Universal consistent truncation for 6d/7d gauge/gravity duals
UUniversal consistent truncationfor 6d/7d gauge/gravity duals
Achilleas Passias, Andrea Rota and Alessandro Tomasiello
Dipartimento di Fisica, Universit`a di Milano–Bicocca, Piazza della Scienza 3, I-20126Milano, ItalyandINFN, sezione di Milano–Bicocca achilleas.passias, andrea.rota, alessandro.tomasiello @unimib.it
Abstract
Recently, AdS solutions of IIA supergravity have been classified; there are infinitelymany of them, whose expression is known analytically, and with internal space of S topology. Their field theory duals are six-dimensional (1 ,
0) SCFT’s. In this paper weshow that for each of these AdS solutions there exists a consistent truncation frommassive IIA supergravity to minimal gauged supergravity in seven dimensions. Thistheory has an SU(2) gauge group, and a single scalar, whose value is related to a certaindistortion of the internal S . This explains the universality observed in recent work onAdS and AdS solutions dual to compactifications of the (1 ,
0) SCFT ’s. Thanks toprevious work on the minimal gauged supergravity, the truncation also implies theexistence of holographic RG-flows connecting those solutions to the AdS vacuum, aswell as new classes of IIA AdS solutions. a r X i v : . [ h e p - t h ] D ec ontents solutions in massive IIA supergravity 5 and AdS . . . . . . . . . . . . . . . . . . . 7 and AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Conformal field theories in dimensions higher than four are still comparatively myste-rious; there is usually no Lagrangian description. This is the case for example for the(2 , T gives N = 4 super-Yang–Mills. Reducing it on a Riemann surface produces a vast “classS” of four-dimensional theories with very interesting duality properties [1–3]. One cansimilarly compactify down to three [4] and to two [5] dimensions.It is reasonable to expect similar phenomena with different six-dimensional CFT’s.This might teach us something about the (2 ,
0) theory, but also about the dynamicsof CFT’s in lower dimensions. Perhaps the simplest generalization of the (2 ,
0) theoryoccurs when one introduces orbifold singularities [6–8]; the study of their compactifi-cations on Riemann surfaces is just starting [9–11]. From the holographic perspective,1owever, these theories are not very different from the (2 ,
0) theory: their dual is simplyAdS × S / Z k [12, 13].Nevertheless, an interesting further generalization can be obtained via NS5–D6–D8-brane systems [14, 15]. This class consists of (1 ,
0) SCFT’s which are non Lagrangian,but which can be described by a quiver on a “tensor branch”. Their holographic dualswere found relatively recently: first numerically in [19], then analytically in [20]. Theirinterpretation as the duals of the SCFT’s described above was given in [21]. Up toorbifolds and orientifolds, these are the most general AdS solutions in perturbativetype II supergravity.Although the compactifications of these theories to lower dimensions are not yetknown, they can already be studied holographically: the corresponding AdS and AdS solutions were found respectively in [22] and [23]. These solutions are similar in spiritto the duals of the compactifications of the (2 ,
0) theory [24–26]: namely, AdS getsreplaced by AdS × Σ or AdS × Σ , and the internal space gets distorted in a certainway. What is perhaps nicer than expected is that this distortion is “universal”. Namely,even though there are infinitely many AdS solutions, the map to obtain the AdS andAdS metric is always the same. Moreover, the two maps are very similar to each other:they differ only by the value of certain numerical factors.In this paper, we greatly extend this universality. We promote the maps to a moregeneral Ansatz, where AdS gets replaced by any seven-dimensional metric g µν , and theinternal space gets distorted in a way that depends on a single scalar parameter X . ThisAnsatz in fact becomes nothing but a reduction to a seven-dimensional effective theory.Its bosonic fields are X and g µν themselves, together with a three-form potential, andan SU(2) gauge field which is related to the fibration of the internal space over theseven external dimensions.This effective theory is the so-called minimal gauged supergravity in seven dimen-sions [27,28], which describes the dynamics of (a gauged version of) the gravity multipletwith sixteen supercharges. It is a subsector of the bigger “maximal” [29] theory, whichdescribes the gravity multiplet with thirty-two supercharges and has gauge group SO(5).Both theories can be obtained [30,31] as consistent truncations from eleven dimensions.Here we find that the minimal theory can also be obtained from massive IIA, in infinitely many ways . In each of these reductions, the supersymmetric AdS vacuumis one of the solutions in [19, 20]. This is perhaps surprising, but the idea is that, inreducing, we are only using the ordinary differential equation (ODE) that the internalgeometry has to solve in the vacuum, and not the details of the individual solution. One can engineer six-dimensional field theories also in F-theory [16–18]. × Σ [24] and AdS × Σ [25] solutions. They uplift to those of [22, 23]. In this sense we are explaining andextending the universality noticed in those papers. Minimal gauged supergravity alsohas “Renormalization Group (RG) flow” solutions that connect the above backgroundsto the AdS maximally supersymmetric vacuum. This shows conclusively that thesolutions of [22, 23] are indeed dual to compactifications on Σ and Σ of the six-dimensional (1 ,
0) SCFT’s.Minimal gauged supergravity also admits AdS × Σ solutions, preserving N = 1 and N = 2 supersymmetry. In the latter case Σ is a K¨ahler–Einstein manifold of negativeconstant curvature, while in the former case Σ is (a compact quotient of) hyperbolicspace H . The corresponding CFT duals are two-dimensional (0 ,
2) and (0 ,
1) SCFTs.Uplifting these solutions yields new AdS solutions of massive IIA supergravity. On thefield theory side, this implies that all the six-dimensional SCFT’s of [14, 15, 21] can becompactified on four-manifolds Σ to produce two-dimensional SCFT’s.Finally, minimal gauged supergravity has a second vacuum, which is not supersym-metric. This means that there are also non-supersymmetric analytical AdS solutionsin massive IIA. Although we will not discuss these solutions in this paper, it would beinteresting to analyze them further, for example by comparing them with the numericalnon-supersymmetric solutions of [32].This paper is organized as follows. In section 2, we will review the seven-dimensionalminimal gauged supergravity. In section 3 we will review the IIA AdS solutions foundnumerically in [19] and analytically in [22], and their AdS and AdS compactifications.In section 4 we will perform the reduction from massive IIA to seven-dimensional min-imal gauged supergravity. Finally, in section 5 we will discuss some supersymmetricsolutions to seven-dimensional minimal gauged supergravity, which thanks to our resultscan be lifted to supersymmetric massive IIA solutions. The bosonic fields of seven-dimensional minimal gauged supergravity [27] are the gravi-ton, a triplet of one-forms A i , i = 1 , ,
3, transforming in the adjoint representation of This solution was actually obtained in the maximal theory, with SO(5) gauge group, but it ispossible to show that it survives in the minimal theory. ϕ and a three-form A . The corresponding Lagrangian is L = R − ∗ dϕ ∧ dϕ − V ( ϕ ) ∗ − e √ ϕ ∗ F ∧ F − e − √ ϕ ∗ F i ∧ F i (2.1)+ F i ∧ F i ∧ A − h F ∧ A , where V ( ϕ ) is the scalar potential V ( ϕ ) = 2 h e − √ ϕ − √ hge − √ ϕ − g e √ ϕ . (2.2) F i = d A i − g(cid:15) ijk A j ∧A k and F = d A are the field strengths of A i and A respectively. g is the gauge coupling constant whereas the constant h is referred to as the topologicalmass.If h/g > e − √ ϕ = √ gh anda minimum at e − √ ϕ = √ gh ; only the former is supersymmetric [28].There is a dual formulation of the theory with a two- instead of a three-form. Inthis case, the topological mass and the corresponding term in the Lagrangian are absentand the scalar potential has no critical points. In [33] it was shown that this versioncan be embedded in ten-dimensional type I supergravity.The fermionic fields are the gravitino ψ µa , µ = 0 , . . . , λ a . They aresymplectic-Majorana spinors transforming as SU(2) doublets; a = 1 , δ ξ ψ µa = ( ∇ µ + ig ( A µ ) ab ) ξ b + i √ e − √ ϕ ( γ µα α − δ µα γ α ) ( F α α ) ab ξ b + e √ ϕ (cid:0) γ µα α α α − δ µα γ α α α (cid:1) F α α α α ξ a + mγ µ ξ a , (2.3a) δ ξ λ a = √ (cid:19)(cid:19) ∂ϕξ a − i √ e − √ ϕ ( (cid:26)(cid:26) F ) ab ξ b + √ e √ ϕ (cid:26)(cid:26) F ξ a − √ m + h e − √ ϕ ) ξ a , (2.3b)where m = − h e − √ ϕ − g √ e √ ϕ . (2.4)Furthermore, ( A ) ab = A i ( T i ) ab , ( F ) ab = F i ( T i ) ab . (2.5) T i = σ i are the generators of SU(2), σ i being the Pauli matrices. The scalar and the form fields of the original paper have being rescaled by a factor of √ and theconstant h by a factor of . p -form F p is defined as (cid:0)(cid:0) F p ≡ p ! F pα ...α p γ α ...α p . (2.6) solutions in massive IIA supergravity In this section we review the IIA AdS solutions of [19]. These, according to ourembedding, are the uplift of the supersymmetric AdS vacuum of the seven-dimensionalminimal gauged supergravity. We also discuss compactifications of these solutions toAdS and AdS [20]; these will be instrumental in coming up with an appropriatereduction Ansatz in section 4. While there are infinitely many AdS solutions in IIA supergravity, they all share a fewfundamental features. The internal space M is an S -fibration over an interval, whosecoordinate we call r . The S shrinks at the two endpoints of this interval, so that M has the topology of an S . Metric and fluxes can be written in terms of three functions:the dilaton, the warping, and one function x related to the volume of the S . All threeonly depend on r : φ = φ ( r ) , A = A ( r ) , x = x ( r ) . (3.1)The metric now reads ds = e A ds + ds M , ds M = dr + e A (1 − x ) ds S . (3.2) ds and ds S are unit radius metrics on AdS and S . The expression of the Neveu–Schwarz flux is H = − (cid:0) e − A + F xe φ (cid:1) vol M , (3.3)where F is the Romans mass and φ the dilaton. The expression for the Ramond–Ramond two-form flux is F = 116 e A − φ √ − x (cid:0) F e A + φ x − (cid:1) vol S . (3.4)5he functions φ ( r ), A ( r ), x ( r ) obey a system of ODEs: dφdr = 14 e − A √ − x (cid:0) x + (2 x − F e A + φ (cid:1) , (3.5a) dxdr = − e − A √ − x (cid:0) xF e A + φ (cid:1) , (3.5b) dAdr = 14 e − A √ − x (cid:0) x − F e A + φ (cid:1) . (3.5c)Originally, in [19], the AdS solutions were found by integrating this system numerically.However, it was later found in [22] that the solutions are determined by a single function β ( y ) satisfying a single ODE:( q ) (cid:48) = 29 F , q ≡ − y √ ββ (cid:48) , (3.6)where the new variable y is defined by dr = (cid:0) (cid:1) e A √ β dy , and a prime denotes differenti-ation with respect to y . Now A , φ , and x are determined by e A = 23 (cid:18) − β (cid:48) y (cid:19) / , e φ = ( − β (cid:48) /y ) / √ β − yβ (cid:48) ,x = − yβ (cid:48) β − yβ (cid:48) . (3.7)The ODE (3.6) can be readily solved analytically by writing it as 16 y β ( β (cid:48) ) = F ( y − ˆ y ), with ˆ y a constant; this can now be integrated by quadrature. Without D8-branes,the generic solution [22, Sec. 5.6] has two special points, corresponding to the presenceof two stacks with k and k D6-branes (or one stack of D6-branes and an O6-plane).One special case happens where F = 0: in this case k = − k ≡ k , and the solutionis β = k ( y − y ). (This solution can also be obtained as a reduction from AdS × S in M-theory [19, Sec. 5.1].) Another special case happens when k = 0: here β = F ( y − y )( y + 2 y ) [22, Sec. 5.5].More solutions can be obtained by introducing D8-branes. In this case, the Romansmass flux F jumps as one crosses the D8’s, and correspondingly the metric is contin-uous but has a discontinuous first derivative, as one expects from a domain wall. Thepositions of the D8’s are fixed by various flux quantization conditions. The metric canbe obtained by gluing together pieces of the analytic solutions described earlier; thiscan be done in such a way as to avoid D6-branes, or as to include them, as one wishes.All in all, one has an infinite set of solutions; they are in one-to-one correspondence [21]6ith NS5–D6–D8 systems [14, 15]. The corresponding SCFT ’s are non-Lagrangian,but an effective description is known on their tensor branch.In any case, we will not need to know too many details about the classification ofthe most general solutions, since the reduction to seven dimensions will work much inthe same way for all of them. This is roughly because we will only need to use (3.5),and not the actual expressions for the solutions. All the solutions we just described are N = 1 supersymmetric. The original method tofind them used a formulation of the supersymmetry equations in terms of differentialforms, where the spinors were never explicitly used. However, in order to comparesupersymmetry in ten dimensions to supersymmetry in seven, in section 4.2 we willactually need the supersymmetry parameters, which were given in [23]: (cid:15) = ( ξ ⊗ χ + ξ c ⊗ χ c ) ⊗ |↑(cid:105) , (cid:15) = ( ξ ⊗ χ − ξ c ⊗ χ c ) ⊗ |↓(cid:105) . (3.8)Here ξ is a Killing spinor on AdS , while |↑(cid:105) and |↓(cid:105) are eigenvectors of the Pauli matrix σ , with eigenvalues +1 and − χ and χ are χ = − ie A e − i π σ e i α σ χ S , χ = e A e − i α σ χ S , (3.9)where sin α = x and χ S is a Killing spinor on S . The superscript c denotes charge con-jugation. The SU(2) R-symmetry acts on the doublets ( ξ, ξ c ) t , ( χ , χ c ) t and ( χ , − χ c ) t in the fundamental representation. and AdS It is possible to compactify the AdS solution on H or H to AdS and AdS respec-tively, by associating the functions that determine the solutions, via the map [20] e A → X e A , r → X r , x → x √ w , (3.10) They were also independently computed by I. Bakhmatov (unpublished notes). X is a constant parameter, with the value X = 1 for the AdS solution and w ≡ X (1 − x ) + x . The corresponding geometries read ds = X e A ds + X ds M , ds M = dr + 1 − x w e A Ds S , (3.11) ds = ds + ds H ds + ds H , X = , where ds H and ds H are metrics of unit radius. The S is fibered over H or H , withthe U(1) spin connection of H twisting a U(1) isometry inside the full SU(2) isometryof S in the first case and the SU(2) spin connection of H twisting the whole isometryin the second.One can then quotient H and H by discrete subgroups of PSL(2 , R ) and PSL(2 , C ),so as to obtain respectively a Riemann surface Σ of genus g ≥
2, or a compact hy-perbolic manifold Σ . The holographic interpretation of these solutions is then similarto the familiar Maldacena–N´u˜nez case [24]: they represent twisted compactifications ofSCFT ’s to SCFT ’s and SCFT ’s.The fact that both solutions can be written as (3.11) suggests a reduction Ansatzfor massive IIA supergravity on M : promote X to scalar field in seven dimensions andintroduce seven-dimensional gauge vector fields gauging the SU(2) isometry of M . In this section we present the Ansatz for the Kaluza-Klein reduction of massive IIAsupergravity on M , to the seven-dimensional minimal gauged supergravity. Our ap-proach to verifying the consistency of the reduction (or truncation) is to substitute theAnsatz into the ten-dimensional equations of motion and show that these are satisfiedprovided that the seven-dimensional equations of motion are satisfied. Vice versa, anysolution of the lower-dimensional theory can be uplifted on M to an exact solution ofthe higher-dimensional theory. This is described in subsection 4.1.In subsection 4.2 we take a further step and show that any supersymmetric solutionof the seven-dimensional theory uplifts to a solution that also preserves supersymmetry.We provide a decomposition Ansatz for the ten-dimensional supersymmetry parameters The dilaton transforms as e φ → X e φ √ w . ξ a such that the lower-dimensional super-symmetry transformations (2.3) vanish can be uplifted so that the higher-dimensionalsupersymmetry transformations vanish as well. The Ansatz for the ten-dimensional metric is (cid:96) − ds = g X − e A ds + X ds M , ds M = dr + 1 − x w e A Ds S , (4.1)where (cid:96) ≡ √ g and w ≡ X (1 − x ) + x . (4.2)The parameter X is promoted in this section to a scalar in seven dimensions; it will turnout to be related to the scalar ϕ of section 2. It was a constant for the AdS solutionsof (3.11). The covariantized metric Ds S on the two-sphere is Ds S ≡ Dy i Dy i , Dy i ≡ dy i + (cid:15) ijk y j g A k . (4.3) y i parametrize S ∈ R as the locus y i y i = 1; explicitly y i = (sin θ cos ψ, sin θ sin ψ, cos θ ) . (4.4)In angular coordinates, Ds S reads Ds S = ( dθ + K θi g A i ) + sin θ ( dψ + K ψi g A i ) , (4.5)where K = cot θ cos ψ∂ ψ + sin ψ∂ θ , K = cot θ sin ψ∂ ψ − cos ψ∂ θ and K = − ∂ ψ are theKilling vectors generating the SO(3) isometry of S .The Ansatz for the dilaton Φ is e = (cid:96) X w e φ . (4.6)Here and in what follows, φ is the dilaton for the AdS solution presented in section9.1.The Ansatz for the Neveu-Schwarz potential B is (cid:96) − B = 116 e A x √ − x w vol − e A dr ∧ ( a − y i A i ) , (4.7)where vol ≡ (cid:15) ijk y i Dy jk is the volume of the covariantized S and a is defined via da = − vol S . H = dB then reads (cid:96) − H = (cid:8) (2 − X + 4 X ) x − X − X (cid:9) w − e − A vol M − X w − (cid:96)F e φ x vol M − e A dr ∧ y i g F i − w − e A x √ − x g F i ∧ Dy i − X w − e A x (1 − x ) dX ∧ vol . (4.8)The Ans¨atze for the Ramond-Ramond fluxes are F = − q (cid:0) vol + y i g F i (cid:1) + 116 w − (cid:96)F e A x √ − x vol , (4.9a) (cid:96) − F = − q w − e A x √ − x y i g F i ∧ vol − q e A dr ∧ (cid:15) ijk g F i ∧ y j Dy k (4.9b) − q e A dr ∧ X g ∗ F − (cid:96) − e A − φ x F , where q ≡ e A − φ √ − x . F and F must obey the Bianchi identities dF − HF = 0 , dF − H ∧ F = 0 . (4.10)A way to see that this is the case for the above expressions is to note that F − BF = dC , (4.11a) F − F F ∧ F = dC , (4.11b)where C = 2 q ( a − y i A i ) , (4.12a) C = − q F ( (cid:15) ijk g F i y j Dy k + g ω ) − e A − φ x A . (4.12b) ω ≡ A i ∧ F i + g(cid:15) ijk A i ∧ A j ∧ A k , satisfying dω = F i ∧ F i . In deriving (4.11b) one10as to take into account the “odd-dimensional self-duality” equation [34] X ∗ F = − √ g A + ω . (4.13)The next step is to obtain the equations that the seven-dimensional fields satisfy,by substituting the Ans¨atze for the ten-dimensional fields into the equations of motionof IIA supergravity.We employ the democratic formulation [35] of type II supergravity and work in thestring frame. The equations of motion of the fluxes are( d + H ∧ ) ∗ F = 0 , d ( e − ∗ H ) − (cid:88) p ∗ F p ∧ F p − = 0 , (4.14)where F ≡ (cid:80) p =0 , , , , , F p . The Einstein equations are R MN + 2 ∇ M ∇ N Φ − H M · H N − e F M · F N = 0 . (4.15)where F M · F N ≡ p − (cid:80) p F pM M ...M ( p − F pNM ...M ( p − and similarly for H M · H N . Finallythe dilaton equation is ∇ Φ − ( ∇ Φ) + R − H = 0 . (4.16)Substituting the Ans¨atze into the flux and dilaton equations of motion, we arrive atthe following equations for the seven-dimensional fields:0 = d ( X − ∗ dX ) + g ( X − − X − + 2 X )vol (4.17a) − X ∗ F ∧ F + X − ∗ F i ∧ F i , d ( X ∗ F ) + √ g F − F i ∧ F i , (4.17b)0 = D ( X − ∗ F i ) − F i ∧ F . (4.17c)In particular, (4.17b) and (4.17c) come from the equations of motion of F and F respectively, while both equations of motion of H and Φ give rise to (4.17a).In order to reduce the Einstein equations, we compute the Riemann and subse-quently the Ricci tensor via the curvature two-form R AB = dω AB + ω AC ∧ ω C B ; thespin connection ω AB is that of the orthonormal frame introduced in appendix A. Aftera lengthy calculation we find that the ten-dimensional Einstein equations, upon using114.17a), reduce to R µν − X − ∂ µ X∂ ν X − g (cid:0) X − − X − − X (cid:1) g µν − X − (cid:16) F i µ · F i ν − F i g µν (cid:17) − X (cid:0) F µ · F ν − F g µν (cid:1) = 0 . (4.18)Equations (4.17) and (4.18) can be derived from the Lagrangian (2.1) for X = e √ ϕ , h = g √ . (4.19) The supersymmetry transformations of the gravitini of IIA supergravity are δ Ψ M = (cid:0) ∇ M − H M (cid:1) (cid:15) − e Φ F Γ M (cid:15) , δ Ψ M = (cid:0) ∇ M + H M (cid:1) (cid:15) − e Φ λ ( F )Γ M (cid:15) . (4.20)Fermion fields with a subscript 1 have positive chirality, whereas fermion fields with asubscript 2 have negative chirality. The suppressed indices of the fluxes are contractedwith anti-symmetric products of gamma matrices. λ is an operator acting on a p -formas λ ( F p ) = ( − p ] F p , where the square brackets denote the integer part of p . Thesupersymmetry transformations of the dilatini are δλ = (cid:0) ∂ Φ − H (cid:1) (cid:15) − e Φ Γ M F Γ M (cid:15) , δλ = (cid:0) ∂ Φ + H (cid:1) (cid:15) − e Φ Γ M λ ( F )Γ M (cid:15) . (4.21)The decomposition Ansatz for the ten-dimensional supersymmetry parameters is (cid:15) = ( ξ ⊗ χ + ξ c ⊗ χ c ) ⊗ |↑(cid:105) , (cid:15) = ( ξ ⊗ χ − ξ c ⊗ χ c ) ⊗ |↓(cid:105) . (4.22)This is analogous to (3.8), but now ξ is a generic seven-dimensional spinor, rather thana Killing one; the symplectic-Majorana doublet ξ a is ( ξ, ξ c ) t . The expressions for χ and χ are formally identical to (3.9), χ = − ie A e − i π σ e i α σ χ S , χ = e A e − i α σ χ S ; (4.23)however, sin α deviates from its vacuum value, sin α = x , following the map (3.10): i.e.sin α = w − x . Accordingly, cos α ≡ w − X √ − x .We can decompose the ten-dimensional supersymmetry transformations by splitting12liff(1 ,
9) as Γ α = γ α ⊗ I ⊗ σ , Γ a +6 = I ⊗ σ a ⊗ σ , (4.24)and substituting for (4.22). Setting (4.21) to zero amounts to0 = X − (cid:19)(cid:19) ∂Xξ a + X (cid:26)(cid:26) F ξ a − i √ X − ( (cid:26)(cid:26) F i ) ab ξ b − √ g ( X − − X ) ξ a , (4.25)whereas setting (4.20) to zero amounts to the above equation for the internal compo-nents and0 = ( ∇ µ + ig ( A iµ ) ab ) ξ b + i √ X − ( γ µα α − δ µα γ α ) ( F i α α ) ab ξ b + X (cid:0) γ µα α α α − δ µα γ α α α (cid:1) F α α α α ξ a − g (cid:16) √ X − + √ X (cid:17) γ µ ξ a . (4.26)for the external ones. These constraints on ξ a are no other than those that one obtainsby setting (2.3) to zero, for X = e √ ϕ and h = g √ .Thus, preserved supersymmetry in seven dimensions guarantees preserved super-symmetry in ten. In this section we discuss (supersymmetric) anti-deSitter solutions of seven-dimensionalminimal gauged supergravity, along with holographic renormalization group (RG)flows, interpolating between the supersymmetric AdS vacuum and lower-dimensionalanti-deSitter vacua. All these uplift to massive IIA in ten dimensions via the formu-las presented in the previous section. In particular, we consider the AdS and AdS solutions which uplift to the ten-dimensional ones reviewed in section 3.3, and morenotably, AdS solutions which uplift to new AdS solutions of massive IIA supergravitywith N = 1 and N = 2 supersymmetry. and AdS N = 1 and N = 2 supersymmetric AdS × H solutions were first found in [24], ina certain truncation of the maximal gauged supergravity in seven dimensions, keeping α = 0 , . . . , a = 1 , , The parameter h is set equal to g √ , in accordance to the result of the reduction presented in theprevious section. N = 1 solution, thetwo scalars and the two gauge vector fields are set to be equal and thus, the solutioncan also be embedded in the minimal theory of section 2. The AdS × H geometry is a subset of warped product geometries ds = e f ( r ) ( dr + ds R , ) + e f ( r ) ds H , (5.1)with a boundary condition for f and f as r → f ∼ f ∼ log r . That is, asymptoti-cally or in the UV the metric approaches AdS with an R , × H boundary. In orderto preserve supersymmetry, the U(1) gauge field is identified with the spin connectionof H while f and f (as well as the scalar) are subject to a set of ODEs — these canbe found in [24, Eq. (27)].The latter admit an AdS × H solution, which (in our language) reads ds = 8 g e √ ϕ (cid:0) ds + ds H (cid:1) , e √ ϕ = 34 , (5.2)with the field strength of the U(1) gauge field g F i = − vol H δ i , while the three-formpotential is equal to zero. In [36], it was shown numerically (within a broader context)that the AdS × H solution arises as the IR fixed point of an RG flow that connects itto the AdS region.An N = 1 supersymmetric AdS × H solution of seven-dimensional minimal gaugedsupergravity was first found in [25]. The metric and the scalar field of the solution read ds = 8 g e √ ϕ (cid:0) ds + ds H (cid:1) , e √ ϕ = 58 . (5.3)The SU(2) gauge field is identified with the SU(2) spin connection ω ij of H via g A i = (cid:15) ijk ω jk . (5.4)The field strength is then g F i = (cid:15) ijk R jk , where R jk is the curvature two-form of thespin connection, while the three-form potential is zero.It was later shown numerically [26] — in an analogous analysis to that for theAdS × H solution — that this solution also arises as the IR fixed point of an “RGflow geometry”, ds = e f ( r ) ( dr + ds R , ) + e f ( r ) ds H , (5.5) The translation between the languages of [24, appendix 7.3] and section 2 is: m ≡ g √ , λ = λ = − φ/ ≡ ϕ √ . f ∼ f ∼ log r in the UV and the corresponding values for the AdS × H solutionin the IR.The existence of the above RG flow solutions in the seven-dimensional minimalgauged supergravity, in conjunction with the consistent truncation of massive IIA su-pergravity presented in this paper, shows that the AdS and AdS solutions of [22, 23]are connected to the AdS ones of [19] by RG flows. This proves that the solutionsof [22, 23] are dual to compactifications of six-dimensional (1 ,
0) theories on Σ and Σ manifolds of negative curvature. We now turn to the supersymmetric AdS solutions. The first one is AdS × H pre-serving two (real) supercharges. The metric and the scalar field of the solution read ds = 2 g e − √ ϕ (cid:0) ds + ds H (cid:1) , e √ ϕ = 712 . (5.6)The SU(2) gauge field equals the self-dual part of the SO(4) spin connection of H . g A i = (cid:15) ijk ω jk + ω i . (5.7)The field strength is then g F i = (cid:15) ijk R jk + R i . Finally, the four-form flux is propor-tional to the volume of H : F = 3 √ g vol H . (5.8)The second one is AdS × M , where M is K¨ahler–Einstein of constant negativecurvature − H × H ), preserving four supercharges. The metric and thescalar field of the solution read ds = 2 g e − √ ϕ (cid:0) ds + ds M (cid:1) , e √ ϕ = 43 . (5.9)Only a U(1) ⊂ SU(2) gauge field is non-zero and is identified with the center U(1)component of the U(2) spin connection of M , or equivalently with the K¨ahler connec-tion on the canonical bundle of M . Taking the spin connection of the center U(1) tobe the truncation of the self-dual part of the spin connection we can write g A i = ( ω + ω ) δ i . (5.10)The field strength is then identified with the Ricci form of M . Finally, the four-form15ux is proportional to the volume of M : F = √ g vol M . (5.11)The above AdS solutions were also found in [37] as the IR fixed points of RGflows constructed in certain truncations of the maximal seven-dimensional gauged su-pergravity. When uplifted to M-theory, the AdS × M solution arises from M5-braneswrapping K¨ahler four-cycles in Calabi–Yau four-folds while the AdS × H one fromM5-branes wrapping Cayley four-cycles in manifolds of Spin(7) holonomy. The scalarand gauge field sector of the truncations can be identified with the corresponding onesof the minimal theory, while the three-form potential sector is formulated in a dualframe, via (4.13). The AdS × M solution was also constructed with different methodsin [38].Let us conclude with a few words on the field theory duals of the solutions wedescribed in this section. In the first case, (5.6), the SU(2) R-symmetry of the originalAdS solution is completely broken by the gauge fields (5.7). Since no R-symmetry isleft, the dual field theory should be a two-dimensional (0 ,
1) SCFT. In the second case,(5.9), only a U(1) gauge field is switched on; its commutant in SU(2) R is the U(1) itself.This signals that the IIA uplift still has a U(1) isometry; this is the R-symmetry of thedual theory, which should then be a (0 ,
2) SCFT this time. It would be interesting tostudy these theories, perhaps generalizing [5].We can also use AdS/CFT to compute the number of degrees of freedom in thesetheories, along the lines of [22, Sec. 5.8], [23, Sec. 4.8]. In fact, the formalism in thispaper allows us to write a general formula. Let F , be the coefficient in the scalingof the free energy F = F , T V with temperature T and volume V , for a SCFT in 2dimensions. Then, the coefficient F , for an (1 ,
0) theory dual to massive IIA and thecoefficient for a theory obtained by compactifying it on a 4-dimensional space Σ arerelated by F , F , = 1(2 X IR ) Vol(Σ ) , (5.12)where X IR is the value of X for the lower-dimensional AdS solution (recall that X = e √ ϕ ). For example, for the solution (5.9), we get F , / F , = 1 / · / ). The corresponding formula for both the AdS and AdS solutions is F , − d F , = ( X IR ) Vol(Σ d ) , d = 3 , . (5.13) Concluding remarks
We have constructed a consistent truncation of massive IIA supergravity on M , toseven-dimensional minimal gauged supergravity, where M is the internal manifold ofthe AdS solutions of [19, 20]. The truncation is universal: it applies to the wholeinfinite family of Riemannian metrics on M . These exhaust the supersymmetric AdS backgrounds of IIA supergravity. The outcome of this truncation is that any solutionof the seven-dimensional theory uplifts to a solution of massive IIA supergravity in tendimensions. Working at the level of the supersymmetry variations, we have also showedthat supersymmetry is preserved in this process.As an application of our result, we focused on RG flows in seven dimensions, whichin ten dimensions connect the AdS and AdS solutions of [22] and [23] to the AdS ones. Furthermore, AdS vacua in seven dimensions produce new N = 1 and N = 2supersymmetric AdS solutions of massive IIA supergravity, dual to (0 ,
1) and (0 , backgrounds to five and four dimensions.In [21] it was argued that the AdS solutions of massive IIA supergravity are thegravity duals of six-dimensional (1 ,
0) SCFT’s, engineered by NS5–D6–D8-brane inter-sections [14, 15]. The universal character of the present truncation implies that super-gravity in seven dimensions describes a sector common to all these theories — includingalso the (2 ,
0) theory itself, described by the original M-theory reduction of [30].A similar “common sector” phenomenon is witnessed in five dimensions, where itwas found that for every AdS solutions of IIB there is a consistent truncation downto minimal five-dimensional supergravity [39]. In the same paper, it was conjecturedthat this phenomenon should hold in any dimensions; our results prove their conjecturein dimension seven. For certain internal manifolds, it is possible to excite more modesand get bigger theories, e.g. for Sasaki–Einstein reductions [41].Beyond this common sector, discerning finer differences between the CFT ’s wouldrequire more sophisticated reduction procedures, where one keeps more internal modes.These might be gravity modes, or they could come from the D6- and D8-branes whichare present in all the IIA vacua of [19, 20]. In both cases, one would end up couplingthe minimal theory to vector multiplets. Via the gauge/gravity duality, our work paves the way for a broader study of the For an earlier example, concerning a reduction from M-theory, see [40]. [42] argues however that the massive IIA vacua cannot be truncated either to the maximal theory,with gauge group SO(5), nor to a theory with gauge group SO(4) [43] (which can be obtained asreduction from M-theory [44, 45]). × M backgrounds,beyond the massless modes, can be used to analyze the spectrum of the dual operators.Finally, since the minimal seven-dimensional gauged supergravity can also be embed-ded in M-theory [30], lessons learned from the more extensively studied AdS /CFT correspondence stemming from the dynamics of M5-branes can guide us in the studyof its (1 ,
0) cousin in the massive IIA theory.
Acknowledgments
We would like to thank G. Dibitetto and O. Varela for interesting discussions. Weare supported in part by INFN. A.P. and A.T. are also supported by the EuropeanResearch Council under the European Union’s Seventh Framework Program (FP/2007-2013) – ERC Grant Agreement n. 307286 (XD-STRING). The research of A.T. is alsosupported by the MIUR-FIRB grant RBFR10QS5J “String Theory and FundamentalInteractions”.
A Orthonormal frame and spin connection
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