rst 2 = 0 implying that X1 = X2 =

X1/23. This truncation is consistent provided F 1 = F 2. Furthermore, one can take an additional step and set 1 = 0 leading to minimal gauged supergravity in ve dimensions.

Dimensional reduction. As it turns out, this dimensional reduction can be performed consistently at the level of the action. Simply put, this means that we can adopt the space-time metric Ansatzds25 = e4Cds23 + e2Cds2( g) (3.2)

where g is a constant curvature Riemann surface of genus g and we have used C to denote the scalar warp factor in ve dimensions. In addition, we have orchestrated the warp factors so that we arrive directly in Einstein frame in three dimensions.

The metric on the Riemann surface may be expressed as

ds2 ( g) = e2h dx2 + dy2

, (3.3)

where the function h depends on the curvature of the Riemann surface. It is respectively, h = log (1 + x2 + y2)/2

( = 1), h = log(2)/2 ( = 0) and h = log(y) ( = 1),

depending on whether the genus is g = 0, g = 1, or g > 1. In addition, one takes the following Ansatz for the eld strengths,

F i = Gi ai vol ( g) , (3.4) where closure of F i ensures that ai are constants and Gi is closed, Gi = dBi.

In doing the reduction at the level of the action the following expression for the ve-dimensional Ricci scalar is useful

R 1 = R 3 1 6dC 3dC + 2e6C 3 1. (3.5) The resulting three-dimensional action in Einstein frame is

L(3) = R 3 1 6dC 3dC

1 + L(3)top , (3.6)

where the topological term takes the form

L(3)top = a1B2 G3 + a2B3 G1 + a3B1 G2. (3.7)

We remark that the reduction and the resulting potential appeared previously in [78]. In appendix C, we have conrmed that it is indeed consistent.

JHEP10(2013)094

Xi=1di 3di 1 2e4C

Xi=1

X2iGi 3Gi

+ 3 4g2e4CX1i 12e8Ca2iX2i + 2e6C

Dualising the action. Now that we have the action, we would like to rewrite it in the form of a three-dimensional non-linear sigma model coupled to supergravity so that we can make contact with three-dimensional gauged supergravities in the literature [9]. We take our rst steps in that direction by dualising the gauge elds, or more appropriately, their eld strengths, and replacing them with scalars:

G1 = X21e4C DY1, DY1 = dY1 + a3B2 + a2B3, G2 = X22e4C DY2, DY2 = dY2 + a1B3 + a3B1,

G3 = X23e4C DY3, DY3 = dY3 + a1B2 + a2B1. (3.8)

Through these redenitions, we can recast the action (3.6) in the following form

L(3) = R 1 6dC dC

JHEP10(2013)094

Xi=1di di 1 2e4C

Xi=1X2iDYi DYi

+ L(3)pot + a1B2 G3 + a2B3 G1 + a3B1 G2, (3.9)

where we have omitted the explicit form of the potential as it will play no immediate role. We have also dropped all subscripts for Hodge duals on the understanding that we are now conning our interest to three dimensions. Note that the Chern-Simons terms are untouched and when we vary with respect to Bi we recover the duality conditions (3.8), so it should be clear that the equations of motion are the same and we have just rewritten the action.

At this point, before blindly stumbling on, we will attempt to motivate the expected gauged supergravity. Firstly, we know from the Killing spinor analysis in [15] that the AdS3 solutions generically preserve four supersymmetries, meaning we are dealing with N = 2

supersymmetry in three dimensions. Indeed, for N = 2, we have precisely an SO(2) R

symmetry group under which the gravitini transform and in this case the target space is a Kahler manifold with the scalars pairing into complex conjugates. Naturally, a prerequisite for a Kahler manifold is that we have an even number of scalars, and we observe that after dualising, this is indeed the case. So, we will now push ahead and identify some features of the N = 2 gauged supergravity.

To identify the scalar manifold it is good to diagonalise the scalars by redening them in the following way

W1 = 2C + 161 +

1 22,

W2 = 2C + 1

1 22,

261 . (3.10)

In terms of the original Xi these new scalars are simply eWi = e2CX1i.

With these redenitions, the Kahler manifold now assumes the simple form

L(3)scalar =

W3 = 2C

3 dWi dWi + e2WiDYi DYi

(3.11)

and we are in a position to identify it as [SU(1, 1)/U(1)]3. The Kahler structure of the scalar target space can be made fully explicit through the introduction of the Kahler potential of the form

K =

Xi=1log (zi) , (3.12)

where we have introduced complex coordinates zi = eWi + iYi. This means that the metric for the manifold is gi = iiK = 14e2Wi, where i = zi,

i = zi.

Having identied the scalar manifold and the Kahler potential, we turn our attention to the scalar potential. In the language of three-dimensional N = 2 gauged supergravity [9],

the scalar potential is comprised of two components, a T tensor and a superpotential W :

L(3)pot = 8T 2 8giiT

iT + 8eK|W |2 2gieKDiW D

i W , (3.13)

where the Kahler covariant derivative is DiW iW + iKW and W is holomorphic, so

i W = iW = 0. While W plays a natural role when eleven-dimensional supergravity is reduced on S2 CY3 to three dimensions [11], whenever the R symmetry is gauged, consis

tency demands that W = 0. Thus, to make contact with the literature, we face the simpler task of identifying the correct T tensor and making sure that the potential is recovered.

After rewriting the scalars, the potential takes the more symmetric form

L(3)pot = 4g2h

eW1W3 + eW2W3 + eW1W2

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+ 2eW1W2W3

. (3.14)

Note that in performing the reduction we have not been picky about supersymmetry and a priori, neglecting the gauge coupling g, which can be set to one, the constants and ai are unrelated. However, setting g = 1 for simplicity, one can nd the appropriate T tensor

T =

1 4

ha21 e2(W2+W3) + a22 e2(W1+W3) + a23 e2(W1+W2)

ha1eW2W3 + a2eW1W3 + a3eW1W2

+ 12

heW1 + eW2 + eW3

i, (3.15)

and check that it reproduces the potential on the nose provided (2.14) is satised. This is precisely the condition identied in [14, 15] for supersymmetry to be preserved. Though it happens that the existence of what is commonly referred to as a superpotential, in this case T , could conceivably be related to some fake supersymmetry structure for the theory, the fact that we recover the supersymmetry condition is reassuring. In fact, in appendix C.1 we reduce some of the Killing spinor equations and show that they also lead to the same T tensor. Thus, once the potential (and also T ) is extremised, the Killing spinor equations are satised.

Central charge and exact R symmetry. At this stage it should be obvious that we have a potential with a supersymmetric critical point provided condition (2.14) holds. Furthermore, once we extremise T , we in turn extremise the potential and arrive at the supersymmetric AdS3 vacuum. As discussed in [16], the extremization of the T tensor o ers a natural supergravity counterpart for c-extremization [14, 15]. Recall that c-extremization has been proposed for SCFTs with N = (0, 2) supersymmetry as a means to identify the

exact central charge and R symmetry where ambiguities exist due to the U(1) R symmetry mixing with other global U(1) symmetries that may be present.

Like the trial c-function proposed in [14, 15], T is also quadratic and comes from squaring the moment maps Vi T = 2Vi ijVj, (3.16)

contracted with the embedding tensor ij [9], where the index i ranges over the various U(1) symmetries, which for the immediate example, i = 1, 2, 3. In addition, since the embedding tensor also appears in the Chern-Simons terms in the action, it also related to the t Hooft anomaly coe cients which appear in the trial c-function for c-extremization [14, 15]. Indeed, for the class of wrapped D3-brane geometries discussed in [14, 15] this can all be made precise through the relations [16]

cR = 3 dGT 1, R = 2 ViT 1Qi , (3.17)

where cR is the exact central charge, R is the exact R symmetry, is related to the volume of the Riemann surface, = 2 vol( g), dG is the dimension of the gauge group and Qi denotes the charges corresponding to the U(1) currents.

All that remains to do is simply to identify the minimum of the potential by extremising T . The critical point of T corresponds to the following values for the scalars:

W1 = ln

. (3.18)

Once written in terms of C, 1 and 2 or in terms of C and Xi, this precisely gives the AdS3 critical point of [14]. Then, slotting the critical value of T into the (3.17), we arrive at the exact central charge and R symmetry,

cR = 12 N2

, (3.19)

R = 2ai (2ai + )

, (3.20)

where we have made use of (2.27) to display the result. In deriving (3.19) we have used the fact that the dimension of the gauge group at large N is dG = N2, while for (3.20) it is good to use the fact that the moment map is Vi =

14 eWi. The central charge and R symmetry agree with those quoted in [14, 15] and reproduce the coe cients of the Killing vector corresponding to the R symmetry (2.24).

4 Less generic reductions

Experience suggests that it is much easier to construct KK reduction Ansatze for direct product solutions than those that are warped products. This should come as no surprise since warped products are often more involved and consequently it may not be easy to identify a symmetry principle to guide the construction of a tting Ansatz. For dimensional

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a2a3

a2 + a3 a1

a1a2

a1 + a2 a3

a1a3

a1 + a3 a2

, W2 = ln

W3 = ln

a1a2a3

reductions from ten or eleven dimensions to ve-dimensional gauged supergravities admitting AdS5 vacua, the restrictions are quite clear. Starting with coset reductions [58, 79, 80], through generic Sasaki-Einstein reductions [8184] to the more general cases, the richness of the reduced theory gradually decreases until one is left with minimal gauged supergravity [71, 72]. For warped AdS5 solutions, only reductions to minimal gauged supergravity are known, with a notable exception being KK reductions [85] based on Y p,q spaces [86, 87], which when uplifted to eleven dimensions, the vacua correspond to warped solutions.

In this section we will discuss KK reductions to three dimensions conned to the special case where the Kahler manifold is a product of Kahler-Einstein spaces. As a direct consequence, (1.1) simplies to

R2 = 2RijRij. (4.1)

A nice treatment of this special case can be found in [22] which we follow. We take the internal Kahler manifold to be a product of a set of two-dimensional Kahler-Einstein metrics

ds2 (M6) =

JHEP10(2013)094

Xi=1 ds2

KE(i)2

. (4.2)

Since M6 now has constant curvature, it is easy to satisfy (4.1). The Ricci form for M6

takes the form

R =

Xi=1liJi , (4.3)

where Ji are the Kahler forms of the constituent metrics and the constants li are zero, positive or negative depending on whether the metric is locally that on T 2, S2 or H2. We also have the one-form connection P =

Pi Pi with dP =

Pi liJi. Slotting (4.3) into (4.1)

we nd a single constraint on the li

l1l2 + l1l3 + l2l3 = 0 , (4.4)

and discover that the overall warp factor is determined,

e4A = 18R =

1 4

Xili . (4.5)

Finally, the expression for the ve-form ux (2.11) simplies and assumes the following form

F5 = (1 + ) L4 vol (AdS3)

Pi li

J1 (l2 + l3) + J2 (l1 + l3) + J3 (l1 + l2)

. (4.6)

We now can make some comments. Demanding that the ten-dimensional space-time has the correct signature, we require R > 0 from (4.5). In the light of (4.4), this means that the potential solutions are constrained to be either S2 T 4 or S2 S2 H2. The rst

option here corresponds to the famous intersecting D3-branes solution, while the second case was considered in [67]. We note that when the KE(i)2 space is H2, it is a well-known fact that one can quotient the space without breaking supersymmetry leading to a compact

Riemann surface with genus g > 1. The Ricci tensor for these solutions can be found in appendix D.

4.1 Twists of SCFTs with Sasaki-Einstein duals

In this section we will discuss KK reductions on the rst class of products of Kahler-Einstein spaces by conning our attention to spaces with curvature, li 6= 0. For simplicity,

we will take l1 = l2, and the requirement that the scalar curvature of the internal M6 be

positive (4.5) subject to (4.4) means that there is only one case, namely M6 = H2 KE4, where KE4 is a positively curved Kahler-Einstein manifold.10 For concreteness, we take (l1, l2, l3) = (2, 2, 1) so that the H2 is canonically normalised.

Our next task is to construct a ten-dimensional Ansatz. While we could begin from scratch, we can incorporate some results from the literature as, in the end, a natural question concerns how they may be related. So we opt to kill two birds with one stone by simply reducing the IIB reduction on a generic Sasaki-Einstein ve-manifold SE5 [8184]

further to three dimensions on a constant curvature Riemann surface (H2). We will follow the notation of [82] and subsequent comments are in the context of that work.

To achieve our goal, we make two simplications. Firstly, we truncate out the complex two-form L2, since as our internal space is now six-dimensional, a complex (2, 0)-form, 2, is less natural. We can easily replace it with a eld coupling to the complex (3, 0)-form 3 via the ve-form ux, but this will simply give us an additional complex scalar. More importantly, one can ask what is the fate of the complex scalars and under dimensional reduction. Recall that they feature prominently in embeddings of holographic superconductors [88] (see also [89, 90]). However, since , couple to the graviphoton A1,

it is not possible to twist A1 in the usual way to produce a supersymmetric AdS3 vacuum without truncating out and . As such, we will have nothing to say about models for holographic superconductivity here. Moreover, as the same elds support the nonsupersymmetric Romans vacuum in ve dimensions, we do not expect to nd an analogue in three dimensions that follows from the reduction procedure.

The ve-dimensional action in Einstein frame can be found in (3.10) of [82]. With the above simplications taken onboard, for completeness, we reproduce the kinetic term

L(5)kin = R vol5

3 dU dU

JHEP10(2013)094

83dU dV

43dV dV 12e2da da (4.7)

12d d 2e8UK1 K1 e4UH1 H1 e4U+G1 G1

12e

83 (U+V )F2 F2 e

43 (U+V )K2 K2 12e

43 (2UV )H2 H2

12e

43 (2UV )+G2 G2 12e

43 (4U+V )H3 H3 12e

43 (4U+V )+G3 G3 ,

the scalar potential

L(5)pot = h

24e

3 (7U+V )

83 (4U+V )

43 (5U+V ) 8e

vol5 , (4.8)

10Suitable choices for KE4 include S2 S2, CP 2 and del Pezzo dPk, k = 3, . . . , 8.

and the topological terms are given by the expression

L(5)top = A1 K2 K2 (dk 2E1 2A1) [dB2 (dc 2C1) + (db 2B1) dC2]+ A1 (dk 2E1) [(db 2B1) dC1 dB1 (dc 2C1)]

+ 2A1 dE1 (db 2B1) (dc 2C1) + A1 (db 2B1) (dc 2C1) F2 4C2 dB2 . (4.9)

In turn, the above elds can be written in terms of various potentials and scalars in ve dimensions

G1 = dc 2C1 adb + 2aB1,

H1 = db 2B1,

K1 = dk 2E1 2A1,

F2 = dA1,

G2 = dC1 adB1,

H2 = dB1,

K2 = dE1 + 12 (db 2B1) (dc 2C1) , (4.10)

thus ensuring the that ten-dimensional Bianchi identities (appendix A) for the uxes hold. In total we have 7 scalars U, V, k, b, c including the axion a and dilaton , 4 one-form potentials A1, B1, C1, E1 and 2 two-form potentials B2, C2.

Dimensional reduction. Having introduced the ve-dimensional theory, we are in a position to push ahead with the same reduction as section 3 to three dimensions on a constant curvature Riemann surface g. We consider the usual metric Ansatz11

ds25 = e4Cds23 + e2Cds2 ( g) , (4.11)

where warp factors have been chosen so that we end up in Einstein frame, and for the moment, we will assume that we have a constant curvature Riemann surface and not specify its curvature . Supersymmetry will later dictate that < 0. As for the rest of the elds, the ve-dimensional scalars reduce to three-dimensional scalars. The fact that the eld strengths H1, G1 appear in the Einstein equation mean that we cannot twist with respect to B1 and C1 since such a twisting is inconsistent with the assumption that the Riemann surface is constantly curved. This leaves A1 and E1, or their eld strengths, which we twist in the following way

K2 = vol ( g) +

F2 = vol ( g) + ~F2 , (4.12)

where tildes denote three-dimensional eld strengths. is dictated to be a constant through F2 = dA1 and no twisting along K1 imposes the requirement that we twist K2 in the

11Here C without subscript will denote the scalar warp factor and C1 is a one-form.

JHEP10(2013)094

~K2 ,

opposite way. This latter point is also in line with our expectation that one can further truncate the theory to minimal gauged supergravity through K1 = 0, K2 = F2 in ve dimensions [82].

Since we are not twisting B1, C1, the eld strengths G1, H1, G2, H2 reduce directly to three dimensions. On the contrary, we can consider a decomposition for the three-form eld strengths G3, H3 on the condition that we respect the symmetries of g. So we can decompose

C2 = e vol ( g) +2, B2 = f vol ( g) + ~B2 , (4.13)

leading to two new scalars e, f in the process. The corresponding eld strengths can then be written as

G3 = M1 vol ( g) + g vol3, M1 = de adf +

2 (dc 2C1 adb + 2aB1) ,

H3 = N1 vol ( g) + h vol3, N1 = df +

2 (db 2B1) . (4.14)

One can check that this choice is consistent with the closure of the Bianchi identities.

The scalars g, h are, up to an integration constants 1, 2, set by the equations of motion

g = 4e

43 (4U+V )8C (1 + f)

h = 4e

JHEP10(2013)094

43 (4U+V )+8C (2 + e a (1 + f)) . (4.15)

We will normalise these so that i = 1.

We now reduce directly at the level of the action and take care to check in appendix E that one gets the same result from reducing the equations of motion, thus guaranteeing the consistency of the reduction. Dropping tildes, as only the three-dimensional elds remain, the resulting kinetic terms are

L(3)kin = R vol3 6dCdC

3 dU dU

83dU dV

43dV dV

2e2da da (4.16)

2d d 2e8UK1 K1 e4U+G1 G1 e4UH1 H1

43 (4U+V )+4C M1 M1

43 (4U+V )4C N1 N1e

43 (U+V )+4C K2 K2

83 (U+V )+4C F2 F2

43 (2UV )++4C G2 G2

43 (2UV )+4C H2 H2 ,

while those of the scalar potential take the form

L(3)pot = e4C

2e2C + 24e

3 (7U+V )

43 (5U+V ) 8e

83 (4U+V )

22e4C

e83 (U+V ) + 2e43 (U+V )

43 (4U+V )4C (1 + f)2

43 (4U+V )+4C (1 + e a (1 + f))2

vol3 . (4.17)

The topological term is then given by the expression

L(3)top = 2A1 K2 4 (1 + e) A1 dB1 + 4 (1 + f) A1 dC1

E1 K2+2E1 df (dc2C1)de (db2B1)+34 (db2B1) (dc2C1)

+ 2k

df + 12(db 2B1)

dC1

de + 12 (dc 2C1) dB1

. (4.18)

Now is an opportune time to identify the supersymmetric AdS3 vacuum. This can be done by comparing directly with (6.9) of [22] (see also [23]). For concreteness we can take KE4 = S2 S2 to exhibit the explicit solution, but one can consider other choices. The

form of the space-time metric before rescaling is

ds2 = L2 "

23ds2 (AdS3) +

JHEP10(2013)094

3 2

dx2 + dy2y2 + 322

Xi=112 d2i + sin2 id2i

dxy

Xicos idi

, (4.19)

where AdS3 is normalised to unit radius and all normalisations for the H2, parametrised by (x, y), and two S2s, parametrised by (i, i) are now explicit. We have also reintroduced an overall scale factor L. We omit the ve-form ux as it will not provide any new information and it is enough to compare the ten-dimensional metrics.

To make meaningful comparison with the KK reduction Ansatz of [82], we need to compare with the following space-time Ansatz

ds2 = e

3 (4U+V )

e4Cds23 + e2Cds2 ( g) + e2Uds2 (KE4) + e2V ( + A1)2 , (4.20)

where d = 2J and the Kahler-Einstein metric gij with positive curvature is normalised so that Rij = 6gij. To make the connection, we rst rescale the KE4 factor in (4.19) by a factor of three, take L2 = 2/ 33

and make the following identications

( + A1) = 1

dz cos 1d1 cos 2d2 dx y

. (4.21)

The supersymmetric AdS3 vacuum can then be identied

U = V = 0, C =

2 log 3, e = f = 1 , (4.22)

where = 1, since the H2 was normalised to unit radius, and = 13 follows from (4.21).

One can indeed check that this choice leads to a critical point of the potential and that the AdS3 radius of the three-dimensional space-time is = 29.

Further truncation & supergravity. In this subsection we consider the above action with the three-form uxes truncated out by setting b = c = B1 = C1 = B2 = C2 = 0,

e = f = 1. Even from the ten-dimensional perspective, it is known that it is always

consistent to perform this truncation to just the metric, elds in the ve-form ux and the axion and dilaton.12

We now recast the simpler action in the more familiar language of three-dimensional gauged supergravity. In part this will involve dualising the one-form potentials. To do so we redene the following elds

K2 = e

43 (U+V )4C DY2 , DY2 = dY2 +

~B2

F2 = e

83 (U+V )4C DY3 , DY3 = dY3 +

~B3 , (4.23)

while, at the same time, adding the following additional CS terms

L(3)top = 2

~B2 K2 +

~B3 F2 . (4.24)

The covariant derivatives are chosen so that the equations of motion are still satised once ~Bi are integrated out. We can then redene K1

K1 = 1

2DY1, DY1 = (dY1 4E1 4A1) , (4.25) and nally introduce the following scalars

W1 = 4U, W2 =

3 (U + V ) 2C, W3 =

43 (U + V ) 2C . (4.26)

The scalar manifold is now [SU(1, 1)/U(1)]4, which should be familiar from previous analysis, and the kinetic term for the action becomes

Lkin =

2dW1 dW1

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2e2W1DY1 DY1 dW2 dW2 e2W2DY2 DY2

2dW3 dW3

2e2W3DY3 DY3

2d d

2e2da da . (4.27)

We can thus introduce the complex coordinates

zi = eWi + iYi , i = 1, 2, 3, z4 = e + ia , (4.28)

allowing us explicitly to write the Kahler potential K as

K = log (z1) 2 log (z2) log (z3) log (z4) . (4.29) While we could have made this point earlier, it is now clear that the axion a and the dilaton decouple completely and can be truncated out. They also do not feature in the scalar potential.

In terms of the other scalars the potential takes the form

Lpot =

2e2W2+W3 + 24eW1+W2+W3 4e2(W1+W2) 8e2(W1+W3) (4.30)

e4W2 + 2e2(W2+W3) vol3 .

12In performing this truncation we remove the six scalars coming from the RR and NS three-form uxes. In general, it is possible to see that one always has an SU(1, 1)/U(1) factor, but it is not clear if the remaining twelve scalars constitute a Kahler manifold. It is also possible that the vacuum spontaneously breaks N = 4 supersymmetry to N = 2, for example [82] in ve dimensions. We leave this point to future work.

We can then work out the corresponding T tensor in terms of and ,

T =

4e2W2

2eW2+W3 eW1+W2 + eW1+W3

2eW3 . (4.31)

We note that and are not independent and we require = 3 so that the T tensor reproduces the potential. Once they are identied in this way, and taking into account the fact that < 0, < 0, one nds a vacuum at

W1 = 0, W2 = W3 = log () U = V = 0, C =

2 log () . (4.32)

Setting = 13, we arrive at the result quoted previously.

Central charge and R symmetry. In fact we have already discussed the central charge for this case as it corresponds to a particular example in section 3, namely ai = 13, = 1, thus ensuring that (2.14) is satised. However, to avoid the onerous task of rescaling metrics and comparing solutions, we can simply recalculate the central charge using the standard holographic prescription [91, 92]

cR = 32G(3) , (4.33)

where is the AdS3 radius and G(3) the three-dimensional Newtons constant. Using the conventions of [14, 15] where G(3) = 1/(4 N2), one can check that the result agrees

with (3.19) when ai = 13.

It is also of interest here to ask about the R symmetry? The ten-dimensional origin of our reduction makes it clear that there is only a single U(1) R symmetry, so there is no ambiguity. However, without this insight, we can ask what the three-dimensional theory can tell us about the R symmetry. Once we truncate out K1, we have essentially two U(1)

symmetries and the moment maps Vi associated to these can be worked out by comparing

the T tensor (3.16) with the CS term in the action. We nd that the components of the embedding tensor are 23 = 2, 22 = 2 and, for agreement, the moment maps are

V2 =

14eW2, V3 =

JHEP10(2013)094

14eW3 , (4.34)

where i = 2, 3 label the U(1)s associated to the gauge elds E1 and A1 respectively. We can then extract the R symmetry

R =

3U(1)2 +

3U(1)3 , (4.35)

where we have again used indices to distinguish the U(1)s. We can now compare to our earlier result (3.20) by inserting ai = 1

3 and one arrives at the same numbers, up to a relative sign. This relative sign can be traced to the relative sign in (4.12) and by simply changing the sign of A1 in ten dimensions, one can nd perfect agreement.

4.2 Intersecting D3-branes

In this section we discuss dimensional reductions to three dimensions for intersecting D3-branes. Some of the work presented here will not be new and will recover the recent work of [38]. Although we could approach this task directly from a ten-dimensional Ansatz, it is handier to make use of an intermediate reduction to six dimensions on a Calabi-Yau two-fold [93], details of which can be found in appendix F.

As such, we adopt the same strategy as [38], but an important distinction is that we will not impose truncations directly in six dimensions and then reduce. Instead, we will reduce directly so that we can unify the reductions presented in [38]. In addition, we will make statements about the underlying gauged supergravity, an aspect that was overlooked in [38]. Note that it is expected that the three-dimensional gauged supergravity be a theory with N = 4 supersymmetry, so that the scalar manifold is a product of

quaternionic manifolds [9], but this falls outside of our scope here and we hope to address this question in future work. Finally, we remark that these reductions are related to those of [11] via T-duality and uplift, a point that is eshed out in appendix B.

So the task now is to perform the reduction on S3, written as a Hopf-bration, from the six-dimensional theory presented in [93] to extract a three-dimensional gauged supergravity. Strictly speaking we are then doing a reduction on the D1-D5 near-horizon or its S-dual F1-NS5, so further T-dualities along CY2 = T 2 T 2 will be required to recover the intersecting

D3-brane vacuum discussed previously. We will come to this point in due course.

Dimensional reduction. Starting from the six-dimensional theory in appendix F, we adopt the natural space-time Ansatz

ds26 = e4U2V ds23 + 14e2Uds2 S2

4e2V (dz + P + A1) , (4.36) where U, V are warp factors and A1 is a one-form with legs on the three-dimensional space-time. Our Ansatz ts into the overarching description for supersymmetric AdS3 solutions from wrapped D3-branes presented earlier with choice (l1, l2, l3) = (0, 0, 4). In contrast

to [38], this means that P = cos d so that dP = vol(S2) = 4J3. In addition, A = 0

follows from (4.5).For the three-form uxes, we consider the following Ansatz

F3 = G0 12 (dz + P + A1) J3 + G1 J3 + G2

H3 = sin (dz + P + A1) J3 + H1 J3 + H2

JHEP10(2013)094

+ 1

2 (dz + P + A1) + ge6U3V vol3 (4.37)

12 (dz + P + A1) + he6U3V vol3 , where the Bianchi identities (see appendix A) determine the following:

G0 = 2 (cos sin 1) ,

G1 = dc 1db 2 (C1 1B1) (cos sin 1) A1,

G2 = dC1 1dB1,

H1 = db 2B1 sin A1,

H2 = dB1 . (4.38)

Here 1 is the scalar axion of type IIB supergravity and we have introduced the constant , scalars b, c and one-form potentials B1, C1. The remaining scalars, i, i = 1, 2, of the six-dimensional theory simply descend to become three-dimensional scalars.

We now plug our Ansatz into the equations of motion of the six-dimensional theory (F.3)(F.9), the details of which can be found in appendix F. In the process one determines the form for g, h:

g = 2e1+2V 2U (cos + sin 2) , (4.39)

h = 2e1+2V 2U [sin cos 2 + (cos + sin 2) 1] , (4.40)

where we have normalised the integration constants for later convenience.

One nds that the equations of motion all come from varying the following three-dimensional action:

L(3) = L(3)kin + L(3)pot + L(3)top , (4.41) where the kinetic term is

L(3)kin = R vol3

2d1 d1

2e21d1 d1

2d2 d2

2e124UH1 H1

2e124UG1 G1

2e12+4UH2 H2

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2e22d2 d2 6dU dU 4dU dV 2dV dV

18e4U+4V F2 F2 , (4.42)

and the scalar potential takes the form

L(3)pot =

8e6U2V 2e8U 2e1+28U4V [sin cos 2 + (cos + sin 2) 1]2

2e1+28U4V (cos + sin 2)2 2e128U4V sin2

2e128U4V (cos sin 1)2

2e12+4UG2 G2

vol3 . (4.43)

Finally, the topological term takes the simple form

L(3)top = 2 (H1 G2 G1 H2) (cos C1 + sin B1) F2 . (4.44) When U = V = i = i = c = b = A1 = B1 = C1 = 0, the above scalar potential

has a critical point corresponding to either the D1-D5 near-horizon, its S-dual, or a one parameter interpolating vacuum. We have chosen the integration constants so that an SL(2, R) transformation, parametrised by the constant ,

C2 B2

cos sin

sin cos

C2 B2

(4.45)

takes one from the vacuum supported by a RR three-form ux ( = 0) to the vacuum supported by a NS three-form ux ( =

2 ). In each case the AdS3 radius is unity. It is known more generally that the e ect of an SL(2, R) transformation is simply to rotate the

Killing spinors [94],13 so supersymmetry is una ected.

13In this immediate context, see [65].

Ten-dimensional picture. As we have reached our three-dimensional theory through the result of two steps, a reduction on a Calabi-Yau two-fold [93] and a further reduction generalising the recent work of [38], here we wish to pause to consider the higher-dimensional picture. We would also like to recast the KK reduction Ansatz in terms of the generic form of wrapped D3-branes. Specialising to CY2 = T 2 T 2, we can perform two T-dualities along the second T 2 leading to the following NS sector with the metric in string frame:

ds210 = e

2 (1+2)

e4U2V ds23 + 14e2Uds2 S2

+ 14e2V (dz + P + A1)2

(4.46)

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+ e

2 (12)ds2 T 21

+ e

2 (12)ds2 T 22

H3 = [2 sin J3 + H2]

2 (dz + P + A1) + H1 J3 + he6U3V vol3, (4.47)

= 1

~ 2 (1 + 2) , (4.48)

where ~

is the new ten-dimensional dilaton. Note that the three-form ux H3 is not a ected by the T-duality. The accompanying RR uxes then take the form

F5 =

hG0J2 J3 + ge12+2U+V J1 J3i 12 (dz + P + A1) + e12+4U G2 J1 J3

+ G1 J2 J3 +

hG2 J2 e124U G1 J1i 12 (dz + P + A1)

+ G0e128U4V vol3 J1 + ge6U3V vol3 J2 ,

F3 = d1 J2 d2 J1 , (4.49)

where J1 =vol(T 21), J2 =vol(T 22) and, as before, J3 = 14 vol(S2) and there is no axion, F1 =0.

Further truncations. Even if we dualise the gauge elds in the action (4.42), since we have an odd number of scalars and N = 2 supergravity in three dimensions has a Kahler

scalar manifold, one will need to truncate out some elds to nd a gauged supergravity description. In this subsection we consider some further truncations and make contact with the work of [38] in the process.

Setting = i = c = A1 = C1 = 0, 1 = 2 = , U = V , and nally employing the

following identication

B1 = (4.50)

one can check that our action can be brought to the form of (4.7) of [38]:

L(3) = R vol3 + 4e4U 2e8U

vol3 d d 4dU dU

2e2+4UH2 H2 . (4.51)

Note we have set = 1 for simplicity, but this can be reinstated if one rescales the radius of the Hopf-bre S3 correctly. We have also retained the scalar eld b, which one is required to set to zero to make direct connection with [38].

2e24UH1 H1

The reduction of [38], where the six-dimensional space-time is supported solely by RR ux, involves setting 1 =2 =, i =b==B1 =0. Making the further identications

C1 =

, A1 = 2A , (4.52)

one arrives at

L(3) = R vol3 d d 6dU dU 4dU dV 2dV dV

2e24U

dc+2 A dc+2

2e2+4U

2e4U+4V F F

+ h8e6U2V 2e8U 2e28U4V 2e28U4V ivol3 +2 F . (4.53)

Once one sets c = 0 one can again conrm this is the same action as (4.17) of [38]. A further truncation of action ( = 0 = A, U = V ) permits warped black string solutions,

the holographic interpretation of which was considered in [95].14

An obvious truncation not discussed in [38] is the truncation to just the NS sector. In some sense this may be regarded as the S-dual of the truncation we have just discussed. We can do this by setting = 2 , i = c = C1 = 0 and 1 = 2 = ~

. The resulting action is

L(3) = R vol3 d

6dU dU 4dU dV 2dV dV

2e2~4UH1 H1

2e2~+4UH2 H2

18e4U+4V F2 F2

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+ h8e6U2V 2e8U 2e28U4V 2e28U4V ivol3 B1 F2 . (4.54)

Up to a rewriting, b = c, A1 = 2A, B1 =

, ~

= , this action is identical to (4.53).

Rewriting the supergravity. Here we identify the underlying gauged supergravities. As a warm-up we consider the action (4.51), but make a conversion from the three-dimensional Yang-Mills (YM) Lagrangian to a Chern-Simons Lagrangian following general prescriptions given in [9] (see also [10, 96]). This procedure replaces every YM gauge eld with two gauge elds and a new scalar eld. This allows us to trade the following Yang-Mills term in the action

L(3)YM =

2e2+4UH2 H2 (4.55)

with the terms

L(3)CS =

2e24UD

+ H2

~B1 , (4.56)

where D~

= d~

~B1 and we now have two gauge elds B1, ~B1 and an additional scalar ~

Varying with respect to ~B1 we get

H2 + e24U D

= 0 , (4.57)

which, on choosing the gauge ~

= 0, we can integrate out ~B1 to recover the original Lagrangian. The equation of motion following from varying B1 now reads

d ~B1 + 2e24U H1 = 0 , (4.58)

14It is easier to start with the action in [95] and use the EOM for to nd the form for the action above.

which can be shown to be equivalent to that of the original Lagrangian once one imposes (4.57). The equation of motion for ~

is trivially satised through (4.57).

With these changes, the scalar kinetic term of the full Lagrangian (4.51) is given by

L(3)kin = d d 4dU dU

where as before H1 = db 2B1. We redene all of the scalars throughY1 = ~

, Y2 = b, W1 = 2U, W2 = 2U, (4.60) so that the scalar kinetic term becomes

L(3)kin =

vol3 . (4.62)

(4.63)

with only one critical point at W1 = W2 = 0. Here it is not immediately obvious that this is the only option. Recall that for N = 2 gauged supergravity, when the R symmetry is

gauged, no holomorphic superpotential can appear [9]. Now when the R symmetry is not gauged, as is the case here, one can consider replacing the T tensor with the free energy F = T eK/2W . However, since eK = eW1+W2, we can see that a problem arises with W

being holomorphic, so this does not appear to be an option.

We now move onto the second action that results from truncating out all the NS three-form ux elds. Referring to (4.46), (4.49), this means that we set = b = B1 = i = 0.

With this simplication, one further observes that it is consistent to set 1 = 2 = . This is simply (4.53) with the scalar c reinstated and A1 and C1 rewritten accordingly,

A1 = 2A, C1 =

We can now diagonalise the scalars by redening them

W1 = 2U, W2 = 2U, W3 = 2U 2V , (4.64) leading to canonically normalised kinetic terms:

L(3)kin =

2e24UH1 H1

2e24UD

(4.59)

2 dWi dWi + e2WiDYi DYi . (4.61)

The corresponding scalar manifold is clearly [SU(1, 1)/U(1)]2 and the Kahler potential is

K =

Pi log(zi), where zi = eWi + iYi. In terms of Wi, the scalar potential becomes

L(3)pot = h

4eW1+W2 2e2(W1+W2)i

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The corresponding T tensor is found to be

T = 1

2 eW1 + eW2 eW1+W2

. (4.65)

In the process we have redened Y2 = c so that DY2 = dY2 + 2( A) and in addition

dualised the one-form potentials, A, so that

F = e24U DY1, DY1 = dY1 + B1,F = e4U4V DY3, DY3 = dY3 + B3. (4.66)

3 dWi dWi + e2WiDYi DYi

As should be customary at this stage, we have to add a corresponding CS term so the new topological term is

L(3)top = 2

F + B1

F + B3 F . (4.67)

Introducing complex coordinates in the usual fashion, zi = eWi + iYi, i = 1, 2, 3, the Kahler potential for the scalar manifold is K =

Pi log(zi).

In terms of our new scalars Wi, the potential takes a simple form and is symmetric in all the scalars Wi:

L(3)pot = 2 h

4eW1+W2+W3 e2(W1+W3) e2(W1+W2) e2(W2+W3)i

A suitable choice for the corresponding T tensor is

T = eW2 +

eW1+W2 eW1+W3 + eW2+W3 , (4.69)

though symmetry dictates that there are other choices and we can send W1 W2 W3 W1 to uncover the other options. Regardless of how we choose T , the critical point is located at Wi = 0. Since the R symmetry is gauged, we do not expect a holomorphic superpotential.

5 Null-warped AdS3 solutions

Recently, it has been noted that null-warped AdS3 solutions, or equivalently geometries exhibiting Schrdinger symmetry with z = 2, can be found in three-dimensional theories that arise as consistent reductions based on the D1-D5 (or its S-dual) near-horizon geometries of type IIB supergravity [38]. In section 4.2, we identied the relevant theories in the gauged supergravity literature and here we will discuss some of the solutions. Prior to [38], it was noted that non-relativistic geometries with dynamical exponent z = 4 could be found in an N = 2 gauged supergravity that is the consistent KK reduction of eleven-

dimensional supergravity on S2 CY3 [11].15 We will now address a natural question by

scanning the other gauged supergravities we have identied for non-relativistic solutions with dynamical exponent z.

Before doing so, we recall some facts about Schrdinger solutions in three dimensions. Starting from an AdS3 vacuum, solutions with dynamical exponent z arise as solutions to

Chern-Simons theories where the relevant equation is

d 3 F +

F = 0 , (5.1) with F = dA and denotes the AdS3 radius. Taking the derivative of (5.1), we see that must be a constant. Adopting the usual form of the space-time Ansatz

ds2 = 2

2rzdu2 + 2rdudv +dr2 4r2

22 , R++ =

vol3 . (4.68)

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, (5.2)

the Einstein equation, through the components of the Ricci tensor:16

R+ =

2 2z (z 1) rz1, R = 0 , (5.3)

15These were mistakenly labelled null-warped AdS3, but this label should be reserved solely for the z = 2 case in the literature.

16We have used the dreibein e+ = r 12 du, e = r 12[notdef][notdef]dv 12 2rz1du[parenrightbig] , er = dr2r .

determines the constant in terms of the dynamical exponent, = z. Observe here that is an arbitrary constant that can either be set to unity through rescaling the metric, or when set to zero, one recovers the unwarped AdS3 vacuum.

Now the task of searching for new solutions becomes a very accessible goal; one simply has to identify and compare the equations of motion of the theory with (5.1) to extract and thus z. For the gauged supergravity discussed in section 3, namely the theory given by the action (3.6), the AdS3 radius is

= 1

2T =

2a1a2a3

, (5.4) which in general depends on the parameters ai. For simplicity, we conne our search to the case where Gi = G, i.e. they are all equal. After changing frame to Einstein frame, consistency of the three equations (C.5) then places constraints on ai:

{a1 = a2 = a3} ,

a1 = a2 = 2 7a3

a2 = a3 = 27a1 . (5.5)

Combining these with the condition for a supersymmetric vacuum (2.14), one reaches the conclusion that good AdS3 solutions exist only for g = H2.17 The two independent choices we nd are

(a1, a2, a3) =

13,13,1 3

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, a1 = a3 = 27a2 ,

, (a1, a2, a3) = 711,211,2 11

, (5.6)

where one is free to consider various cyclic permutations of the latter. The rst choice leads to the non-integral value z = 43 with =

29 . The second choice does produce an integer,

namely z = 18 with = 8

11 . Thus, within the limited scope of our search, we do not nd any null-warped AdS3 (z = 2) solutions here.

Moving on, we can turn to the gauged supergravity corresponding to twisted compactications of N = 1 SCFTs, namely (4.16). A particular case of this we have already

covered above. Referring the reader to equations (E.1) and (E.6), if one truncates consistently to just K1, K2 and F2, and regardless of how one further truncates to an equation bearing resemblance to (5.1), one nds the dynamical exponent z = 43 . This should not come as a surprise as once one truncates to these elds, the theory should correspond to ve-dimensional U(1)3 theory where one identies two of the gauge elds and truncates out a scalar.

However, for the action (4.16), we do have other options. As we are considering a null space-time, it is consistent to truncate to just the scalar c and one-form C1 with the various other scalars taking their vacuum values. Obviously, this is not a consistent truncation in general, but since we assume G2 G2 = G1 G1 = M1 M1 = 0 in this case, we do not have to worry about the consistency of equations such as (E.5), (E.7) and (E.8). Note that M1 is not independent and is related to G1, M1 = 2G1. This in turn means that, in addition to the Einstein equation, we only have two ux equations

d G1 = 0, d G2

9 G1 = 0 , (5.7)

17One can compare the values of ai against gure 1 of [15].

where we have used = 29 and e2C = 13. If we further truncate to set G1 = 29G2, then we

can nd null-warped AdS3 solutions with z = 2. This allows us to determine c which can be set consistently to zero. In the notation of section 4.1, the solution may be expressed as

ds2 = 2

rzdu2 + 2rdudv +dr2 4r2

3 r du , (5.8)

where we have rescaled C1 so that = 1.

We can also consider deformations for AdS3 supported by the scalar b and one-form B1. This involves consistently truncating the action (4.16) to N1, H1 and H2 and since this may be regarded as the S-dual of the truncation presented immediately above, we recover the same solution.

For some sense of completeness, we also touch upon the existence of solutions for the theory arising from a dimensional reduction on S2 T 4 from ten dimensions presented

in section 4.2. Schrdinger solutions based on the D1-D5 near-horizon, or its S-dual F1-NS5, have already been the focus of considerable attention in the literature. Not only have solutions been constructed directly in ten dimensions [55], but examples in the three-dimensional setting have also been identied in [38]. Though not mentioned in [38], an S-duality transformation is all that is required to generate an example supported purely by the NS sector provided one starts with the RR supported two-parameter family of [38]. Rather than take this path, we will work directly with our reduced theory and employ an appropriate Ansatz. We will also make use of a further truncation.

Starting from the action in section 4.2, we take = 2 and truncate out various elds U = V = i = i = a = c = C1 = 0. This corresponds to setting the scalars to their AdS3 vacuum ( = 1) values and the choice of is appropriate for a vacuum supported solely by NS ux. Further truncating out A1 leads to the condition H1 = H2, leading to the

equations of motion:

d H2 = 2H2 ,R = 2g + H2H

where we have used the fact that B1 is null. Note that the CS equation is now in the accustomed form (5.1), so we can be condent we have a null-warped solution. It is then a straightforward exercise to provide the explicit form of the solution that satises these equations of motion:

ds2 = rzdu2 + 2rdudv +

C1 = 2

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2 , (5.9)

dr2

4r2 ,

B1 = r du . (5.10)

It would be interesting to see if any solutions can be generated through applying TsT [25] transformations, such as those considered in [97].

6 Outlook

Our primary motivation for this work stems from [16] where ve-dimensional U(1)3 gauged supergravity was dimensionally reduced on a Riemann surface and the lower-dimensional theory re-expressed in terms of the language of three-dimensional gauged supergravity [9]. As explained in section 3, the T tensor presents a natural supergravity counterpart to the quadratic trial function for the central charge presented in [14, 15] and it is a striking feature that the T tensor, through the embedding tensor, knows about the exact R symmetry. Without recourse to the higher-dimensional solution, this provides a natural way to identify the exact central charge and R symmetry directly in three dimensions.

Since any solution to this particular three-dimensional gauged supergravity uplifts to the U(1)3 theory in ve dimensions, which is itself a reduction of type IIB supergravity [75], we have also taken the opportunity to step back and address consistent KK reductions to three dimensions for wrapped D3-brane geometries. As reviewed in section 2, the origin of supersymmetric AdS3 geometries in type IIB can be traced to D3-branes wrapping Kahler two-cycles in Calabi-Yau manifolds, with CFTs of interest to c-extremization, namely those with N = (0, 2) supersymmetry, resulting when a two-cycle in a Calabi-Yau four-fold is

wrapped. All AdS3 solutions of this form fall into the general classication of supersym-metric geometries presented in [20] and at the heart of each supersymmetric geometry is a six-dimensional Kahler manifold M6, satisfying the di erential condition (1.1).

Not only does this condition appear in the ux equations of motion, but the Einstein equation is satised through imposing this condition. This makes the task of nding a fully generic KK reduction, in contrast to the case studied in section 3, where one assumes the presence of a Riemann surface, an inviting problem. It is expected that one can gauge the U(1) R symmetry and reduce to three dimensions in line with the conjecture of [72] that gaugings of R symmetry groups always lead to consistent reductions to lower-dimensional gauged supergravities. What is not clear at this moment is whether a truly generic reduction - one working at the level of the supersymmetry conditions - on M6 exists, thus

mimicking general reductions to ve dimensions discovered in [71, 72], or whether one needs to specify more structure for the M6. An added subtlety here is that since the reduced

theory is expected to t into N = 2 gauged supergravity, it is not enough simply to retain a

gauge eld coming from an R symmetry gauging and an extra degree of freedom is required.

Naturally enough, what we have discussed here just pertains to D3-branes and AdS3 vacua also arise in eleven-dimensional supergravity arising from wrapped M5-branes. It is then tting to consider KK reductions from eleven dimensions to three-dimensional gauged supergravity. While supersymmetric AdS3 solutions can be found by considering twists of seven-dimensional supergravity [15, 98, 99], more general solutions are expected to t into the general classication of supersymmetric solutions presented in [59, 60]. A particular case discussed in [15], namely seven-dimensional supergravity reduced on H2H2, we have already considered18 and we will report on M5-brane analogues in future work [100].

In addition to the c-extremization angle, another thread to our story concerns the search for null-warped AdS3 or Schrdinger (z = 2) solutions. While it is likely that we

18It corresponds to N = 2 supergravity with Kahler manifold [SU(1, 1)/U(1)]4.

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have recovered some known solutions, and found solutions with more general z, we believe that the solutions based on H2 KE4 internal geometries are new. What remains is to

check whether they preserve supersymmetry, and indeed the identication of the Killing spinor equations for the reduced theories in sections 4.1 and 4.2 needs to be considered if one is to discuss supersymmetric solutions. The reduction in section 3 aside, we have simply focused on the bosonic sector and the equations of motion. It may also be interesting to study families of Schrdinger solutions interpolating between the D1-D5 vacuum and F1-NS5 vacuum directly in three dimensions. This would presumably overlap with the higher-dimensional examples presented in [55]. It is expected that some supersymmetry is preserved.

Combining the principle of c-extremization [14, 15], which can be understood in terms of three-dimensional supergravity [16], and the fact that null-warped AdS3 solutions clearly exist, it is worth considering if c-extremization can be extended to warped AdS3. The most immediate setting to address this question is the theory of section 3, however, as we have seen, the simplest solutions appear to preclude solutions with z = 2. A more thorough search for null-warped solutions is warranted. If they do not exist, one can imagine starting from a more involved theory in ve dimensions that includes the U(1)3 gauged supergravity. Evidently, the more involved reductions based on H2 KE4 and S2 T 4 allow solutions,

so it can be expected that this question can be addressed in future work.

It would equally be interesting to look for a holographic analogue of c-extremization in two dimensions.19 Starting from eleven dimensions, one can reduce to four dimensions [75] retaining the Cartan subgroup U(1)4 of the R symmetry group. Relevant solutions are already known [70, 78], and the two-dimensional theory one gets from twisted compactications on Riemann surfaces are likely to be in the literature, for example [101], and may be related to BFSS matrix quantum mechanics [102]. At a quick glance, it looks like we have some of the jigsaw pieces in place.

One of the potentially interesting avenues for future study is to explore the connection between supersymmetric black holes in ve dimensions and null-warped AdS3 space-times.

For non-relativistic geometries with z = 4, it was noted in [11] that these geometries naturally appear when one considers a general class of ve-dimensional supersymmetric black holes and strings and then reduces on an S2. The corresponding picture for the known null-warped solutions can also be worked out. It would be interesting to extend recent studies of the classical motion of strings in warped AdS3 backgrounds [103] to higher-dimensional black holes.

Finally, we are aware of string theory embeddings of holographic superconductors in four and ve dimensions [8890], where an important element in the construction is the presence of charged scalars that couple to the complex form of the internal Kahler-Einstein manifold. To date, there is no example of an embedding of the bottom-up model considered in [104], though strong similarities between the supersymmetric geometries here and Sasaki-Einstein manifolds suggest that this may be a good place to look. So far we have been unable to nd a consistent reduction based on M6 = S2 T 4 or M6 = H2 KE4, but one

could hope to address the problem perturbatively. Such an approach was adopted in [105].19We are grateful to N. Halmagyi for suggesting this possibility.

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Acknowledgments

We would like to thank P. Szepietowski for sharing his interest in generic reductions at an initial stage of this project. We are grateful to N. Bobev, J. P. Gauntlett, N. Halmagyi,J. Jeong, H. Lu and Y. Nakayama for correspondence on related topics. We have also enjoyed further discussions with M. Guica, E. Sezgin and O. Varela. E C wishes to thank the Centro de Ciencias de Benasque Pedro Pasqual and the organisers of the String Theory meeting for hospitality while we were writing up. P. K. is supported by Chulalongkorn University through the Ratchadapisek Sompote Endowment Fund, Thailand Center of Excellence in Physics through the ThEP/CU/2-RE3/11 project, and The Thailand Research Fund (TRF) under grant TRG5680010. E. C acknowledges support from the research grant MICINN-09- FPA2012-35043-C02-02.

A Type IIB supergravity conventions

Our conventions for type IIB supergravity follow those of [82], which for completeness, we reproduce here. Restricting ourselves to the bosonic sector of type IIB supergravity, the eld content consists of RR n-forms F(n), n = 1, 3, 5, the NS form H(3), the dilaton and the metric. The forms satisfy the Bianchi identities

dF(5) + F(3) H(3) = 0 , (A.1) dF(3) + F(1) H(3) = 0 , (A.2)

dF(1) = 0 , (A.3)

dH(3) = 0 , (A.4)

which can be satised through the introduction of potentials C(n1), B(2). In terms of these

potentials, the forms are F(5) = dC(4) C(2) H(3), F(3) = dC(2) C(0)H(3), F(1) = dC(0), H(3) = dB(2). In addition to the self-duality condition on the ve-form, F(5) = F(5), the

equations of motion take the form:

d e F(3)

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F(5) H(3) = 0 , (A.5) d e2 F(1)

+ e H(3) F(3) = 0 , (A.6) d e H(3)

e F(1) F(3) F(3) F(5) = 0 , (A.7)

d d e2 F(1) F(1) +

2e H(3) H(3)

2e F(3) F(3) = 0 , (A.8)

RMN = 12MC(0)NC(0) +

2M N +

1 96FMPQRSF

P QRS N

1 4e

H PQMHNPQ

1 12gMNHPQRHPQR

1 4e

F P Q

M FNPQ

112gMNF PQRFPQR

. (A.9)

B Connection between [11] and [38]

In this section we will discuss the connection between two dimensional reductions from higher-dimensional supergravities to three-dimensional theories that have appeared in the literature. Both theories admit supersymmetric Schrdinger solutions, however, for those based on the D1-D5 near-horizon [38] the dynamical exponent z = 2 appears, while the dynamical exponent quoted in [11] is z = 4.

Recall that these theories support AdS3 vacua whose higher-dimensional manifestations are AdS3 S3 CY2 geometries of type IIB supergravity and AdS3 S2 CY3

geometries of eleven-dimensional supergravity, respectively. Specialising to the case where the Calabi-Yau three-fold is a direct product involving a torus T 2, CY3 = CY2 T 2, it is a well-known fact that the geometries are related via dimensional reduction and T-duality.

This raises a question about the di erence in the quoted dynamical exponents. Here we address that issue and show that a sub-truncation of [38] and [11] is common and that amongst the z = 2 solutions presented in [38], one can also nd a z = 4 solution.

We start by considering the KK reduction Ansatz from eleven-dimensions. The solution appearing in [11] has a higher-dimensional manifestation of the form

ds211 = e4W ds23 + e2W ds2 S2

+ ds2 (CY2) + dx25 + dx26 ,

G4 = vol(S2) + H2

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(JCY2 + dx5 dx6) , (B.1)

where we have consistently truncated out the elds f, V, B1 leaving just a scalar W and one-form potential B2, where H2 = dB2. Here (x5, x6) label coordinates on the T 2 and is a constant. Plugging this Ansatz into the equations of motion of eleven-dimensional supergravity one nds [11]

d e4W 3 H2

= 2H2 , (B.2) d 3 dW =

2eW H2 3H2 + e6W 2e8W

vol3 , (B.3)

and the Einstein equation which we omit.

Dimensional reduction on x6 and T-duality on x5 leads to the following IIB KK reduction Ansatz

ds210 = e4W ds23 + e2W ds2 S2

+ ds2 (CY2) + (dx5 cos d + B2)2 , (B.4)

F5 = (1 + 10)

h vol S2

JCY2 + JCY2 H2

(dx5 cos d + B2) ,

where (, ) parametrise the two-sphere S2 and all other elds, including the dilaton are zero.

At this point it is easier to compare with the ten-dimensional uplift [93] of the six-dimensional Ansatz considered in [38] to get our bearings. After rescaling the metric to make the transition to string frame, the ten-dimensional space-time may be written as

ds210 = e

2 + 22 ds26 + e

2 22 ds2 (CY2) ,

ds26 = e4U2V ds23 + 14e2Uds2 S2

+ 1

4e2V (d + cos d + 2A)2 , (B.5)

where we have set the length-scale corresponding to the AdS3 radius to unity for simplicity. To compare the metrics we note that we require the following identications:

= 1 = 2 = 2V, eW =

2eU, 2x5 = , =

2, B2 = A. (B.6)

While this places us in the class of consistent reductions in section 4.2 of [38], the added condition that the dilaton is zero tells us that the scalars , V appearing in equations(B.25) and (B.29) of [38] are zero. These equations together then tell us that the two gauge elds appearing in [38] should be identied A =

. For CY2 = T 4, the RR-sector is then

simply related via T-duality.

The choice A = immediately leads to the condition F 2 = 0 through (B.25), however there is another option. We can choose A =

with the further relation

A = 1

4e4U 3 F . (B.7)

With this relation one can then satisfy oneself that (B.27) and the U equation from (B.29) of [38] can be identied with (B.2) and (B.3) above, meaning that this particular sub-truncation of both reductions is the same.

Indeed, since the higher-dimensional AdS3 solutions can be related via dimensional reduction and T-duality, it is expected that the KK reductions are also related at some level.

C Details of reduction of D = 5 U(1)3 gauged supergravity

Here we begin by recording the ve-dimensional equations of motion one gets from varying the action (3.1). The equations of motion for the gauge elds Ai, i = 1, 2, 3, are

d X21 F 1

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= F 2 F 3, d X22 F 2

= F 1 F 3, d X23 F 3

= F 1 F 2, (C.1)

and those of the scalars are given by

d d1 =

16 X21F 1 F 1 + X22F 2 F 2 2X23F 3 F 3

46 X11 + X12 2X13

vol5 , (C.2)

d d2 =

12 X21F 1 F 1 X22F 2 F 2

g222 X11 X12

vol5 .

Finally, the Einstein equation reads

R = 12

Xi=1ii + 12

Xi=1 X2i

F iF i 16gF iF i

4 3g2

Xi=1X1i . (C.3)

The reduction at the level of the equations of motion is most simply performed be rst reducing on the internal space, in this case a Riemann surface g, and then rescaling the external space-time to go to Einstein frame. Thus, here we consider the initial Ansatz for the ve-dimensional space-time

ds25 = ds23 + e2Cds2 ( g) , (C.4)

where C is a scalar warp factor depending on the coordinates of the three-dimensional space-time.

To reduce the gauge eld strengths we consider the Ansatz (3.4). The equations of motion for the gauge elds now reduce as

d X21e2C 3 G1

= a3G2 + a2G3

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d X22e2C 3 G2

= a3G1 + a1G3

d X23e2C 3 G3

= a1G2 + a2G1

. (C.5)

From the scalar equations of motion, we nd

d e2C 3 d1

= 1

6e2C

X21 G1 3G1 + a21e4C vol3

+ X22

G2 3G2

+ a22e4C vol3

2X23 G3 3G3 + a23e4C vol3

46e2C X11 + X12 2X13

vol3,

d e2C 3 d2

= 1

2e2C

X21 G1 3G1 + a21e4C vol3

X22

G2 3G2

+ a22e4C vol3

22g2e2C X11 X12

vol3 . (C.6)

The Einstein equation along the Riemann surface presents us with another scalar equation of motion, this time for C:

C 2AC + e2C =

Xi=1 X2i

23a2ie4C 16GiGi 43g23

Xi=1 X1i,

(C.7)

where is the curvature of the Riemann surface.

Finally, the Einstein equation in three dimensions may be written as

R = 2 (C + CC) +

Xi=1ii + 12

Xi=1 X2i

GiGi 1 6gGiGi

1 6g

3 a2ie4CX2i + 8g2X1i

. (C.8)

The above equations can be shown to result from varying the action

L(3) = e2C "

R 3 1 + 2dC 3dC

Xi=1di 3di 1 2

Xi=1X2iGi 3Gi#

+ 3 4g2e2CX1i 12e2Ca2iX2i + 2

! 3

1 + L(3)top , (C.9)

where the topological term is

L(3)top = a1B2 G3 + a2B3 G1 + a3B1 G2 . (C.10)

Here Bi is the one-form potential for Gi, Gi = dBi.

Now, to go to Einstein frame we just need to do a conformal transformation, g =

e4C. This leads to the Einstein frame action (3.6) quoted in the text.

In checking the Einstein equation we have made use of the following Ricci tensor components

R = R 2 (C + CC) , Rmn =

e2C C 2CC mn , (C.11)

where , label space-time directions and m, n correspond to directions on the Riemann surface.

C.1 Killing spinor equations

We would like to conrm that the T tensor (3.16) can be extracted directly from the Killing spinor equations via reduction. In a related context, a similar calculation appeared in [19] and in that context assisted the identication of a ve-dimensional prepotential. Our motivation here is the same.

We adopt the conventions for the Killing spinor equations in D = 5 from (F.1) of [15] (see also [66]), and in some sense, up to some additional elds, the calculation here is almost identical to appendix F of [15]. We work with the natural vielbein

e = e2C, ea = eCa , (C.12)

where = 0, 1, 2 label three-dimensional space-time directions and a = 3, 4 denote directions along the Riemann surface. Our Ansatz for the ux follows from (3.4).

For the Killing spinor we make the choice

= eC , (C.13)

where is a constant we will x later. We use the following decomposition of the ve-dimensional gamma matrices

= 3, 3 = 1 1, 4 = 1 2. (C.14)

As in [15], where one has 34 = i, following decomposition, we have 3 = .

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Inserting the Ansatz into the Killing spinor equations we arrive at

23 =" 3e2CC +3

Xi=1

Xi3 + i3e2CaiX1i4 +i12e4C 3X1iGi #eC ,

(C.15)

6(1) =

182

Xi=1X1i e4CGi 2iaie2C

14X13 e4CG3 2ia3e2C

+ i

2 (X1 X2 + 2X3) i

4 e2C1

eC , (C.16)

2(2) =

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18X11 e4CG1 2ia1e2C

18X12 e4CG2 2ia2e2C

eC . (C.17)

Note, in contrast to [15] where scalars with raised and lowered indices are employed, here our Xi are simply those in (2.22). As a consistency check, (C.15), (C.16), (C.17) agree with (3.20) of [15] when Gi = 0 and i = i(r), C = g(r).

Taking various linear combinations we can write

433 + 2

36i(1) + 22i(2) = 1 ,

433 + 2

36i(1) 22i(2) = 2 ,

433

436i(1) = 3 (C.18)

leading to the variations (constant = 2)

1 =

W1+ i2X11e2CG1 +e4C 2e2CX1a2X12a3X13

+ i

2 (X1 + X2) i

4 e2C2

2 =

W2+ i2X12e2CG2 +e4C 2e2CX2a1X11a3X13

3 =

W3+ i2X31e2CG3 +e4C 2e2CX3a1X11a2X12

. (C.19)

Dualising Gi as instructed in the text, the above equations can be condensed into a single equation

a = 2E ai Dzi 2iT

, (C.20) which is the expected form for the Killing spinor equation for the spinor elds [9, 11] and we see that the T tensor (3.16) features. E ai, a = 1, 2, 3, is the complex dreibein dened through gi = E aiEia, where Eia = (E ai).

D Curvature for Kahler-Einstein space-times

Working in Einstein frame, we adopt the following Ansatz for the space-time

ds210 = e2Ads2 (M3) + e2A

14e2W (dz + P + A1)2 + e2A

Xa=1 e2Vads2

KE(a)2

, (D.1)

where A is a constant overall factor, we have dropped the overall scale L appearing in (2.11) and W , Va, a = 1, 2, 3 denote scalar warp factors. A1 is a one-form living on the three-dimensional space-time M3.

We adopt the natural orthonormal frame

e = eA, ez = eA+W 12 (dz + P + A1) , ei = eA+Vai, (D.2)

where = 0, 1, 2 label AdS3 directions and i = 3, . . . , 8 correspond to directions along the internal Kahler-Einstein spaces.

With constant A, the spin-connection for the metric may be written as

14eA+W (F2)ez,

ij =

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1 4e3A+W2Vala(Ja)ijez,

z = eAW ez

14eA+W (F2)e,

i = eAVaei,

iz =

14e3A+W2Vala(Ja)ijej. (D.3)

Using the above spin-connection one can calculate the Ricci-form

R = e2A

R (W +W W )

Xa=12 (Va+VaVa) 18e2W F2F 2

Rzz = 1

8e6A+2W

Xa=1e4Val2a e2A (W + W W ) 2W

Xa=1 e2AVa

+ 1

16e2A+2W F2F 2,

R11 = R22 = e2A

V1W V12V13

Xi=a Va

+l1e2A2V1 1 8l21e4A+2W4V1,

R33 = R44 = e2A

V2W V22V23

Xi=a Va

+l2e2A2V2 1 8l22e4A+2W4V2,

R55 = R66 = e2A

V3W V32V33

Xa=1 Va

+l3e2A2V3 1 8l23e4A+2W4V3,

Rz =

1 4e2W2(V1+V2+V3)

e3W+2(V1+V2+V3)F

, (D.4)

where all other terms are zero.

E Details of reduction on H2 KE4

In this section we record equations of motion of the dimensionally reduced three-dimensional theory. This will be useful for testing the consistency of the reduction. We

begin with the Bianchi identities. The Bianchi identities for the three-form uxes F(3) and H(3) are trivially satised using the expressions in the text. The Bianchi for F(5) is partially satised, with the remaining equations being:

d e43 (U+V )+4C K2

4e8U K1 + (K2 F2) N1 G1 H1 M1 = 0,

d e8U K1

+ 1

2N1 G2

2H2 M1 = 0 . (E.1)

The equations of motion for F(3) and H(3) give respectively the equations

d e43 (4U+V )+4C M1 4h vol3 +2H2 K1 2H1 K2 = 0 ,

d e43 (2UV )++4C G2 4e4U+ G1 e43 (4U+V )+4C M1

+ ge

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43 (4U+V )++8C F2 + 2N1 K1 + 2e

43 (U+V )+4C H1 K2 = 0 ,

e4U+ G1

N1 K2 + h vol3 +e43 (U+V )+4CH2 K2 + 2e8UH1 K1 = 0 ,(E.2)

and

e43 (4U+V )4C N1 + 4g vol3 2G2 K1 + 2G1 K2 e43 (4U+V )+4C da M1 = 0,

e43 (2UV )+4C H2 4e4U H1 e43 (4U+V )4C N1 + he43 (4U+V )+8CF2

2M1 K1 2e

43 (U+V )+4C G1 K2 e

43 (2UV )++4C da G2 = 0,

e4U H1 + M1 K2 g vol3 e43 (U+V )+4C G2 K2 2e8UG1 K1

e4U+da G1 = 0 . (E.3)

The axion and dilaton equation are respectively

d e2 da +e43 (4U+V )+4C N1 M1e43 (4U+V )++8C gh vol3+e43 (2UV )++4C H2 G2

2e4U+H1 G1 = 0 , (E.4)

and

d d e2da da +

43 (4U+V )4C

heN1 N1 eM1 M1i

43 (4U+V )+8C

heh2 eg2ivol3 +12e43 (2UV )+4C

heH2 H2 eG2 G2i

+ e4U

heH1 H1 eG1 G1i= 0 . (E.5)

The equations of motion for A1, U and V are d

e83 (U+V )+4C F2 2K2 8e8U K1 + e43 (4U+V )+8C

ehH2 + egG2 = 0, (E.6)

d dU + e8UK1 K1

1 8e4

3 (4U+V )4C

eN1 N1 + eM1 M1

(E.7)

+ 1

8e4

3 (4U+V )+8C

eh2 + eg2 vol3 18e43 (2UV )+4C

eH2 H2 + eG2 G2

+ 1

4e4U

eH1 H1+eG1 G1

+e4C

6e23 (7U+V ) +2e43 (5U+V ) +4e83 (4U+V )

=0,

d dV

1 8e4

3 (4U+V )4C

eN1 N1 + eM1 M1 + 18e43 (4U+V )+8C

eh2 + eg2 vol3

2e8

3 (U+V )+4C F2 F2 +

2e4

3 (U+V )+4C K2 K2 e8UK1 K1

22e8

3 (U+V )8C vol3 +122e4

3 (U+V )8C vol3 +38e4

3 (2UV )+4C

eH2 H2 + eG2 G2

1 4e4U

eH1 H1 + eG1 G1 + e4C

4e43 (5U+V ) + 4e83 (4U+V ) vol3 = 0 . (E.8)

F Details of reduction on S2 T 4

IIB reduced on CY2. Here we briey review the KK reduction Ansatz of type IIB on a Calabi-Yau two-fold that featured in [93]. The KK Ansatz in Einstein frame is

ds210 = e

2 2ds26 + e

JHEP10(2013)094

2 2ds2 (CY2) ,

F(5) = vol (CY2) d2 + e22 6 d2 , (F.1)

and all other elds of type IIB supergravity simply reduce to six dimensions. This Ansatz thus leads to extra scalars in addition to the axion 1 and dilaton 1 of type IIB super-gravity, one corresponding to a breathing mode 2, and another axion 2 coming from the self-dual ve-form ux. The six-dimensional action is

e1L = R

Xi=112 (i)2

Xi=112e2i (i)2 1 12e12H23

112e12F 23 2dB2 dC2, (F.2)

where H3 = dB2 and F3 = dC2 1dB2. Some sign changes relative to [93] follow from

the di erence in conventions. The equations of motion are:

e12 6 F3 d2 dB2 = 0, (F.3)

e12 6 H3 e12d1 6F3 + d2 F3 = 0, (F.4)

e21 6 d1 + e12dB2 6F3 = 0, (F.5)

e22 6 d2 dB2 dC2 = 0, (F.6)

d 6 d1 e21d1 6d1 +

2e12H3 6H3

2e12F3 F3 = 0, (F.7)

d 6 d2 e22d2 6d2 +

2e12H3 6H3 +

2e12F3 F3 = 0, (F.8)

R = 1

2 ii + e2iii

+ 14e12

H312H 12

16gH3123H 1233

+ 1

4e12

F312F 12

1 6gF3123F

123 3

. (F.9)

Reduction to three dimensions. To reduce the above equations of motion to three dimensions we substitute in our six-dimensional space-time Ansatz

ds26 = ds23 + 14e2Uds2 S2

+ 1

4e2V (dz + P + A1) , (F.10)

and expressions for the three-form eld strengths (4.37). From (F.3) and (F.4) we get

d e12+2U+V g 22 sin = 0, (F.11)

d e12+V 2U G1 + d2 H2 = 0, (F.12)

d e12V +2U G2 2e12+V 2U G1+12ge12+2U+V F2d2 H1 = 0 , (F.13)

and

d e12+V +2Uh e12+V +2Ugd1 + G0d2 = 0, (F.14)

d e12+V 2U H1 e12+V 2Ud1 G1 d2 G2 = 0, (F.15)

d e12V +2U H2 2e12+V 2U H1 +12he12+V +2UF2

e12V +2Ud1 G2 + d2 G1 = 0 . (F.16)

We can now solve (F.11) and (F.14) to determine g and h

g = 2e1+2V 2U (cos + sin 2) , (F.17)

h = 2e1+2V 2U [sin cos 2 + (cos + sin 2)1] . (F.18)

In the process we have chosen the integration constants for convenience.From (F.5) and (F.6) we get the following two equations:

d e21+2U+V d1 +

h2 sin G0e122UV ghe12+2U+V i vol3

+ e12+V 2UH1 G1 + e12V +2UH2 G2 = 0 , (F.19)

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e21+2U+V d2 + [hG0 2 sin g] vol3 +H1 G2 G1 H2 = 0 . (F.20)

The nal two scalar equations give

d e2U+V d1

e21+2U+V d1 d1 +

2e22UV

h4e1 sin2 e1G20i vol3

+ 12e22U+V

he1H1H1e1G1G1i+ 12e2+2UV

he1H2H2e1G2 G2i

2e2+2U+V

he1h2 e1g2ivol3 = 0, (F.21)

d e2U+V d2

e22+2U+V d2 d2 +

2e22UV

h4e1 sin2 + e1G20i vol3

+ 12e22U+V

he1H1H1+e1G1G1i+ 12e2+2UV

he1H2H2+e1G2G2i

2e2+2U+V

he1h2 + e1g2

vol3 = 0 . (F.22)

We now only have to work out the Einstein equation. Taking into account a change in how we dene scalars, namely W V, V1 U, we can use the Ricci tensor appearing

in (D.4). We simply have to take note of the fact that the S2 is normalised so that l1 = 4, in which case A = 0.

From the Einstein equation, we get the following equations:

2e2V 4U vol3 d dV dV dV 2dV dU +

18e2V F2 F2 (F.23)

14e22V 4U

4e1 sin2 + e1G20 + 1 4e2

e1h2 + e1g2

vol3

+ 1

4e22V

he1H2H2+e1G2G2i1 4e24U

he1H1 H1+e1G1 G1i

4e2U 2e2V 4U

vol3 d dU dU dV 2dU dU (F.24)

14e22V 4U

4e1 sin2 + e1G20 + 1 4e2

e1h2 + e1g2 vol3

1 4e22V

he1H2H2+e1G2G2i+ 14e24U

he1H1 H1+e1G1 G1i

2e2U2V d e3V +2U F2

= 2 sin e124UV H1 + G0e124UV G1

e12V hH2 e12V gG2 . (F.25)

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References

[1] H. Samtleben, Lectures on Gauged Supergravity and Flux Compactications, http://dx.doi.org/10.1088/0264-9381/25/21/214002

Web End =Class. Quant. Grav. 25 (2008) 214002 [arXiv:0808.4076] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0808.4076

Web End =INSPIRE ].

[2] B. de Wit and H. Nicolai, The Consistency of the S7 Truncation in D = 11 Supergravity, http://dx.doi.org/10.1016/0550-3213(87)90253-7

Web End =Nucl. Phys. B 281 (1987) 211 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B281,211

Web End =INSPIRE ].

[3] H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistent nonlinear K K reduction of 11 D supergravity on AdS7 S4 and selfduality in odd dimensions,

http://dx.doi.org/10.1016/S0370-2693(99)01266-6

Web End =Phys. Lett. B 469 (1999) 96 [http://arxiv.org/abs/hep-th/9905075

Web End =hep-th/9905075 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9905075

Web End =INSPIRE ].

[4] H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistency of the AdS7 S4 reduction

and the origin of selfduality in odd dimensions, http://dx.doi.org/10.1016/S0550-3213(00)00193-0

Web End =Nucl. Phys. B 581 (2000) 179 [http://arxiv.org/abs/hep-th/9911238

Web End =hep-th/9911238 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9911238

Web End =INSPIRE ].

[5] A. Khavaev, K. Pilch and N.P. Warner, New vacua of gauged N = 8 supergravity in ve-dimensions, http://dx.doi.org/10.1016/S0370-2693(00)00795-4

Web End =Phys. Lett. B 487 (2000) 14 [http://arxiv.org/abs/hep-th/9812035

Web End =hep-th/9812035 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9812035

Web End =INSPIRE ].

[6] M. Cveti et al., Embedding AdS black holes in ten-dimensions and eleven-dimensions, http://dx.doi.org/10.1016/S0550-3213(99)00419-8

Web End =Nucl. Phys. B 558 (1999) 96 [http://arxiv.org/abs/hep-th/9903214

Web End =hep-th/9903214 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9903214

Web End =INSPIRE ].

[7] H. L, C. Pope and T.A. Tran, Five-dimensional N = 4, SU(2) U(1) gauged supergravity

from type IIB, http://dx.doi.org/10.1016/S0370-2693(00)00073-3

Web End =Phys. Lett. B 475 (2000) 261 [http://arxiv.org/abs/hep-th/9909203

Web End =hep-th/9909203 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9909203

Web End =INSPIRE ].

[8] M. Cveti, H. L, C. Pope, A. Sadrzadeh and T.A. Tran, Consistent SO(6) reduction of type IIB supergravity on S5, http://dx.doi.org/10.1016/S0550-3213(00)00372-2

Web End =Nucl. Phys. B 586 (2000) 275 [http://arxiv.org/abs/hep-th/0003103

Web End =hep-th/0003103 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0003103

Web End =INSPIRE ].

[9] B. de Wit, I. Herger and H. Samtleben, Gauged locally supersymmetric D = 3 nonlinear -models, http://dx.doi.org/10.1016/j.nuclphysb.2003.08.022

Web End =Nucl. Phys. B 671 (2003) 175 [http://arxiv.org/abs/hep-th/0307006

Web End =hep-th/0307006 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0307006

Web End =INSPIRE ].

[10] H. Nicolai and H. Samtleben, Chern-Simons versus Yang-Mills gaugings in three-dimensions, http://dx.doi.org/10.1016/S0550-3213(03)00569-8

Web End =Nucl. Phys. B 668 (2003) 167 [http://arxiv.org/abs/hep-th/0303213

Web End =hep-th/0303213 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0303213

Web End =INSPIRE ].

[11] E. Colgin and H. Samtleben, 3D gauged supergravity from wrapped M5-branes with AdS/CMT applications, http://dx.doi.org/10.1007/JHEP02(2011)031

Web End =JHEP 02 (2011) 031 [arXiv:1012.2145] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.2145

Web End =INSPIRE ].

[12] E. Gava, P. Karndumri and K. Narain, 3D gauged supergravity from SU(2) reduction of N = 1 6D supergravity, http://dx.doi.org/10.1007/JHEP09(2010)028

Web End =JHEP 09 (2010) 028 [arXiv:1006.4997] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1006.4997

Web End =INSPIRE ].

[13] H. L, C.N. Pope and E. Sezgin, SU(2) reduction of six-dimensional (1,0) supergravity, http://dx.doi.org/10.1016/S0550-3213(03)00534-0

Web End =Nucl. Phys. B 668 (2003) 237 [http://arxiv.org/abs/hep-th/0212323

Web End =hep-th/0212323 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0212323

Web End =INSPIRE ].

[14] F. Benini and N. Bobev, Exact two-dimensional superconformal R-symmetry and c-extremization, http://dx.doi.org/10.1103/PhysRevLett.110.061601

Web End =Phys. Rev. Lett. 110 (2013) 061601 [arXiv:1211.4030] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.4030

Web End =INSPIRE ].

[15] F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, http://dx.doi.org/10.1007/JHEP06(2013)005

Web End =JHEP 06 (2013) 005 [arXiv:1302.4451] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1302.4451

Web End =INSPIRE ].

[16] P. Karndumri and E. Colgin, Supergravity dual of c-extremization, http://dx.doi.org/10.1103/PhysRevD.87.101902

Web End =Phys. Rev. D 87 (2013) 101902 [arXiv:1302.6532] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1302.6532

Web End =INSPIRE ].

[17] K.A. Intriligator and B. Wecht, The exact superconformal R symmetry maximizes a, http://dx.doi.org/10.1016/S0550-3213(03)00459-0

Web End =Nucl. Phys. B 667 (2003) 183 [http://arxiv.org/abs/hep-th/0304128

Web End =hep-th/0304128 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0304128

Web End =INSPIRE ].

[18] Y. Tachikawa, Five-dimensional supergravity dual of a-maximization, http://dx.doi.org/10.1016/j.nuclphysb.2005.11.010

Web End =Nucl. Phys. B 733 (2006) 188 [http://arxiv.org/abs/hep-th/0507057

Web End =hep-th/0507057 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0507057

Web End =INSPIRE ].

[19] P. Szepietowski, Comments on a-maximization from gauged supergravity, http://dx.doi.org/10.1007/JHEP12(2012)018

Web End =JHEP 12 (2012) 018 [arXiv:1209.3025] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1209.3025

Web End =INSPIRE ].

[20] N. Kim, AdS3 solutions of IIB supergravity from D3-branes, http://dx.doi.org/10.1088/1126-6708/2006/01/094

Web End =JHEP 01 (2006) 094 [http://arxiv.org/abs/hep-th/0511029

Web End =hep-th/0511029 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0511029

Web End =INSPIRE ].

[21] J.P. Gauntlett and O.A. Mac Conamhna, AdS spacetimes from wrapped D3-branes, http://dx.doi.org/10.1088/0264-9381/24/24/009

Web End =Class. Quant. Grav. 24 (2007) 6267 [arXiv:0707.3105] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0707.3105

Web End =INSPIRE ].

[22] J.P. Gauntlett, N. Kim and D. Waldram, Supersymmetric AdS3, AdS2 and Bubble Solutions, http://dx.doi.org/10.1088/1126-6708/2007/04/005

Web End =JHEP 04 (2007) 005 [http://arxiv.org/abs/hep-th/0612253

Web End =hep-th/0612253 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0612253

Web End =INSPIRE ].

[23] J.P. Gauntlett, O.A. Mac Conamhna, T. Mateos and D. Waldram, New supersymmetric AdS3 solutions, http://dx.doi.org/10.1103/PhysRevD.74.106007

Web End =Phys. Rev. D 74 (2006) 106007 [http://arxiv.org/abs/hep-th/0608055

Web End =hep-th/0608055 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0608055

Web End =INSPIRE ].

[24] J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS5 solutions of type IIB supergravity, http://dx.doi.org/10.1088/0264-9381/23/14/009

Web End =Class. Quant. Grav. 23 (2006) 4693 [http://arxiv.org/abs/hep-th/0510125

Web End =hep-th/0510125 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0510125

Web End =INSPIRE ].

[25] O. Lunin and J.M. Maldacena, Deforming eld theories with U(1) U(1) global symmetry

and their gravity duals, http://dx.doi.org/10.1088/1126-6708/2005/05/033

Web End =JHEP 05 (2005) 033 [http://arxiv.org/abs/hep-th/0502086

Web End =hep-th/0502086 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0502086

Web End =INSPIRE ].

[26] K. Pilch and N.P. Warner, A new supersymmetric compactication of chiral IIB supergravity, http://dx.doi.org/10.1016/S0370-2693(00)00796-6

Web End =Phys. Lett. B 487 (2000) 22 [http://arxiv.org/abs/hep-th/0002192

Web End =hep-th/0002192 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0002192

Web End =INSPIRE ].

[27] M. Gabella, D. Martelli, A. Passias and J. Sparks, N=2 supersymmetric AdS4 solutions of M-theory, arXiv:1207.3082 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.3082

Web End =INSPIRE ].

[28] N. Halmagyi, K. Pilch and N.P. Warner, On Supersymmetric Flux Solutions of M-theory, arXiv:1207.4325 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.4325

Web End =INSPIRE ].

[29] D. Anninos, W. Li, M. Padi, W. Song and A. Strominger, Warped AdS3 Black Holes, http://dx.doi.org/10.1088/1126-6708/2009/03/130

Web End =JHEP 03 (2009) 130 [arXiv:0807.3040] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.3040

Web End =INSPIRE ].

JHEP10(2013)094

[30] S. Deser, R. Jackiw and S. Templeton, Three-Dimensional Massive Gauge Theories, http://dx.doi.org/10.1103/PhysRevLett.48.975

Web End =Phys. Rev. Lett. 48 (1982) 975 [http://inspirehep.net/search?p=find+J+Phys.Rev.Lett.,48,975

Web End =INSPIRE ].

[31] S. Deser, R. Jackiw and S. Templeton, Topologically Massive Gauge Theories,http://dx.doi.org/10.1016/0003-4916(82)90164-6

Web End =Annals Phys. 140 (1982) 372 [Erratum ibid. 185 (1988) 406][Annals Phys. 281 (2000) 409] [http://inspirehep.net/search?p=find+J+AnnalsPhys.,140,372

Web End =INSPIRE ].

[32] K.A. Moussa, G. Clement and C. Leygnac, The black holes of topologically massive gravity, Class. Quant. Grav. 20 (2003) L277 [http://arxiv.org/abs/gr-qc/0303042

Web End =gr-qc/0303042 ] [http://inspirehep.net/search?p=find+EPRINT+gr-qc/0303042

Web End =INSPIRE ].

[33] A. Bouchareb and G. Clement, Black hole mass and angular momentum in topologically massive gravity, http://dx.doi.org/10.1088/0264-9381/24/22/018

Web End =Class. Quant. Grav. 24 (2007) 5581 [arXiv:0706.0263] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0706.0263

Web End =INSPIRE ].

[34] G. Clement, Warped AdS3 black holes in new massive gravity,

http://dx.doi.org/10.1088/0264-9381/26/10/105015

Web End =Class. Quant. Grav. 26 (2009) 105015 [arXiv:0902.4634] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0902.4634

Web End =INSPIRE ].

[35] E.A. Bergshoe , O. Hohm and P.K. Townsend, Massive Gravity in Three Dimensions, http://dx.doi.org/10.1103/PhysRevLett.102.201301

Web End =Phys. Rev. Lett. 102 (2009) 201301 [arXiv:0901.1766] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0901.1766

Web End =INSPIRE ].

[36] M. Gary, D. Grumiller and R. Rashkov, Towards non-AdS holography in 3-dimensional higher spin gravity, http://dx.doi.org/10.1007/JHEP03(2012)022

Web End =JHEP 03 (2012) 022 [arXiv:1201.0013] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1201.0013

Web End =INSPIRE ].

[37] U. Gran, J. Greitz, P.S. Howe and B.E. Nilsson, Topologically gauged superconformal Chern-Simons matter theories, http://dx.doi.org/10.1007/JHEP12(2012)046

Web End =JHEP 12 (2012) 046 [arXiv:1204.2521] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1204.2521

Web End =INSPIRE ].

[38] S. Detournay and M. Guica, Stringy Schrdinger truncations, http://dx.doi.org/10.1007/JHEP08(2013)121

Web End =JHEP 08 (2013) 121 [arXiv:1212.6792] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1212.6792

Web End =INSPIRE ].

[39] D. Son, Toward an AdS/cold atoms correspondence: A geometric realization of the Schrdinger symmetry, http://dx.doi.org/10.1103/PhysRevD.78.046003

Web End =Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0804.3972

Web End =INSPIRE ].

[40] K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, http://dx.doi.org/10.1103/PhysRevLett.101.061601

Web End =Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0804.4053

Web End =INSPIRE ].

[41] C.P. Herzog, M. Rangamani and S.F. Ross, Heating up Galilean holography, http://dx.doi.org/10.1088/1126-6708/2008/11/080

Web End =JHEP 11 (2008) 080 [arXiv:0807.1099] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.1099

Web End =INSPIRE ].

[42] J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, http://dx.doi.org/10.1088/1126-6708/2008/10/072

Web End =JHEP 10 (2008) 072 [arXiv:0807.1100] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.1100

Web End =INSPIRE ].

[43] A. Adams, K. Balasubramanian and J. McGreevy, Hot Spacetimes for Cold Atoms, http://dx.doi.org/10.1088/1126-6708/2008/11/059

Web End =JHEP 11 (2008) 059 [arXiv:0807.1111] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.1111

Web End =INSPIRE ].

[44] S.A. Hartnoll and K. Yoshida, Families of IIB duals for nonrelativistic CFTs, http://dx.doi.org/10.1088/1126-6708/2008/12/071

Web End =JHEP 12 (2008) 071 [arXiv:0810.0298] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0810.0298

Web End =INSPIRE ].

[45] Y. Nakayama, M. Sakaguchi and K. Yoshida, Non-Relativistic M2-brane Gauge Theory and New Superconformal Algebra, http://dx.doi.org/10.1088/1126-6708/2009/04/096

Web End =JHEP 04 (2009) 096 [arXiv:0902.2204] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0902.2204

Web End =INSPIRE ].

[46] K.-M. Lee, S. Lee and S. Lee, Nonrelativistic Superconformal M2-Brane Theory, http://dx.doi.org/10.1088/1126-6708/2009/09/030

Web End =JHEP 09 (2009) 030 [arXiv:0902.3857] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0902.3857

Web End =INSPIRE ].

[47] O. Aharony, O. Bergman, D.L. Ja eris and J. Maldacena, N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, http://dx.doi.org/10.1088/1126-6708/2008/10/091

Web End =JHEP 10 (2008) 091 [arXiv:0806.1218] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.1218

Web End =INSPIRE ].

[48] E. Colgin and H. Yavartanoo, NR CFT(3) duals in M-theory, http://dx.doi.org/10.1088/1126-6708/2009/09/002

Web End =JHEP 09 (2009) 002 [arXiv:0904.0588] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0904.0588

Web End =INSPIRE ].

JHEP10(2013)094

[49] H. Ooguri and C.-S. Park, Supersymmetric non-relativistic geometries in M-theory, http://dx.doi.org/10.1016/j.nuclphysb.2009.08.021

Web End =Nucl. Phys. B 824 (2010) 136 [arXiv:0905.1954] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0905.1954

Web End =INSPIRE ].

[50] J. Jeong, H.-C. Kim, S. Lee, E. Colgin and H. Yavartanoo, Schrdinger invariant solutions of M-theory with Enhanced Supersymmetry, http://dx.doi.org/10.1007/JHEP03(2010)034

Web End =JHEP 03 (2010) 034 [arXiv:0911.5281] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0911.5281

Web End =INSPIRE ].

[51] M. Guica, K. Skenderis, M. Taylor and B.C. van Rees, Holography for Schrdinger backgrounds, http://dx.doi.org/10.1007/JHEP02(2011)056

Web End =JHEP 02 (2011) 056 [arXiv:1008.1991] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1008.1991

Web End =INSPIRE ].

[52] S. Janiszewski and A. Karch, String Theory Embeddings of Nonrelativistic Field Theories and Their Holographic Hoava Gravity Duals, http://dx.doi.org/10.1103/PhysRevLett.110.081601

Web End =Phys. Rev. Lett. 110 (2013) 081601 [arXiv:1211.0010] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.0010

Web End =INSPIRE ].

[53] S. Janiszewski and A. Karch, Non-relativistic holography from Hoava gravity, http://dx.doi.org/10.1007/JHEP02(2013)123

Web End =JHEP 02 (2013) 123 [arXiv:1211.0005] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.0005

Web End =INSPIRE ].

[54] P. Hoava, Quantum Gravity at a Lifshitz Point, http://dx.doi.org/10.1103/PhysRevD.79.084008

Web End =Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0901.3775

Web End =INSPIRE ].

[55] N. Bobev and B.C. van Rees, Schrdinger Deformations of AdS3 S3, http://dx.doi.org/10.1007/JHEP08(2011)062

Web End =JHEP 08 (2011) 062

[arXiv:1102.2877] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1102.2877

Web End =INSPIRE ].

[56] P. Kraus and E. Perlmutter, Universality and exactness of Schrdinger geometries in string and M-theory, http://dx.doi.org/10.1007/JHEP05(2011)045

Web End =JHEP 05 (2011) 045 [arXiv:1102.1727] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1102.1727

Web End =INSPIRE ].

[57] S. El-Showk and M. Guica, Kerr/CFT, dipole theories and nonrelativistic CFTs, http://dx.doi.org/10.1007/JHEP12(2012)009

Web End =JHEP 12 (2012) 009 [arXiv:1108.6091] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.6091

Web End =INSPIRE ].

[58] S. Detournay, T. Hartman and D.M. Hofman, Warped Conformal Field Theory, http://dx.doi.org/10.1103/PhysRevD.86.124018

Web End =Phys. Rev. D 86 (2012) 124018 [arXiv:1210.0539] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1210.0539

Web End =INSPIRE ].

[59] J.P. Gauntlett, O.A. Mac Conamhna, T. Mateos and D. Waldram, AdS spacetimes from wrapped M5 branes, http://dx.doi.org/10.1088/1126-6708/2006/11/053

Web End =JHEP 11 (2006) 053 [http://arxiv.org/abs/hep-th/0605146

Web End =hep-th/0605146 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0605146

Web End =INSPIRE ].

[60] P. Figueras, O.A. Mac Conamhna and E. Colgin, Global geometry of the supersymmetric AdS3/CF T2 correspondence in M-theory, http://dx.doi.org/10.1103/PhysRevD.76.046007

Web End =Phys. Rev. D 76 (2007) 046007 [http://arxiv.org/abs/hep-th/0703275

Web End =hep-th/0703275 ]

[http://inspirehep.net/search?p=find+EPRINT+hep-th/0703275

Web End =INSPIRE ].

[61] O.A. Mac Conamhna and E. Colgin, Supersymmetric wrapped membranes, AdS2 spaces and bubbling geometries, http://dx.doi.org/10.1088/1126-6708/2007/03/115

Web End =JHEP 03 (2007) 115 [http://arxiv.org/abs/hep-th/0612196

Web End =hep-th/0612196 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0612196

Web End =INSPIRE ].

[62] U. Gran, J. Gutowski and G. Papadopoulos, IIB backgrounds with ve-form ux, http://dx.doi.org/10.1016/j.nuclphysb.2008.01.015

Web End =Nucl. Phys. B 798 (2008) 36 [arXiv:0705.2208] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0705.2208

Web End =INSPIRE ].

[63] H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, http://dx.doi.org/10.1088/1126-6708/2004/10/025

Web End =JHEP 10 (2004) 025 [http://arxiv.org/abs/hep-th/0409174

Web End =hep-th/0409174 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0409174

Web End =INSPIRE ].

[64] N. Berkovits and J. Maldacena, Fermionic T-duality, Dual Superconformal Symmetry and the Amplitude/Wilson Loop Connection, http://dx.doi.org/10.1088/1126-6708/2008/09/062

Web End =JHEP 09 (2008) 062 [arXiv:0807.3196] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.3196

Web End =INSPIRE ].

[65] E. Colgin, Self-duality of the D1 D5 near-horizon, http://dx.doi.org/10.1007/JHEP04(2012)047

Web End =JHEP 04 (2012) 047

[arXiv:1202.3416] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1202.3416

Web End =INSPIRE ].

[66] J.M. Maldacena and C. Nez, Supergravity description of eld theories on curved manifolds and a no go theorem, http://dx.doi.org/10.1142/S0217751X01003937

Web End =Int. J. Mod. Phys. A 16 (2001) 822 [http://arxiv.org/abs/hep-th/0007018

Web End =hep-th/0007018 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0007018

Web End =INSPIRE ].

[67] M. Naka, Various wrapped branes from gauged supergravities, http://arxiv.org/abs/hep-th/0206141

Web End =hep-th/0206141 [http://inspirehep.net/search?p=find+EPRINT+hep-th/0206141

Web End =INSPIRE ].

JHEP10(2013)094

[68] J.P. Gauntlett, O.A. Mac Conamhna, T. Mateos and D. Waldram, Supersymmetric AdS3 solutions of type IIB supergravity, http://dx.doi.org/10.1103/PhysRevLett.97.171601

Web End =Phys. Rev. Lett. 97 (2006) 171601 [http://arxiv.org/abs/hep-th/0606221

Web End =hep-th/0606221 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0606221

Web End =INSPIRE ].

[69] A. Almuhairi and J. Polchinski, Magnetic AdS R2: Supersymmetry and stability,

arXiv:1108.1213 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.1213

Web End =INSPIRE ].

[70] S. Cucu, H. L and J.F. Vazquez-Poritz, A supersymmetric and smooth compactication of M-theory to AdS5, http://dx.doi.org/10.1016/j.physletb.2003.05.002

Web End =Phys. Lett. B 568 (2003) 261 [http://arxiv.org/abs/hep-th/0303211

Web End =hep-th/0303211 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0303211

Web End =INSPIRE ].

[71] J.P. Gauntlett, E. Colgin and O. Varela, Properties of some conformal eld theories with M-theory duals, http://dx.doi.org/10.1088/1126-6708/2007/02/049

Web End =JHEP 02 (2007) 049 [http://arxiv.org/abs/hep-th/0611219

Web End =hep-th/0611219 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0611219

Web End =INSPIRE ].

[72] J.P. Gauntlett and O. Varela, Consistent Kaluza-Klein reductions for general supersymmetric AdS solutions, http://dx.doi.org/10.1103/PhysRevD.76.126007

Web End =Phys. Rev. D 76 (2007) 126007 [arXiv:0707.2315] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0707.2315

Web End =INSPIRE ].

[73] J.P. Gauntlett and O. Varela, D = 5 SU(2) U(1) Gauged Supergravity from D = 11

Supergravity, http://dx.doi.org/10.1088/1126-6708/2008/02/083

Web End =JHEP 02 (2008) 083 [arXiv:0712.3560] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0712.3560

Web End =INSPIRE ].

[74] J. Jeong, O. Kelekci and E. Colgin, An alternative IIB embedding of F(4) gauged supergravity, http://dx.doi.org/10.1007/JHEP05(2013)079

Web End =JHEP 05 (2013) 079 [arXiv:1302.2105] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1302.2105

Web End =INSPIRE ].

[75] M. Cveti et al., Embedding AdS black holes in ten-dimensions and eleven-dimensions, http://dx.doi.org/10.1016/S0550-3213(99)00419-8

Web End =Nucl. Phys. B 558 (1999) 96 [http://arxiv.org/abs/hep-th/9903214

Web End =hep-th/9903214 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9903214

Web End =INSPIRE ].

[76] E. Colgin, M.M. Sheikh-Jabbari and H. Yavartanoo, unpublished work.

[77] R. Fareghbal, C. Gowdigere, A. Mosa a and M. Sheikh-Jabbari, Nearing 11d Extremal Intersecting Giants and New Decoupled Sectors in D = 3, 6 SCFTs,http://dx.doi.org/10.1103/PhysRevD.81.046005

Web End =Phys. Rev. D 81 (2010) 046005 [arXiv:0805.0203] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0805.0203

Web End =INSPIRE ].

[78] S. Cucu, H. L and J.F. Vazquez-Poritz, Interpolating from AdSD2 S2 to AdSD,

http://dx.doi.org/10.1016/j.nuclphysb.2003.10.041

Web End =Nucl. Phys. B 677 (2004) 181 [http://arxiv.org/abs/hep-th/0304022

Web End =hep-th/0304022 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0304022

Web End =INSPIRE ].

[79] D. Cassani and A.F. Faedo, A supersymmetric consistent truncation for conifold solutions, http://dx.doi.org/10.1016/j.nuclphysb.2010.10.010

Web End =Nucl. Phys. B 843 (2011) 455 [arXiv:1008.0883] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1008.0883

Web End =INSPIRE ].

[80] I. Bena, G. Giecold, M. Graa, N. Halmagyi and F. Orsi, Supersymmetric Consistent Truncations of IIB on T 1,1, http://dx.doi.org/10.1007/JHEP04(2011)021

Web End =JHEP 04 (2011) 021 [arXiv:1008.0983] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1008.0983

Web End =INSPIRE ].

[81] D. Cassani, G. DallAgata and A.F. Faedo, Type IIB supergravity on squashed Sasaki-Einstein manifolds, http://dx.doi.org/10.1007/JHEP05(2010)094

Web End =JHEP 05 (2010) 094 [arXiv:1003.4283] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1003.4283

Web End =INSPIRE ].

[82] J.P. Gauntlett and O. Varela, Universal Kaluza-Klein reductions of type IIB to N = 4 supergravity in ve dimensions, http://dx.doi.org/10.1007/JHEP06(2010)081

Web End =JHEP 06 (2010) 081 [arXiv:1003.5642] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1003.5642

Web End =INSPIRE ].

[83] J.T. Liu, P. Szepietowski and Z. Zhao, Consistent massive truncations of IIB supergravity on Sasaki-Einstein manifolds, http://dx.doi.org/10.1103/PhysRevD.81.124028

Web End =Phys. Rev. D 81 (2010) 124028 [arXiv:1003.5374] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1003.5374

Web End =INSPIRE ].

[84] K. Skenderis, M. Taylor and D. Tsimpis, A Consistent truncation of IIB supergravity on manifolds admitting a Sasaki-Einstein structure, http://dx.doi.org/10.1007/JHEP06(2010)025

Web End =JHEP 06 (2010) 025 [arXiv:1003.5657] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1003.5657

Web End =INSPIRE ].

[85] E. Colgin and O. Varela, Consistent reductions from D = 11 beyond Sasaki-Einstein, http://dx.doi.org/10.1016/j.physletb.2011.07.063

Web End =Phys. Lett. B 703 (2011) 180 [arXiv:1106.4781] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1106.4781

Web End =INSPIRE ].

[86] J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS5 solutions of M-theory, http://dx.doi.org/10.1088/0264-9381/21/18/005

Web End =Class. Quant. Grav. 21 (2004) 4335 [http://arxiv.org/abs/hep-th/0402153

Web End =hep-th/0402153 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0402153

Web End =INSPIRE ].

JHEP10(2013)094

[87] J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Sasaki-Einstein metrics on S2 S3,

Adv. Theor. Math. Phys. 8 (2004) 711 [http://arxiv.org/abs/hep-th/0403002

Web End =hep-th/0403002 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0403002

Web End =INSPIRE ].

[88] S.S. Gubser, C.P. Herzog, S.S. Pufu and T. Tesileanu, Superconductors from Superstrings, http://dx.doi.org/10.1103/PhysRevLett.103.141601

Web End =Phys. Rev. Lett. 103 (2009) 141601 [arXiv:0907.3510] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0907.3510

Web End =INSPIRE ].

[89] J.P. Gauntlett, J. Sonner and T. Wiseman, Holographic superconductivity in M-theory, http://dx.doi.org/10.1103/PhysRevLett.103.151601

Web End =Phys. Rev. Lett. 103 (2009) 151601 [arXiv:0907.3796] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0907.3796

Web End =INSPIRE ].

[90] J.P. Gauntlett, J. Sonner and T. Wiseman, Quantum Criticality and Holographic Superconductors in M-theory, http://dx.doi.org/10.1007/JHEP02(2010)060

Web End =JHEP 02 (2010) 060 [arXiv:0912.0512] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0912.0512

Web End =INSPIRE ].

[91] J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,http://dx.doi.org/10.1007/BF01211590

Web End =Commun. Math. Phys. 104 (1986) 207 [http://inspirehep.net/search?p=find+J+Comm.Math.Phys.,104,207

Web End =INSPIRE ].

[92] M. Henningson and K. Skenderis, The Holographic Weyl anomaly, http://dx.doi.org/10.1088/1126-6708/1998/07/023

Web End =JHEP 07 (1998) 023 [http://arxiv.org/abs/hep-th/9806087

Web End =hep-th/9806087 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9806087

Web End =INSPIRE ].

[93] M. Du , H. L and C. Pope, AdS3 S3 (un)twisted and squashed and an O(2,2,Z) multiplet

of dyonic strings, http://dx.doi.org/10.1016/S0550-3213(98)00810-4

Web End =Nucl. Phys. B 544 (1999) 145 [http://arxiv.org/abs/hep-th/9807173

Web End =hep-th/9807173 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9807173

Web End =INSPIRE ].

[94] T. Ortn, SL(2, R) duality covariance of Killing spinors in axion-dilaton black holes, http://dx.doi.org/10.1103/PhysRevD.51.790

Web End =Phys. Rev. D 51 (1995) 790 [http://arxiv.org/abs/hep-th/9404035

Web End =hep-th/9404035 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9404035

Web End =INSPIRE ].

[95] M. Guica, Decrypting the warped black strings, arXiv:1305.7249 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1305.7249

Web End =INSPIRE ].

[96] M. Berg, M. Haack and H. Samtleben, Calabi-Yau fourfolds with ux and supersymmetry breaking, http://dx.doi.org/10.1088/1126-6708/2003/04/046

Web End =JHEP 04 (2003) 046 [http://arxiv.org/abs/hep-th/0212255

Web End =hep-th/0212255 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0212255

Web End =INSPIRE ].

[97] I. Bena, M. Guica and W. Song, Un-twisting the NHEK with spectral ows, http://dx.doi.org/10.1007/JHEP03(2013)028

Web End =JHEP 03 (2013) 028 [arXiv:1203.4227] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1203.4227

Web End =INSPIRE ].

[98] J.P. Gauntlett, N. Kim and D. Waldram, M Five-branes wrapped on supersymmetric cycles, http://dx.doi.org/10.1103/PhysRevD.63.126001

Web End =Phys. Rev. D 63 (2001) 126001 [http://arxiv.org/abs/hep-th/0012195

Web End =hep-th/0012195 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0012195

Web End =INSPIRE ].

[99] J.P. Gauntlett and N. Kim, M ve-branes wrapped on supersymmetric cycles. 2., http://dx.doi.org/10.1103/PhysRevD.65.086003

Web End =Phys. Rev. D 65 (2002) 086003 [http://arxiv.org/abs/hep-th/0109039

Web End =hep-th/0109039 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0109039

Web End =INSPIRE ].

[100] P. Karndumri, E. Colgin, 3D supergravity from wrapped M5-branes, to appear.

[101] T. Ortiz and H. Samtleben, SO(9) supergravity in two dimensions, http://dx.doi.org/10.1007/JHEP01(2013)183

Web End =JHEP 01 (2013) 183 [arXiv:1210.4266] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1210.4266

Web End =INSPIRE ].

[102] T. Banks, W. Fischler, S. Shenker and L. Susskind, M theory as a matrix model: AConjecture, http://dx.doi.org/10.1103/PhysRevD.55.5112

Web End =Phys. Rev. D 55 (1997) 5112 [http://arxiv.org/abs/hep-th/9610043

Web End =hep-th/9610043 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9610043

Web End =INSPIRE ].

[103] T. Kameyama and K. Yoshida, String theories on warped AdS backgrounds and integrable deformations of spin chains, http://dx.doi.org/10.1007/JHEP05(2013)146

Web End =JHEP 05 (2013) 146 [arXiv:1304.1286] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1304.1286

Web End =INSPIRE ].

[104] J. Ren, One-dimensional holographic superconductor from AdS3/CF T2 correspondence,http://dx.doi.org/10.1007/JHEP11(2010)055

Web End =JHEP 11 (2010) 055 [arXiv:1008.3904] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1008.3904

Web End =INSPIRE ].

[105] F. Denef and S.A. Hartnoll, Landscape of superconducting membranes,http://dx.doi.org/10.1103/PhysRevD.79.126008

Web End =Phys. Rev. D 79 (2009) 126008 [arXiv:0901.1160] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0901.1160

Web End =INSPIRE ].

JHEP10(2013)094

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SISSA, Trieste, Italy 2013

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(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

AdS ^sub 3^ solutions dual to ... = (0, 2) SCFTs arise when D3-branes wrap Kähler two-cycles in manifolds with SU(4) holonomy. Here we review known AdS ^sub 3^ solutions and identify the corresponding three-dimensional gauged supergravities, solutions of which uplift to type IIB supergravity. In particular, we discuss gauged supergravities dual to twisted compactifications on Riemann surfaces of both ... = 4 SYM and ... = 1 SCFTs with Sasaki-Einstein duals. We check in each case that c-extremization gives the exact central charge and R symmetry. For completeness, we also study AdS ^sub 3^ solutions from intersecting D3-branes, generalise recent KK reductions of Detournay & Guica and identify the underlying gauged supergravities. Finally, we discuss examples of null-warped AdS ^sub 3^ solutions to three-dimensional gauged supergravity, all of which embed in string theory.

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Title

3D supergravity from wrapped D3-branes

Author

Karndumri, Parinya; Colgáin, Eoin Ó

Pages

1-48

Publication year

2013

Publication date

Oct 2013

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP10(2013)094

ProQuest document ID

1652924857

SISSA, Trieste, Italy 2013

3D supergravity from wrapped D3-branes

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