Supersymmetric AdS_6 Solutions of Type IIB Supergravity
SSupersymmetric AdS solutionsof type IIB supergravity Hyojoong Kim a , Nakwoo Kim a and Minwoo Suh ba Department of Physics and Research Institute of Basic ScienceKyung Hee University, Seoul 130-701, Korea b Department of PhysicsSogang University, Seoul 121-742, Korea [email protected], [email protected], [email protected]
Abstract
We study the general requirement for supersymmetric AdS solutions in type IIB su-pergravity. We employ the Killing spinor technique and study the differential and algebraicrelations among various Killing spinor bilinears to find the canonical form of the solutions.Our result agrees precisely with the work of Apruzzi et. al. [1] which used the pure spinortechnique. We also obtained the four-dimensional theory through the dimensional reduc-tion of type IIB supergravity on AdS . This effective action is essentially a nonlinear sigmamodel with five scalar fields parametrizing SL(3 , R ) / SO(2 , ⊂ SL(3 , R ) in a way analogous to gaugedsupergravity. June, 2015 a r X i v : . [ h e p - t h ] O c t ontents solutions 3 B.1 Gamma matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14B.2 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
C Spinor bilinears 16
C.1 Algebraic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16C.2 Differential relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
D Fierz identities 19
D.1 Relations of scalar bilinears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19D.2 Inner products of vector bilinears . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
E Killing vectors of five-dimensional target space 21 Introduction
In recent years, there have been renewed interests in supersymmetric AdS solutions in D = 10supergravity. Via the gauge/gravity correspondence [2], such solutions should be dual to certain D = 5 superconformal field theories. Five-dimensional gauge theories are perturbatively non-renormalizable. Seiberg nonetheless argued that N = 1 supersymmetric Sp ( N ) gauge theorieswith hypermultiplets of N f < SO ( N f ) × U (1) globalsymmetry is enhanced to E N f +1 [3, 4, 5]. Such fixed point theories have string theory construc-tion: in terms of the near-horizon limit of D4-D8 brane configurations. Based on the AdS /CFT correspondence [6], Brandhuber and Oz identified the gravity dual as supersymmetric AdS × w S solution of massive type IIA supergravity [7]. More recently this correspondence was generalizedto quiver gauge theories and AdS × w S / Z n orbifolds in [8].Thanks to the development of the localization technique [9] and its generalization to five-dimensional gauge theories [10, 11], some BPS quantities can be calculated exactly. The conjec-tured enhancement of global symmetry to E N f +1 was verified from the analysis of superconformalindex in [12]. Furthermore, the S free energy and also the -BPS circular Wilson loop operatorsare calculated and shown to agree with the gravity side computations [13, 14, 15, 16].Encouraged by the successful application of localization technique on the field theory side, it isnatural for us to look for new supersymmetric AdS solutions. In massive type IIA supergravity, itwas proved that the Brandhuber-Oz solution is the unique one [17]. In type IIB supergravity, theT-dual version of the Brandhuber-Oz solution has been known for a long time [18]. A new solutionwas obtained more recently employing the technique of non-Abelian T-dual transformation in[19]. The dual gauge theory was investigated in [20], but it is not completely understood yet.For a thorough study, the authors of [1] investigated the general form of supersymmetricAdS solutions of type IIB supergravity, using the pure spinor approach. They found that thefour-dimensional internal space is a fibration of S over a two-dimensional space, and also showedthat the supersymmetry conditions boil down to two coupled partial differential equations. Ofcourse any solution of the PDEs provides a supersymmetric AdS solution at least locally. Inparticular, the two explicit solutions mentioned above can be reproduced as specific solutions tothe PDEs. But otherwise these non-linear coupled PDEs are so complicated that currently itlooks very hard, if not impossible, to obtain more AdS solutions by directly solving the PDEs.The objective of this article is to procure additional insight into this problem, using alternativemethods. In the first part we use the Killing spinor approach which is probably more well-knownand has been successfully applied to many similar problems, see e.g. [21, 22, 23]. Followingthe standard procedure we work out the algebraic and differential constraints which should besatisfied by various spinor bilinears and derive the supersymmetric conditions. In the end, we2onfirm that our results are in precise agreement with that of [1]. Secondly, via dimensionalreduction of the bosonic sector of the D = 10 action on AdS , we present a four-dimensionaleffective theory action, which turns out to be a non-linear sigma model of five scalar fieldscoupled to gravity. The scalar fields parametrize the coset space SL(3 , R ) / SO(2 , sl (3 , R ) symmetry to a certain subalgebra.Although in this paper we do not present new solutions, we believe the identification of the D = 4effective action will prove useful in the construction of explicit solutions and their classifications.This paper is organized as follows. Section 2 contains an analysis on the supersymmetryconditions for AdS solutions. In section 3, we study the four-dimensional effective theory fromdimensional reduction on AdS . In section 4 we conclude. Technical details are relegated toappendices. solutions We consider the most general supersymmetric AdS solutions of type IIB supergravity. We takethe D=10 metric as a warped product of AdS with a four-dimensional Riemannian space M ds = e U ds AdS + ds M , (2.1)where U is a warp factor. To respect the symmetry of AdS , we should set the five-form fluxto zero. The complex three-form flux G is non-vanishing only on M . The warp factor U , thedilation φ and the axion C , are functions on M and independent of coordinates in AdS .To preserve some supersymmetry, we require the vanishing of supersymmetry transformationsof the gravitino and the dilatino i.e. δψ M = 0 , δλ = 0. With the gamma matrix decomposition(B.1) and the spinor ansatz (B.8), we reduce the ten-dimensional Killing spinor equations tofour-dimensional ones. There are two differential and four algebraic-type equations: D m ξ ± + 196 G npq ( γ m γ npq + 2 γ npq γ m ) ξ ± = 0 , (2.2)¯ D m ξ ± + 196 G ∗ npq ( γ m γ npq + 2 γ npq γ m ) ξ ± = 0 , (2.3) ime − U ξ ∓ + ∂ n U γ n ξ ± − G npq γ npq ξ ± = 0 , (2.4) ime − U ξ ∓ + ∂ n U γ n ξ ± − G ∗ npq γ npq ξ ± = 0 , (2.5) P n γ n ξ ± + 124 G npq γ npq ξ ± = 0 , (2.6) P ∗ n γ n ξ ± + 124 G ∗ npq γ npq ξ ± = 0 , (2.7)3here D m ξ ± = ( ∇ m − i Q m ) ξ ± , ¯ D m ξ ± = ( ∇ m + i Q m ) ξ ± . (2.8)With the assumption that there exists at least one nowhere-vanishing solution to the equationsin the above, we can construct various spinor bilinears. Then the supersymmetric condition istranslated into various algebraic and differential relations between the spinor bilinears. We haverecorded them in appendix C.1 and C.2. We first need to study the isometry of the four-dimensional Riemannian space M . We note thatthe following two complex vectors satisfy the Killing equation ∇ ( m K n ) = 0. ξ γ n ξ − + ξ γ n ξ − , ξ c γ n ξ − + ξ c γ n ξ − . (2.9)If these vectors are to provide a true symmetry of the full ten-dimensional solution as well, weneed to check if L K U = ( d i K + i K d ) U = K m ∂ m U = 0 , (2.10)where L K is a Lie derivative along the Killing vector K . From (C.4) and (C.6), we find that infact only three of them satisfy the above condition. Hence, the true Killing vectors are K n ≡ Re ( ξ c γ n ξ − + ξ c γ n ξ − ) , (2.11) K n ≡ Im ( ξ c γ n ξ − + ξ c γ n ξ − ) , (2.12) K n ≡ Re ( ξ γ n ξ − + ξ γ n ξ − ) . (2.13)Using (2.6) and (2.7), we have P m K mi = 0, which implies that L K i φ = L K i C = 0 , (2.14)where i = 1 , ,
3. Also we obtain i K ∗ G = 0 from (2.40) and (2.41), and i k d ∗ G = 0 using theequation of the motion for G, thus L K i ∗ G = 0 . (2.15)Hence, we conclude that K i describe symmetries of the full ten-dimensional solutions.Now let us study the Lie bracket of the Killing vectors. Using (C.13) and (C.19), the Fierzidentities (D.2) and the normalization (C.28), we show that the three Killing vectors satisfy an SU (2) algebra, [ K i , K j ] = (cid:15) ijk K k . (2.16) The equation of the motion for G is d ∗ G = ( − dU + iQ ) ∧ ∗ G + P ∧ ∗ G ∗ . SU (2) isometry of the four-dimensional Riemannian space corresponds to the SU (2) R R-symmetry of dual five-dimensional field theory. Then we construct a 3 × K i · K j ) = 0 . (2.17)This guarantees that K i are the Killing vectors of S . The radius l of the two-sphere is given by2 l = ( K ) + ( K ) + ( K ) = 2 (cid:20) m e U − ξ ξ )( ξ ξ ) (cid:21) . (2.18) We have showed that once we require the supersymmetry conditions, then the four-dimensionalRiemannian space should contain S . Now we focus on the remaining two-dimensional space.We start with two one-forms L n and L n from (C.11) L n ≡ e U + φ ( ξ ξ + ξ ξ ) ∂ n C − me − φ L n , = − i∂ n (cid:16) e U − φ ( ξ ξ − ξ ξ ) (cid:17) , (2.19) L n ≡ Im (cid:0) ξ γ n ξ − + ξ γ n ξ − (cid:1) = 1 m e − φ ∂ n (cid:16) e U + φ ( ξ ξ + ξ ξ ) (cid:17) , (2.20)where L n = Re (cid:0) ξ γ n ξ − − ξ γ n ξ − (cid:1) . (2.21)Using the Fierz identities, one can show that the one-forms L and L are orthogonal to theKilling vectors K i · L = K i · L = 0 . (2.22)Together with L K i C = 0, the one-form L is also orthogonal to the Killing vectors. Then, weintroduce coordinates z and y , z = − mi e U − φ ( ξ ξ − ξ ξ ) , (2.23) y = 3 m e U + φ ( ξ ξ + ξ ξ ) . Since L K i z = i K i dz ∼ K i · L = 0 and similarly L K i y = 0, the coordinates z and y are independentof the sphere coordinates. In terms of the coordinates z and y , the one-forms are L = 13 m ydC − me − φ L = 13 m dz, (2.24) L = 13 m e − φ dy. (2.25)5hen we calculate inner products of the one-forms L and L , hoping to be able to fix theremaining two-dimensional metric. However, we cannot immediately calculate the inner productsinvolving L , because it includes dC . The resolution is that we consider the one-form L definedin (2.21) instead. From (C.15) and (C.16), we have d ( e U − φ L ) = e U + φ dC ∧ L , (2.26) d ( e U + φ L ) = 0 . (2.27)We introduce another coordinate w and write L as L = 13 m e − U − φ dw. (2.28)Then we can calculate inner products of L and L using the Fierz identities and read off thetwo-dimensional metric components in w and y coordinates, ds = 1 m ( e U + φ − y − e φ z ) (cid:104) e − U + φ ( e U − φ − z ) dy (2.29)+ e − U − φ ( e U + φ − y ) dw − e − U y z dy dw (cid:105) . At this stage, z is an unknown function of y and w . The details are in appendix D.2.We would like to express dC in terms of the coordinate z instead of w . From the Killingspinor equations (2.4)–(2.7), we have L · dC = e − φ d (4 U + φ ) · L − e − U − φ z, (2.30) L · dC = e − φ d (4 U − φ ) · L + 43 e − U − φ y. (2.31)The integrability conditions d ( dz ) = d ( dy ) = 0 from (2.24), (2.25), when combined with (2.26),(2.27) give L ∧ dC + e − φ d (4 U + φ ) ∧ L = 0 , (2.32) L ∧ dC + e − φ d (4 U − φ ) ∧ L = 0 . (2.33)Summarising, from (2.30)–(2.33), we find that dC = 12 yz (cid:104) ( e U − φ − e − φ y ) d (4 U − φ )+( e U − φ − z ) d (4 U + φ ) − e − U − φ z dw +4 e − φ y dy (cid:105) . (2.34)If we plug this into (2.24), we can express dw in terms of dy and dz . Then, we can write themetric and dC in the y and z coordinates.Now we are ready to present our main result. We introduce a new coordinate x defined by x = e U − e U − φ y − e U + φ z . (2.35)6hen, we can have all fields and functions in terms of coordinates x and y only. We have themetric of the four-dimensional Riemannian space, ds = 19 m (cid:104) e − U x ds S (2.36)+ e − U e U + φ − e φ x − e U y (cid:104) ( e U + φ − y ) dx + 9 ( e U − x ) dy + 6 x y dx dy (cid:105)(cid:105) . Similarly dC is written as dC = e − U − φ y (cid:112) e U + φ − e φ x − e U y (cid:104) e U + φ + e φ x ) dU (2.37) −
12 ( e U + φ − e φ x − e U y ) dφ − e φ xdx (cid:105) . The consistency conditions (2.32) and (2.33) give two partial differential equations,4 e φ x = 12 (cid:16) e U + φ + e φ x − e U y (cid:17) ∂ x U + 8 e φ xy ∂ y U − e φ (cid:16) e U − x (cid:17) ∂ x φ + 2 e φ xy ∂ y φ, (2.38) − e U + φ xy = 12 e U y (cid:16) e U + φ − e φ x − e U y (cid:17) ∂ x U + 4 e φ x (cid:16) e U + x (cid:17) ∂ y U + e φ x (cid:16) − e U + φ + e φ x + 2 e U y (cid:17) ∂ y φ − ye U + φ (cid:16) e U − x (cid:17) ∂ x φ. (2.39)The complex three-form flux is obtained by using (2.4)–(2.7) rather straightforwardly, ∗ Re G = − y e − U − φ/ (2.40) × (cid:20) ( e U + φ + e φ x + 2 e U y ) dU −
14 ( e U + φ − e φ x ) dφ − e φ xdx − e U ydy (cid:21) , ∗ Im G = 2 e − U − φ/ (cid:112) e U + φ − e φ x − e U y (2.41) × (cid:20) (3 e U + φ − e φ x − e U y ) dU + 14 ( e U + φ − e φ x ) dφ + 13 e φ xdx + 2 e U ydy (cid:21) . Here we used γ mnpq = √ g (cid:15) mnpq γ .To summarize, we have employed the Killing spinor analysis in Einstein frame and obtainedthe most general supersymmetric AdS solutions for the metric and the fluxes in terms of thewarping factor U and the dilation φ . This implies that, when we have solutions U and φ tothe two PDEs (2.38) and (2.39), then we can completely determine the metric (2.36), the one-form flux (2.37) and the three-form flux (2.40), (2.41). Our analysis shows a perfect agreementwith the work of [1], where the authors used the pure spinor approach in string frame. We canreproduce their results with the following identification of our fields to theirs. g mn → e − φ g mn , U → A − φ , dC → F , Re G → e − φ H , Im G → − e φ F . (2.42)Also our coordinates ( x, y ) correspond to ( p, q ) defined in (4.17) of [1].7 .4 Equations of motion From the equations of motion and the Bianchi identities of D=10 type IIB supergravity, we obtainthe four-dimensional ones via dimensional reduction. Let us start with dualizing the complexthree-form flux G into real scalars f and g ∗ Re G = 12 e − U + φ ( Cdf − f dC + d ˜ g ) , = 12 e − U + φ ( dg + 2 Cdf ) , (2.43) ∗ Im G = e − U − φ df. (2.44)where g = ˜ g − f C . They satisfy the equation of motion for G automatically. Also the Bianchiidentity for P is satisfied by (A.4). Then the Einstein equation, the equation for P and theBianchi identity for G give the following six equations. R mn = 6 ∇ m ∇ n U + 6 ∂ m U ∂ n U + 12 e φ ∂ m C∂ n C + 12 ∂ m φ∂ n φ − (cid:104) e − U + φ (cid:16) ( ∂ m g + 2 C∂ m f )( ∂ n g + 2 C∂ n f ) −
34 ( ∂g + 2 C∂f ) g mn (cid:17) +4 e − U − φ (cid:16) ∂ m f ∂ n f −
34 ( ∂f ) g mn (cid:17)(cid:105) , (2.45) (cid:3) U + 6( ∂U ) + 5 e − U − e − U − φ ( ∂f ) − e − U + φ ( ∂g + 2 C∂f ) = 0 , (2.46) (cid:3) φ + 6 ∂U · ∂φ − e φ ( ∂C ) − e − U − φ ( ∂f ) + 18 e − U + φ ( ∂g + 2 C∂f ) = 0 , (2.47) (cid:3) C + 6 ∂U · ∂C + 2 ∂φ · ∂C + 12 e − U − φ ( ∂f ) · ( ∂g + 2 C∂f ) = 0 , (2.48) ∂ (cid:16) √ g e − U − φ (cid:16) ∂f + 12 e φ C ( ∂g + 2 C∂f ) (cid:17)(cid:17) = 0 , (2.49) ∂ (cid:16) √ g e − U + φ ( ∂g + 2 C∂f ) (cid:17) = 0 . (2.50)One can study the integrability conditions of the Killing spinor equations and check whether thesupersymmetry conditions satisfy the equations of motion and the Bianchi identities automati-cally. Instead, here we checked that the metric (2.36) and the solutions to the BPS equations(2.37)–(2.41), do satisfy the above equations of motion.8 Four-dimensional effective action
In this section we study AdS solutions of type IIB supergravity from a different perspective i.e. by performing a dimensional reduction of type IIB supergravity on AdS space to a four-dimensional theory. From the equations of motion obtained in the previous section, we constructa four-dimensional effective Lagrangian as L = √ g e U (cid:104) R + 30( ∂U ) −
12 ( ∂φ ) − e φ ( ∂C ) (3.1)+ 12 e − U − φ ( ∂f ) + 18 e − U + φ ( ∂g + 2 C∂f ) − e − U (cid:105) . By rescaling the metric g mn = e − U ˜ g mn , we have the Einstein frame Lagrangian L = (cid:112) ˜ g (cid:104) ˜ R − ∂U ) −
12 ( ∂φ ) − e φ ( ∂C ) + 12 e − U − φ ( ∂f ) + 18 e − U + φ ( ∂g + 2 C∂f ) − e − U (cid:105) , (3.2)= (cid:112) ˜ g (cid:104) ˜ R − G IJ ∂ Φ I ∂ Φ J − V (Φ) (cid:105) , (3.3)where Φ I , I = 1 , · · · , , are the five scalar fields U, φ, C, f and g . This is a non-linear sigmamodel of five scalar fields coupled to gravity with a non-trivial scalar potential. Note that thesign of the kinetic terms of the dualized scalars f and g is reversed. However it is well knownthat when we perform dimensional reduction on an internal space including time, the sign ofcertain kinetic terms come out reversed, e.g. [24]. We study properties of the five-dimensional target space. The metric is given by ds = 48 dU + dφ + e φ dC − e − U + φ ( dg + 2 Cdf ) − e − U − φ df . (3.4)This space is Einstein, which satisfies R IJ = − G IJ .The dilaton φ and the axion C form a complex one-form P . Also g and f originate from thecomplex three-form flux G . Hence, we turn to the four-dimensional sub-manifold spanned by φ, C, g, f . We choose the orthonormal frame as e = dφ, e = e φ dC,e = 12 e − U + φ/ ( dg + 2 Cdf ) , e = e − U − φ/ df, (3.5)9nd construct a (1 , J and a (2 , J = e ∧ e + e ∧ e , Ω = ( e + ie ) ∧ ( e + ie ) , (3.6)which satisfy J ∧ J = 12 Ω ∧ ¯Ω , J ∧ Ω = 0 . (3.7)By taking an exterior derivative to these two-forms, we have dJ = 0 , (3.8) d Ω = iP ∧ Ω , (3.9)where P = − e . Hence, we find that the four-dimensional submanifold is K¨ahler. Its Ricciform is obtained by R = dP = − e ∧ e .To investigate the isometry of the target space, we solved the Killing equation ∇ ( I K J ) = 0,and found eight Killing vectors in (E.1). These Killing vectors generate an sl (3 , R ) algebra. Thedetails can be found in appendix E.One can explicitly check that the five-dimensional target space is in fact the cosetSL(3 , R ) / SO(2 , We construct the coset representative V in Borel gauge by exponentiatingCartan generators H , H and positive root generators E α , E α , E α , V = e √ φH e − √ UH e CE α e fE α e gE α . (3.10)With the basis of SL(3 , R ) introduced in (E.2), one can obtain the coset representative V in a3 × so (2 ,
1) in sl (3 , R ), P µ ( ij ) = V a ( i | ∂ µ ( V − ) ka η k | j ) . (3.11)Here i, j, k = 1 , , ,
1) and a = 1 , , , R ) index. An invariantmetric of SO(2 ,
1) is η ij = diag(1 , , − . (3.12)Finally, the kinetic terms of the scalar fields of the Lagrangian (3.2) is L kinetic = − Tr( P µ P µ ) . (3.13) Having a coset after dimensional reduction is of course a very familiar story in supergravity. As it is verywell known, Kaluza-Klein reduction of D = 4 Einstein gravity on a circle leads to SL(2 , R ) / SO(2), and its biggerversions appear in various supergravity theories [24, 25, 26, 27]. .3 Scalar potential Now let us consider the scalar potential V = 30 e − U in the Lagragian. Its existence mustobviously break the SL(3 , R ) global symmetry into a nontrivial subalgebra. Among the eightgenerators in appendix E, this scalar potential is invariant under the action of five Killing vectors K , K , K , K and K . With the following identification e = K , e = √ K , e = K , e = K , e = − K (3.14)one can see that they form a certain five-dimensional Lie algebra so-called A , in table II of[28]. This algebra is isomorphic to the semi-direct sum sl (2 , R ) (cid:110) R [29].The scalar potential here comes from the curvature of internal space AdS . Certainly the sit-uation is very similar to gauged supergravities where the higher-dimensional origin of the gaugingprocess is related to the curvature of the internal space. Within the context of lower-dimensionalsupergravity itself, compared to the un-gauged action, a subgroup of the global symmetry is madelocal and the associated vector fields acquire non-abelian gauge interactions. A new parameter,say g , should be introduced as gauge coupling. To preserve supersymmetry, the action and thesupersymmetry transformations are modified and importantly for us in general a scalar potentialshould be added at order g . Although our theory is not a supergravity theory per se , and thereare no vector fields, we borrow the idea of gauged supergravity and write the scalar potential interms of the coset representative of non-linear sigma model, through the so-called T -tensor. Thismay be justified because our four-dimensional action also has Killing spinor equations which arecompatible with the field equations. In other words the integrability condition of Killing spinorequations should imply the fields satisfy the Euler-Lagrange equations. It is the T -tensor whichencodes the gauging process and determines the modification of supersymmetry transformationrules and the action in gauged supergravity.For a class of the maximal supergravity theories with a global symmetry group SL( n, R ), itis well known that the gauged supergravity can be obtained by gauging the SO( n ) subgroup.This gauging can be generalized to the non-compact subgroup SO( p, q ) with p + q = n andthe non-semi-simple group CSO( p, q, r ) with p + q + r = n , which was introduced in [30, 31].CSO( p, q, r ) = SO( p, q ) (cid:110) R ( p + q ) · r is a subgroup of SL( n, R ), e.g. (6.8) of [32], and preserve themetric q ab = diag(1 , · · · (cid:124) (cid:123)(cid:122) (cid:125) p , − , · · · (cid:124) (cid:123)(cid:122) (cid:125) q , , · · · (cid:124) (cid:123)(cid:122) (cid:125) r ) . (3.15)Let us focus on the non-semi-simple group CSO(1 , , T -tensor as (ap-parently in the same way as in the gauged supergravity) T ij = V ai V bj q ab , (3.16)11here q ab = diag(1 , − , . (3.17)Then one can easily check that the scalar potential is V = − (cid:16) (Tr T ) − Tr( T ) (cid:17) . (3.18)It should be possible to re-write the Killing spinor equations (2.2)–(2.7) as well as the action tomake the symmetry SL(3 , R ) and the choice of CSO(1 , ,
1) more manifest. We plan to do thisconstruction, based on Killing spinor equations and their compatibility with the field equations,for all possible choices of compact and non-compact maximal subgroups of SL( n, R ) in a separatepublication. We have studied AdS solutions of type IIB supergravity theory in this paper. In the first part,we have employed the Killing spinor analysis and revisited supersymmetric AdS solutions, whichwas studied in [1] using the pure spinor approach. We have constructed three Killing vectors,which satisfy SU (2) algebra and give S factor in the four-dimensional internal space M . Inother words, the SU (2) symmetry, which corresponds to SU (2) R R-symmetry in the dual fieldtheory, appears as isometries of the background if we impose the supersymmetric conditions.Also we have found two one-forms which are orthogonal to the Killing vectors. Using theseone-forms, we have introduced the coordinates and determined the metric of the remaining two-dimensional space, and two coupled PDEs defined on it. Also the scalar fields and three-formfluxes have been found. Once we are given the solution to the PDEs, then the metric and thefluxes can be determined. Our results completely agree with the work of [1].Although the result of [1] makes a significant progress in the classification of the supersym-metric AdS solutions in type IIB supergravity theory, there still remain a couple of importantproblems to be studied further. To be sure, the most important but difficult task is to solvethe PDEs (2.38), (2.39) and find a new AdS solution. Also it is very important to constructthe field theories dual to AdS solutions of IIB supergravity, which is still unknown. In [20], theproperties of the dual field theory were studied through their AdS solution. For the general classof solutions studied in [1], the authors suggested that ( p, q ) five-brane webs [33] play a crucialrole. They conjectured that ( p, q ) five-brane webs might be somehow related to the PDEs andthe supergravity solutions could be obtained in the near-horizon limit.Our independent analysis adds credence to the fact that the nonlinear PDEs found in [1]provide necessary and sufficient conditions for supersymmetric AdS in IIB supergravity. Oneshould however admit that the PDEs in the present form are far from illuminating. As it is12ometimes the case, the study of the general form of supersymmetric solutions in supergravity isnot always very efficient in constructing new solutions. However, identifying the canonical form ofthe metric and form-fields as done in [1] and in this paper are equivalent to having the completeinformation on Killing spinors. So they become very useful for the study of supersymmetricprobe consideration, for instance in the study of supersymmetric Wilson loops from D-branes.We thus think that a less technical, and more intuitive way of understanding the supersym-metric AdS solutions would be very desirable. We hope our analysis in the second half of thispaper is a modest first step towards such framework. There we have presented a four-dimensionaltheory via a dimensional reduction on AdS space. The problem of finding AdS solutions oftype IIB supergravity is reduced to a four-dimensional non-linear sigma model, i.e. a gravitytheory coupled to five scalars with a non-trivial scalar potential. The scalar kinetic terms pa-rameterize SL(3 , R ) / SO(2 , , ,
1) which is a subgroup of SL(3 , R ) is relevant to thescalar potential at hand, and presented the analogue of T -tensor. We hope the knowledge of thesymmetry structure in the effective four-dimensional action will become useful to get a deeperinsight into the existing solutions [18, 19], for the identification of their gauge theory duals, andeventually also for constructing more explicit solutions.The D = 4 effective action at hand is purely bosonic and it is not expected to be part of asupergravity action. But it enjoys a nice property that it is equipped with an associated set ofKilling spinor equations which allows BPS solutions. When the Killing spinor equations (2.2)–(2.7) are re-written in a covariant way where the coset symmetry and the choice of gauging groupCSO(1 , ,
1) is more manifest, we expect we can generalize the construction to a bigger symmetrySL( n, R ) with n > e.g. for bottom-up model building in the AdS/CFT inspired study of condensedmatter physics. A similar generalization of BPS systems was successfully performed startingwith AdS solutions in IIB supergravity and AdS solutions in eleven dimensional supergravityin the line of works reported in [34, 35, 36]. We plan to report on such generic analysis in aseparate publication. Acknowledgements
We are grateful to Dario Rosa for explaining the work [1] to us. We thank Hiroaki Nakajimaand Hoil Kim for comments and discussions. MS thanks Changhyun Ahn, Kimyeong Lee, andJeong-Hyuck Park for encouragement and support. The work of MS was mostly done when hewas a post-doctoral fellow at Korea Institute for Advanced Study. This research was supportedby a post-doctoral fellowship grant from Kyung Hee University (KHU-20131358, HK and NK),13ational Research Foundation of Korea (NRF) grants funded by the Korea government (MEST)with grant No. 2010-0023121 (HK, NK, MS), No. 2012046278 (NK), No. 2013064824 (HK),2013R1A1A1A05005747 (MS), and No. 2012-045385/2013- 056327/2014-051185 (MS).
A Type IIB supergravity
We follow the conventions of [23]. In type IIB supergravity, the bosonic fields are the graviton g MN , five-form flux F (5) , complex three-form flux G (3) , dilaton φ and axion C . For the fermionicfields, there are gravitino ψ M and dilatino λ . The supersymmetry variation of the fermionic fieldsare given by δ ψ M = D M (cid:15) + 196 (Γ M Γ NP Q G NP Q + 2 Γ
NP Q G NP Q Γ M ) (cid:15) c (A.1)+ i NP QRS F NP QRS Γ M (cid:15) ,δ λ = i Γ M P M (cid:15) c + i
24 Γ
MNP G MNP (cid:15) . (A.2)where the covariant derivative is D M (cid:15) = ( ∇ M − i Q M ) (cid:15). (A.3)The fields P M and Q M are written in terms of the dilaton and axion as P = i e φ dC + 12 dφ,Q = − e φ dC. (A.4)The chirality conditions areΓ ψ = − ψ , Γ λ = λ , Γ (cid:15) = − (cid:15) . (A.5) B Gamma matrices and spinors
B.1 Gamma matrices
We follow the conventions of [37]. We decompose the ten-dimensional gamma matrices by writingΓ µ = ρ µ ⊗ γ , Γ m = 1 ⊗ γ m , (B.1)where µ = 0 , , , , , m = 1 , , ,
4. Then the chirality matrix is given by Γ = ρ ⊗ γ .14 η − + − δ − − +Table 1: The values of η and δ in various dimensions.In even dimensions, we introduce the intertwiners, which act on the gamma matrices as A Γ M A − = Γ † M ,C − Γ M C = − Γ TM ,D − Γ M D = − Γ ∗ M , (B.2)with D = C A T . These intertwiners can be chosen to satisfy the following relations at given d dimensions, A d = A † d , C d = η C Td , D d = δ ( D ∗ d ) − , (B.3)where the values of η and δ are given in table 1. We decompose the ten-dimensional intertwinersas A = A ⊗ A , C = C ⊗ C , D = D ⊗ D . (B.4) B.2 Spinors
There are two ten-dimensional Majorana-Weyl spinor (cid:15) i , which satisfyΓ (cid:15) i = − (cid:15) i , (cid:15) ci = (cid:15) i , (B.5)where i = 1 ,
2. We decompose (cid:15) i into six- and four-dimensional spinors, ψ and χ , respectively,as (cid:15) i = ψ + ⊗ χ i − + ψ − ⊗ χ i + + c.c., (B.6)where ± represent the chirality. In our case, the six-dimensional spinors ψ ± satisfy the Killingspinor equation on AdS ∇ µ ψ ± = i m ρ µ ρ ψ ∓ , (B.7)where m is the inverse radius of AdS . Then we have the complexified spinor, (cid:15) ≡ (cid:15) + i(cid:15) , = ψ + ⊗ ξ − + ψ − ⊗ ξ + ψ c + ⊗ ξ c − + ψ c − ⊗ ξ c , (B.8)15here ξ ± = χ ± + iχ ± , ξ c ± = χ c ± + iχ c ± . (B.9)The Dirac adjoint and the charge conjugation are, respectively¯ η = η † A, η c = Dη ∗ . (B.10) C Spinor bilinears
One can construct all the spinor bilinears such as ξ A,i γ ( a ) ξ B,j and ξ cA,i γ ( a ) ξ B,j . Here
A, B = 1 , i, j represent the chirality + , − and γ ( a ) ≡ γ m ··· m a . Some of the spinor bilinears identicallyvanish by the chirality, ξ + ξ − = 0 , ξ + γ m ξ + = 0 , ξ + γ mn ξ − = 0 . (C.1)Also due to the antisymmetry of the charge conjugation matrix C , we have ξ c ξ = ξ c ξ = 0 , ξ c ξ = − ξ c ξ , ξ c γ m ξ − = ξ c − γ m ξ . (C.2) C.1 Algebraic relations
In this section, we study the algebraic relations between the spinor bilinears, which can be derivedfrom the algebraic Killing equations (2.4)–(2.7).If we multiply ξ ∓ to (2.4) and ξ ∓ to a hermitian conjugate of (2.5) , then eliminate thethree-form flux terms, we have ξ ξ − ξ − ξ − = − ξ − ξ − + ξ ξ , (C.3) ∂ m U (cid:16) ξ γ n ξ − + ξ γ n ξ − + ξ − γ m ξ + ξ − γ m ξ (cid:17) = 0 . (C.4)If we multiply the charge conjugate spinor instead and follow the same procedure, we obtain ξ c ξ = ξ c − ξ − , (C.5) ∂ m U (cid:16) ξ c γ m ξ − + ξ c γ m ξ − (cid:17) = 0 . (C.6)Similarly eliminating the terms which have only one gamma matrix, we obtain ξ ξ + ξ − ξ − = 0 , ξ ξ + ξ − ξ − = 0 , (C.7) G mnp (cid:16) ξ γ mnp ξ − + ξ γ mnp ξ − (cid:17) = 0 . (C.8)16 .2 Differential relations Scalar bilinears ∇ m ( ξ ± ξ ± + ξ ± ξ ± ) = ∂ m U ( ξ ± ξ ± + ξ ± ξ ± ) (C.9)+ 12 ime − U ( ξ ± γ m ξ ∓ − ξ ∓ γ m ξ ± + ξ ± γ m ξ ∓ − ξ ∓ γ m ξ ± ) , ∇ m ( ξ ± ξ ± − ξ ± ξ ± ) = − ∂ m U ( ξ ± ξ ± − ξ ± ξ ± ) (C.10) − ime − U ( ξ ± γ m ξ ∓ − ξ ∓ γ m ξ ± − ξ ± γ m ξ ∓ + ξ ∓ γ m ξ ± ) , ∇ m ( ξ ± ξ ± ) = ( iQ m − ∂ m U ) ξ ± ξ ± − P m ξ ± ξ ± (C.11) − ime − U ( ξ ± γ m ξ ∓ − ξ ∓ γ m ξ ± ) , ∇ m ( ξ c ± ξ ± ) = − ∂ m U ξ c ± ξ ± + 2 ∂ m U ξ c ± ξ ± (C.12) − ime − U ( ξ c ± γ m ξ ∓ − ξ c ∓ γ m ξ ± ) + ime − U ( ξ c ± γ m ξ ∓ − ξ c ∓ γ m ξ ± ) . Vector biliears ∇ [ l ( ξ γ m ] ξ − + ξ γ m ] ξ − ) = − ∂ [ l U ( ξ γ m ] ξ − + ξ γ m ] ξ − ) (C.13)+ 32 ime − U ( ξ γ lm ξ + ξ − γ lm ξ − + ξ γ lm ξ + ξ − γ lm ξ − ) , ∇ [ l ( ξ γ m ] ξ − − ξ γ m ] ξ − ) = − ∂ [ l U ( ξ γ m ] ξ − − ξ γ m ] ξ − ) (C.14)+ 12 ime − U ( ξ γ lm ξ + ξ − γ lm ξ − − ξ γ lm ξ − ξ − γ lm ξ − ) , ∇ [ l ( ξ γ m ] ξ − ) = ( iQ [ l − ∂ [ l U ) ξ γ m ] ξ − + P [ l ( ξ γ m ] ξ − ) (C.15)+ 2 ime − U ( ξ γ lm ξ + ξ − γ lm ξ − ) , ∇ [ l ( ξ γ m ] ξ − ) = ( − iQ [ l − ∂ [ l U ) ξ γ m ] ξ − + P ∗ [ l ( ξ γ m ] ξ − ) (C.16)+ 2 ime − U ( ξ γ lm ξ + ξ − γ lm ξ − ) , ∇ [ l ( ξ c γ m ] ξ − ) = ( iQ [ l − ∂ [ l U ) ξ c γ m ] ξ − + P [ l ξ c γ m ] ξ − (C.17)+ ime − U ( ξ c γ lm ξ + ξ c − γ lm ξ − ) , ∇ [ l ( ξ c γ m ] ξ − ) = ( − iQ [ l − ∂ [ l U ) ξ c γ m ] ξ − + P ∗ [ l ξ c γ m ] ξ − (C.18)+ ime − U ( ξ c γ lm ξ + ξ c − γ lm ξ − ) , ∇ [ l ( ξ c γ m ] ξ − + ξ c γ m ] ξ − ) = − ∂ [ l U ( ξ c γ m ] ξ − + ξ c γ m ] ξ − ) (C.19)+ 32 ime − U ( ξ c γ lm ξ + ξ c − γ lm ξ − + ξ c γ lm ξ + ξ c − γ lm ξ − ) , ∇ [ l ( ξ c γ m ] ξ − − ξ c γ m ] ξ − ) = − ∂ [ l U ( ξ c γ m ] ξ − − ξ c γ m ] ξ − ) (C.20)+ 12 ime − U ( ξ c γ lm ξ + ξ c − γ lm ξ − − ξ c γ lm ξ − ξ c − γ lm ξ − ) . wo-form bilinears ∇ [ r ( ξ ± γ st ] ξ ± + ξ ± γ st ] ξ ± ) = − ∂ [ r U ( ξ ± γ st ] ξ ± + ξ ± γ st ] ξ ± ) (C.21) − ime − U ( ξ ± γ rst ξ ∓ − ξ ∓ γ rst ξ ± + ξ ± γ rst ξ ∓ − ξ ∓ γ rst ξ ± ) , ∇ [ r ( ξ ± γ st ] ξ ± − ξ ± γ st ] ξ ± ) = − ∂ [ r U ( ξ ± γ st ] ξ ± − ξ ± γ st ] ξ ± ) (C.22)+ 13 ( G rst ξ ± ξ ± − G ∗ rst ξ ± ξ ± ) − ime − U ( ξ ± γ rst ξ ∓ − ξ ∓ γ rst ξ ± − ξ ± γ rst ξ ∓ + ξ ∓ γ rst ξ ± ) , ∇ [ r ( ξ ± γ st ] ξ ± ) = ( iQ [ r − ∂ [ r U ) ξ ± γ st ] ξ ± + P [ r ξ ± γ st ] ξ ± (C.23) − G rst ( ξ ± ξ ± − ξ ± ξ ± ) − ime − U ( ξ ± γ rst ξ ∓ − ξ ∓ γ rst ξ ± ) , ∇ [ r ( ξ c ± γ st ] ξ ± ) = ( iQ [ r − ∂ [ r U ) ξ c ± γ st ] ξ ± + P [ r ξ c ± γ st ] ξ ± + 16 G rst ( ξ c ± ξ ± − ξ c ± ξ ± ) − ime − U ( ξ c ± γ rst ξ ∓ − ξ c ∓ γ rst ξ ± ) , (C.24) ∇ [ r ( ξ c ± γ st ] ξ ± ) = ( − iQ [ r − ∂ [ r U ) ξ c ± γ st ] ξ ± + P ∗ [ r ξ c ± γ st ] ξ ± − G ∗ rst ( ξ c ± ξ ± − ξ c ± ξ ± ) − ime − U ( ξ c ± γ rst ξ ∓ − ξ c ∓ γ rst ξ ± ) , (C.25) ∇ [ r ( ξ c ± γ st ] ξ ± ) = − ∂ [ r U ξ c ± γ st ] ξ ± − ∂ [ r (2 U ) ξ c ± γ st ] ξ ± (C.26) − ime − U ( ξ c ± γ rst ξ ∓ − ξ c ∓ γ rst ξ ± ) . Normalization of scalar bilinears
From (C.9) and (C.10), we have d [ e − U ( ξ ξ + ξ − ξ − )] = d [ e − U ( ξ ξ + ξ − ξ − )] = 0 . (C.27)Then, we can fix the normalization ξ ξ + ξ − ξ − = ξ ξ + ξ − ξ − = e U m . (C.28)18 Fierz identities
In four dimensions, the Fierz identity is η T η η T η = 14 (cid:0) η T η η T η + η T γ η η T γ η (cid:1) + 14 (cid:0) η T γ m η η T γ m η − η T γ m γ η η T γ m γ η (cid:1) − η T γ mn η η T γ mn η . (D.1)When we calculate the Lie bracket of the Killing vectors, we need to compute contractions ofvectors with two-forms. With the spinors η and η of the same chirality, we find the followingrelation useful η T γ m η η T γ mn η = 2 η T γ n η η T η − η T γ n η η T η − η T γ n γ η η T γ η . (D.2) D.1 Relations of scalar bilinears
We also find useful relations between the scalar bilinears using the Fierz identities. If we choose η T = ξ , η = ξ , η T = ξ , η = ξ , we have ξ ξ ξ ξ = 12 ξ ξ ξ ξ − ξ γ mn ξ ξ γ mn ξ . (D.3)Similarly, if we choose η T = ξ c , η = ξ , η T = ξ , η = ξ c , we have ξ c ξ ξ ξ c = 12 ξ ξ ξ ξ + 18 ξ γ mn ξ ξ γ mn ξ . (D.4)Thus we find that | ξ ξ | + | ξ c ξ | = ξ ξ ξ ξ . (D.5)We also have a similar result with the minus chirality spinors. Then, using (C.5) and (C.7),we obtain ξ ξ ξ ξ = ξ − ξ − ξ − ξ − . (D.6)With the normalization (C.28), we conclude that ξ ξ = ξ − ξ − , ξ ξ = ξ − ξ − . (D.7) D.2 Inner products of vector bilinears
The vectors K , K , K and one-forms L , L play a crucial role in determining the form of thefour-dimensional metric. In this section we explain the procedure in detail. First, we calculate the19orms and the inner products of these vectors using the Fierz identities. For example, choosing η T = ξ , η = ξ , η T = ξ − , η = ξ − , we get ξ γ n ξ − ξ − γ n ξ = 2 ξ ξ ξ − ξ − . (D.8)Similarly, inner products of any vectors can be written as products of scalars. For the threeKilling vectors, we have( K ) = ( ξ ξ − ξ ξ ) − ( ξ c ξ − ( ξ c ξ ) ∗ ) , (D.9)( K ) = ( ξ ξ − ξ ξ ) + ( ξ c ξ + ( ξ c ξ ) ∗ ) , ( K ) = 4 | ξ c ξ | ,K · K = i (( ξ c ξ ) − ( ξ c ξ ) ∗ ) ,K · K = ( ξ ξ − ξ ξ )( ξ c ξ + ( ξ c ξ ) ∗ ) ,K · K = − i ( ξ ξ − ξ ξ )( ξ c ξ − ( ξ c ξ ) ∗ ) . The inner products of L and L are( L ) = ( ξ ξ + ξ ξ ) − ( ξ ξ + ξ ξ ) = 19 m e U (1 − e − U − φ y ) , (D.10)( L ) = ( ξ ξ + ξ ξ ) + ( ξ ξ − ξ ξ ) = 19 m e U (1 − e − U + φ z ) , (D.11) L · L = − i (cid:16) ( ξ ξ ) − ( ξ ξ ) (cid:17) = 19 m e − U y z, (D.12)where we express the scalar bilinears in terms of the coordinates z and y defined in (2.23) at thelast step. 20 Killing vectors of five-dimensional target space
We have found the eight Killing vectors of the five-dimensonal target space of the non-linearsigma model (3.4) as K = √ ∂ φ − C ∂ C + f ∂ f − g ∂ g ) , (E.1) K = 12 √ ∂ U + 6 f ∂ f + 6 g ∂ g ) ,K = − ∂ C + f ∂ g ,K = − C ∂ φ + 2( C − e − φ ) ∂ C + g ∂ f ,K = − g ∂ U + ( g + 4 Cf ) ∂ φ − e − φ ( − f + 2 C e φ f + Ce φ g ) ∂ C + (2 Ce U + φ − f g ) ∂ f − e − φ (4 e U + 4 C e U +2 φ + e φ g ∂ g ) ,K = ∂ g ,K = − f ∂ U − f ∂ φ + ( 12 g + Cf ) ∂ C − ( e U + φ + f ) ∂ f − ( f g − Ce U + φ ) ∂ g ,K = ∂ f . The eight generators of the SL(3 , R ) group are T ≡ H = 1 √ − , T ≡ H = 1 √ − , (E.2) T ≡ E α = , T ≡ E α = , T ≡ E α = ,T ≡ E − α = , T ≡ E − α = , T ≡ E − α = , where H , H are Cartan generators and E α , E α , E α are positive root generators. By identifying K i = T i , the eight Killing vectors satisfy an sl (3 , R ) algebra.21 eferences [1] F. Apruzzi, M. Fazzi, A. Passias, D. Rosa and A. Tomasiello, AdS solutions of type IIsupergravity, JHEP , 099 (2014) arXiv:1406.0852 [hep-th].[2] J. M. Maldacena,
The Large N limit of superconformal field theories and supergravity,
Int.J. Theor. Phys. , 1113 (1999) [Adv. Theor. Math. Phys. , 231 (1998)] [hep-th/9711200].[3] N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynam-ics,
Phys. Lett. B , 753 (1996) [hep-th/9608111].[4] D. R. Morrison and N. Seiberg,
Extremal transitions and five-dimensional supersymmetricfield theories,
Nucl. Phys. B , 229 (1997) [hep-th/9609070].[5] K. A. Intriligator, D. R. Morrison and N. Seiberg,
Five-dimensional supersymmetric gaugetheories and degenerations of Calabi-Yau spaces,
Nucl. Phys. B , 56 (1997) [hep-th/9702198].[6] S. Ferrara, A. Kehagias, H. Partouche and A. Zaffaroni,
AdS(6) interpretation of 5-D su-perconformal field theories,
Phys. Lett. B , 57 (1998) [hep-th/9804006].[7] A. Brandhuber and Y. Oz,
The D-4 - D-8 brane system and five-dimensional fixed points,
Phys. Lett. B , 307 (1999) [hep-th/9905148].[8] O. Bergman and D. Rodriguez-Gomez,
5d quivers and their AdS(6) duals,
JHEP , 171(2012) [arXiv:1206.3503 [hep-th]].[9] V. Pestun,
Localization of gauge theory on a four-sphere and supersymmetric Wilson loops,
Commun. Math. Phys. , 71 (2012) [arXiv:0712.2824 [hep-th]].[10] K. Hosomichi, R. K. Seong and S. Terashima,
Supersymmetric Gauge Theories on the Five-Sphere,
Nucl. Phys. B , 376 (2012) [arXiv:1203.0371 [hep-th]].[11] J. Kallen, J. Qiu and M. Zabzine,
The perturbative partition function of supersymmetric 5DYang-Mills theory with matter on the five-sphere,
JHEP , 157 (2012) [arXiv:1206.6008[hep-th]].[12] H. C. Kim, S. S. Kim and K. Lee,
JHEP , 142 (2012) [arXiv:1206.6781 [hep-th]].[13] D. L. Jafferis and S. S. Pufu,
Exact results for five-dimensional superconformal field theorieswith gravity duals,
JHEP , 032 (2014) [arXiv:1207.4359 [hep-th]].2214] B. Assel, J. Estes and M. Yamazaki,
Wilson Loops in 5d N=1 SCFTs and AdS/CFT,
Annales Henri Poincare , 589 (2014) [arXiv:1212.1202 [hep-th]].[15] L. F. Alday, M. Fluder, P. Richmond and J. Sparks, Gravity Dual of SupersymmetricGauge Theories on a Squashed Five-Sphere,
Phys. Rev. Lett. , no. 14, 141601 (2014)[arXiv:1404.1925 [hep-th]].[16] L. F. Alday, M. Fluder, C. M. Gregory, P. Richmond and J. Sparks,
Supersymmetricgauge theories on squashed five-spheres and their gravity duals,
JHEP , 067 (2014)[arXiv:1405.7194 [hep-th]].[17] A. Passias,
A note on supersymmetric AdS solutions of massive type IIA supergravity, JHEP , 113 (2013) [arXiv:1209.3267 [hep-th]].[18] M. Cvetic, H. Lu, C. N. Pope and J. F. Vazquez-Poritz,
AdS in warped space-times,
Phys.Rev. D , 122003 (2000) [hep-th/0005246].[19] Y. Lozano, E. O Colgain, D. Rodriguez-Gomez and K. Sfetsos, Supersymmetric
AdS via TDuality, Phys. Rev. Lett. , no. 23, 231601 (2013) [arXiv:1212.1043 [hep-th]].[20] Y. Lozano, E. O Colgain and D. Rodriguez-Gomez,
Hints of 5d Fixed Point Theories fromNon-Abelian T-duality,
JHEP , 009 (2014) [arXiv:1311.4842 [hep-th], arXiv:1311.4842].[21] J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram,
Supersymmetric AdS(5) solutionsof M theory,
Class. Quant. Grav. , 4335 (2004) [hep-th/0402153].[22] H. Lin, O. Lunin and J. M. Maldacena, Bubbling AdS space and 1/2 BPS geometries,
JHEP , 025 (2004) [hep-th/0409174].[23] J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram,
Supersymmetric AdS(5) solutionsof type IIB supergravity,
Class. Quant. Grav. , 4693 (2006) [hep-th/0510125].[24] E. Cremmer, I. V. Lavrinenko, H. Lu, C. N. Pope, K. S. Stelle and T. A. Tran, Eu-clidean signature supergravities, dualities and instantons,
Nucl. Phys. B , 40 (1998)[hep-th/9803259].[25] D. Maison,
Ehlers-Harrison type transformations for Jordan’s extended theory of gravitation,
Gen. Rel. Grav. , 717 (1979).[26] S. Giusto and A. Saxena, Stationary axisymmetric solutions of five dimensional gravity,
Class. Quant. Grav. , 4269 (2007) [arXiv:0705.4484 [hep-th]].2327] P. Fre and A. S. Sorin, Supergravity Black Holes and Billiards and Liouville integrable struc-ture of dual Borel algebras,
JHEP , 066 (2010) [arXiv:0903.2559 [hep-th]].[28] J. Patera, R. T. Sharp, P. Winternitz and H. Zassenhaus,
Invariants of Real Low Dimen-sional Lie Algebras,
J. Math. Phys. , 986 (1976).[29] H. Kato, Low dimensional Lie groups admitting left invariant flat projective or affine struc-tures, arXiv:1406.3424.[30] C.M. Hull,
Noncompact Gaugings of N = 8 Supergravity,
Phys. Lett. B , 39 (1984).[31] C.M. Hull,
More Gaugings of N = 8 Supergravity,
Phys. Lett. B , 297 (1984).[32] H. Samtleben and M. Weidner,
The Maximal D=7 supergravities,
Nucl. Phys. B , 383(2005) [hep-th/0506237].[33] O. Aharony and A. Hanany,
Branes, superpotentials and superconformal fixed points,
Nucl.Phys. B , 239 (1997) [hep-th/9704170].[34] N. Kim,
AdS(3) solutions of IIB supergravity from D3-branes,
JHEP , 094 (2006)[hep-th/0511029].[35] N. Kim and J. D. Park,
Comments on AdS(2) solutions of D=11 supergravity,
JHEP ,041 (2006) [hep-th/0607093].[36] J. P. Gauntlett and N. Kim,
Geometries with Killing Spinors and Supersymmetric AdSSolutions,
Commun. Math. Phys. , 897 (2008) [arXiv:0710.2590 [hep-th]].[37] M. F. Sohnius,
Introducing Supersymmetry,
Phys. Rept.128