Comments on supersymmetric solutions of minimal gauged supergravity in five dimensions
OOctober 10, 2018
Comments on supersymmetric solutions ofminimal gauged supergravity in five dimensions
Davide Cassani a , Jakob Lorenzen b , and Dario Martelli ba Sorbonne Universit´es UPMC Paris 06,UMR 7589, LPTHE, F-75005, Paris, FranceandCNRS, UMR 7589, LPTHE, F-75005, Paris, France davide.cassani AT lpthe.jussieu.fr b Department of Mathematics, King’s College London,The Strand, London WC2R 2LS, United Kingdom jakob.lorenzen AT kcl.ac.uk, dario.martelli AT kcl.ac.uk
Abstract
We investigate supersymmetric solutions of minimal gauged supergravity in five di-mensions, in the timelike class. We propose an ansatz based on a four-dimensionallocal orthotoric
K¨ahler metric and reduce the problem to a single sixth-order equa-tion for two functions, each of one variable. We find an analytic, asymptoticallylocally AdS solution comprising five parameters. For a conformally flat boundary,this reduces to a previously known solution with three parameters, representingthe most general solution of this type known in the minimal theory. We discussthe possible relevance of certain topological solitons contained in the latter to ac-count for the supersymmetric Casimir energy of dual superconformal field theorieson S × R . Although we obtain a negative response, our analysis clarifies severalaspects of these solutions. In particular, we show that there exists a unique regulartopological soliton in this family. a r X i v : . [ h e p - t h ] J un ontents SO (4) -symmetric solutions 29B Uplifting topological solitons to type IIB 32 Supersymmetric solutions to five-dimensional supergravity play an important role inthe AdS/CFT correspondence. In particular, solutions to minimal gauged supergrav-ity describe universal features of four-dimensional N = 1 superconformal field theories(SCFT’s). The boundary values of the fields sitting in the gravity multiplet, that is themetric, the graviphoton and the gravitino, are interpreted on the field theory side asbackground fields coupling to the components of the energy-momentum tensor multiplet,namely the energy-momentum tensor itself, the R -current and the supercurrent. Whenthe solution is asymptotically Anti de Sitter (AAdS), and one chooses to work in globalcoordinates, the dual SCFT is defined on the conformally flat boundary S × R . So-lutions that are only asymptotically locally Anti de Sitter (AlAdS) describe SCFT’s on1on conformally flat boundaries. For instance, they can describe SCFT’s on backgroundscomprising a squashed S .The conditions for obtaining a supersymmetric solution to minimal gauged supergrav-ity in five dimensions were presented about a decade ago in [1]. These fall in two distinctclasses, timelike and null. The timelike class is somewhat simpler and takes a canonicalform determined by a four-dimensional K¨ahler base. Using this formalism, in [2] thefirst example of supersymmetric AAdS black hole free of closed timelike curves was con-structed. Other AAdS solutions were obtained by different methods in [3, 4, 5, 6], withthe solution of [6] being the most general in that it encompasses the others as specialsub-cases. The solution of [6] also includes the most general AAdS black hole known inminimal gauged supergravity; in the supersymmetric limit, this was shown to take time-like canonical form in [7]. The formalism of [2] also led to construct AlAdS solutions,see [8, 9, 10] for the timelike class and [11] (based on [12]) for the null class.In this paper, we come back to the problem of finding supersymmetric solutions tominimal gauged supergravity. Our motivation is twofold: on the one hand we would liketo investigate the existence of black holes more general than the one of [6]. Indeed, in thesupersymmetric limit the black hole of [6] has two free parameters, while one may expectthe existence of a three-parameter black hole, whose mass, electric charge, and two angularmomenta are constrained only by the BPS condition (see e.g. [13] for a discussion). Oursecond motivation is to construct supersymmetric A(l)AdS solutions with no horizon, thatare of potential interest as holographic duals of pure states of SCFT’s in curved space.For instance, large N SCFT’s on a squashed S × R , where the squashing only preservesa U (1) × U (1) symmetry, should be described by an AlAdS supergravity solution that isyet to be found. It is reasonable to assume that this would preserve U (1) × U (1) × R symmetry also in the bulk, so it would carry mass and two angular momenta in additionto the electric charge.Within the approach of [1], we introduce an ansatz for solutions in the timelike classbased on an orthotoric K¨ahler metric. This has two commuting isometries and depends ontwo functions, each of one variable. We reduce the problem of obtaining supersymmetricsolutions to a single equation of the sixth order for the two functions. This followsfrom a constraint on the conditions of [1], that we formalise in full generality. We find apolynomial solution to the sixth-order equation depending on three non-trivial parameters.Subsequently, in the process of constructing the rest of the five-dimensional metric weobtain two additional parameters, leading to an AlAdS solution. We show that when thesetwo extra parameters are set to zero, our solution is AAdS and is related to that of [6] by2 change of coordinates. We observe that for specific values of the parameters of [6] thechange of coordinates becomes singular, and interpret this in terms of a scaling limit of theorthotoric ansatz, leading to certain non-orthotoric K¨ahler metrics previously employedin the search for supergravity solutions. This proves that our orthotoric ansatz, togetherwith its scaling limit, encompasses all known supersymmetric solutions to minimal gaugedsupergravity belonging to the timelike class.After having completed the general study, we focus on certain non-trivial geometrieswith no horizon contained in the solution of [6], called topological solitons . These are apriori natural candidates to describe pure states of dual SCFT’s. We report on an attemptto match holographically the vacuum state of an N = 1 SCFT on the cylinder S × R , andin particular the non-vanishing supersymmetric vacuum expectation values of the energyand R -charge [14]. Some basic requirements following from the supersymmetry algebralead us to consider a 1/2 BPS topological soliton presented in [15]. Although a directcomparison of the charges shows that this fails to describe the vacuum state of the dualSCFT, in the process we clarify several aspects of such solution. We also show that the1/4 BPS topological solitons with an S × R boundary contain a conical singularity.The structure of the paper is as follows. In section 2 we review the equations of [1] forsupersymmetric solutions of the timelike class and obtain the general form of the sixth-order constraint on the K¨ahler metric. In section 3 we present our orthotoric ansatz andfind a solution to the constraint; then we construct the full five-dimensional solution andrelate it to that of [6]. The scaling limit of the orthotoric metric is presented in section 4.In section 5 we make the comparison with the supersymmetric vacuum of an SCFT on S × R , and discuss further aspects of the topological soliton solutions. In section 6 we drawour conclusions. In appendix A we prove uniqueness of a supersymmetric solution withinan ansatz with SO (4) × R symmetry, while in appendix B we discuss the obstructions touplift the 1/2 BPS topological solitons of [15] to type IIB supergravity on Sasaki-Einsteinmanifolds and make some comments on the dual field theories. In this section we briefly review the conditions for bosonic supersymmetric solutions tominimal five-dimensional gauged supergravity found in [1], focusing on the timelike class.We also provide a general expression of a constraint first pointed out in an example in [9].3he bosonic action of minimal gauged supergravity is S = 12 κ (cid:90) (cid:20) ( R + 12 g ) ∗ − F ∧ ∗ F + 13 √ A ∧ F ∧ F (cid:21) , (2.1)where R is the Ricci scalar of the five-dimensional metric g µν , A is the graviphoton U (1)gauge field, F = d A is its field strength, g > κ is the gravitational coupling constant. The Einstein and Maxwell equations ofmotion are R (5) µν + 4 g g µν − F µκ F ν κ + 112 g µν F κλ F κλ = 0 , d ∗ F − √ F ∧ F = 0 . (2.2)A bosonic background is supersymmetric if there is a non-zero Dirac spinor (cid:15) satisfying (cid:20) ∇ (5) µ − i √ (cid:0) γ µνκ − δ νµ γ κ (cid:1) F νκ − g (cid:0) γ µ + √ iA µ (cid:1)(cid:21) (cid:15) = 0 , (2.3)where the gamma-matrices obey the Clifford algebra { γ µ , γ ν } = 2 g µν . Bosonic supersym-metric solutions were classified (locally) in [1] by analysing the bilinears in (cid:15) . It was shownthat all such solutions admit a Killing vector field V that is either timelike, or null. Inthis paper we do not discuss the null case and focus on the timelike class.Choosing adapted coordinates such that V = ∂/∂y , the five-dimensional metric canbe put in the form d s = − f (d y + ω ) + f − d s B , (2.4)where d s B denotes the metric on a four-dimensional base B transverse to V , while f and ω are a positive function and a one-form on B , respectively. Supersymmetry requires B tobe K¨ahler. This means that B admits a real non-degenerate two-form X that is closed, i.e. d X = 0, and such that X mn is an integrable complex structure ( m, n = 1 , . . . , B , and we raise the index of X mn with the inverse metric on B ).It will be useful to recall that a four-dimensional K¨ahler manifold also admits a complextwo-form Ω of type (2 ,
0) satisfying ∇ m Ω np + iP m Ω np = 0 , (2.5)where P is a potential for the Ricci form, i.e. R = d P . The Ricci form is a closed two-formdefined as R mn = R mnpq ( X ) pq , where R mnpq is the Riemann tensor on B . Moreover, We work in ( − ++++) signature. Our Riemann tensor is defined as R µνκλ = ∂ κ Γ µνλ +Γ µκσ Γ σνλ − κ ↔ λ ,and the Ricci tensor is R µν = R λµλν . X + iX , the triple of real two-forms X I , I = 1 , ,
3, satisfies the quaternionalgebra: X I mp X J pn = − δ IJ δ mn + (cid:15) IJK X K mn . (2.6)We choose the orientation on B by fixing the volume form as vol B = − X ∧ X . Itfollows that the X I are a basis of anti-self-dual forms on B , i.e. ∗ B X I = − X I .The geometry of the K¨ahler base B determines the whole solution, namely f , ω inthe five-dimensional metric (2.4) and the graviphoton field strength F . The function f isfixed by supersymmetry as f = − g R , (2.7)where R is the Ricci scalar of d s B ; this is required to be everywhere non-zero.The expression for the Maxwell field strength is F = −√ (cid:104) f (d y + ω ) + 13 g P (cid:105) . (2.8)Note that the Killing vector V also preserves F , hence it is a symmetry of the solution.It remains to compute the one-form ω . This is done by solving the equationd ω = f − ( G + + G − ) , (2.9)where the two-forms G ± , satisfying the (anti)-self-duality relations ∗ B G ± = ± G ± , aredetermined as follows. Supersymmetry states that G + is proportional to the tracelessRicci form of B : G + = − g (cid:16) R − R X (cid:17) . (2.10)Expanding G − in the basis of anti-self-dual two-forms as G − = 12 gR ( λ X + λ X + λ X ) , (2.11)one finds that the Maxwell equation fixes λ as λ = 12 ∇ R + 23 R mn R mn − R , (2.12)where ∇ is the Laplacian on B . The remaining two components, λ , λ , only have to becompatible with the requirement that the right hand side of (2.9) be closed,d (cid:2) f − ( G + + G − ) (cid:3) = 0 . (2.13) The traceless Ricci form R = R − R X is the primitive part of R . It has self-duality opposite tothe K¨ahler form. Our λ I are rescaled by a factor of 2 gR compared to those in [1]. (cid:2) Ω mn ( ∂ n + iP n )( λ + iλ ) (cid:3) + Ξ m = 0 , (2.14)where Ξ m = R mn ∂ n R + ∂ m (cid:18) ∇ R + 23 R pq R pq − R (cid:19) . (2.15)Acting on (2.14) with Π pq X qm , where Π = ( + iX ) is the projector on the (1 ,
0) part,one obtains the equivalent form D (1 , ( λ + iλ ) + Θ (1 , = 0 , (2.16)where D (1 , m = Π mn ( ∇ n + iP n ) is the holomorphic K¨ahler covariant derivative, and wedefined Θ (1 , m = Π mn X np Ξ p . Eq. (2.16) determines λ + iλ , and hence G − , up to ananti-holomorphic function. This concludes the analysis of the timelike case as presentedin [1].It was first pointed out in [9] that for equation (2.13) to admit a solution, a constrainton the K¨ahler geometry must be satisfied. Hence not all four-dimensional K¨ahler basesgive rise to supersymmetric solutions. While in [9] this was shown for a specific family ofK¨ahler bases, here we provide a general formulation. Taking the divergence of (2.14) andusing (2.5) we find ∇ m Ξ m = 0 , (2.17)that is ∇ (cid:18) ∇ R + 23 R pq R pq − R (cid:19) + ∇ m ( R mn ∂ n R ) = 0 . (2.18)We thus obtain a rather complicated sixth-order equation constraining the K¨ahler metric. To the best of our knowledge, this has not appeared in the physical or mathematicalliterature before. We observe that the term ( ∇ ) R + 2 ∇ m ( R mn ∂ n R ) corresponds to thereal part of the Lichnerowicz operator acting on R , which vanishes for extremal K¨ahlermetrics (see e.g. [16, sect. 4.1]). Thus in this case (2.18) reduces to ∇ (2 R pq R pq − R ) =0. If the K¨ahler metric has constant Ricci scalar, the constraint simplifies further to ∇ ( R pq R pq ) = 0. Finally, if the K¨ahler metric is homogeneous, or Einstein, then Ξ = 0and the constraint is trivially satisfied. It can also be derived starting from the observation that since D (1 , is a good differential, namely( D (1 , (cid:1) = 0, equation (2.16) has the integrability condition D (1 , Θ (1 , = 0. The latter is an a prioricomplex equation, however one finds that the real part is automatically satisfied while the imaginary partis equivalent to (2.18). Constraints on (six-dimensional and eight-dimensional) K¨ahler metrics involving higher-derivativecurvature terms were also found in the study of AdS and AdS supersymmetric solutions to type IIBand 11-dimensional supergravity, respectively [17, 18, 19].
6o summarise, the five-dimensional metric and the gauge field strength are determinedby the four-dimensional K¨ahler geometry up to an anti-holomorphic function. The K¨ahlermetric is constrained by the sixth-order equation (2.18). Moreover, one needs R (cid:54) = 0. Theconditions spelled out above are necessary and sufficient for obtaining a supersymmetricsolution of the timelike class. The solutions preserve at least 1/4 of the supersymmetry,namely two real supercharges. In this section we construct supersymmetric solutions following the procedure describedabove. We start from a very general ansatz for the four-dimensional base, given by a classof local K¨ahler metrics known as orthotoric . These were introduced in ref. [20], to whichwe refer for an account of their mathematical properties. The orthotoric K¨ahler metric reads g d s B = η − ξ F ( ξ ) d ξ + F ( ξ ) η − ξ (dΦ + η dΨ) + η − ξ G ( η ) d η + G ( η ) η − ξ (dΦ + ξ dΨ) , (3.1)where F ( ξ ) and G ( η ) are a priori arbitrary functions. Note that ∂/∂ Φ, ∂/∂ Ψ are Killingvector fields, hence this is a co-homogeneity two metric. The K¨ahler form has a universalexpression, independent of F ( ξ ), G ( η ): X = 1 g d [( η + ξ )dΦ + ηξ dΨ] . (3.2)The term orthotoric means that the momentum maps η + ξ , ηξ for the Hamiltonian Killingvector fields ∂/∂ Φ, ∂/∂ Ψ, respectively, have the property that the one-forms d ξ , d η areorthogonal. As a consequence, the K¨ahler metric does not contain a d η d ξ term.It is convenient to introduce an orthonormal frame E = 1 g (cid:115) η − ξ F ( ξ ) d ξ , E = 1 g (cid:115) F ( ξ ) η − ξ (dΦ + η dΨ) ,E = 1 g (cid:115) η − ξ G ( η ) d η , E = 1 g (cid:115) G ( η ) η − ξ (dΦ + ξ dΨ) , (3.3) This ansatz was also considered in [21], however only the case F ( x ) = −G ( x ), where these are cubicpolynomials, was discussed there. In this case the metric (3.1) is equivalent to the Bergmann metric on SU (2 , /S ( U (2) × U (1)). Orthotoric metrics also appear in Sasaki-Einstein geometry: as shown in [22],the K¨ahler-Einstein bases of L p,q,r Sasaki-Einstein manifolds [23] are of this type. B = − E ∧ E ∧ E ∧ E . Then the K¨ahler form can be written as X = E ∧ E + E ∧ E . (3.4)For the complex two-form Ω we can takeΩ = X + iX = ( E − iE ) ∧ ( E − iE ) . (3.5)This satisfies the properties (2.5), (2.6), with the Ricci form potential given by P = F (cid:48) ( ξ )(dΦ + η dΨ) + G (cid:48) ( η )(dΦ + ξ dΨ)2( ξ − η ) . (3.6)Other formulae that we will need are the Ricci scalar R = g F (cid:48)(cid:48) ( ξ ) + G (cid:48)(cid:48) ( η ) ξ − η , (3.7)and its Laplacian ∇ R = g η − ξ [ ∂ ξ ( F ∂ ξ R ) + ∂ η ( G ∂ η R )] . (3.8) To construct the solution we plug our orthotoric ansatz in the supersymmetry equationsof section 2. Eq. (2.7) gives for the function ff = 24( η − ξ ) F (cid:48)(cid:48) ( ξ ) + G (cid:48)(cid:48) ( η ) . (3.9)In order to solve eq. (2.9) for ω , we need to first construct G + , G − . From eq. (2.10) weobtain G + = 18 g ( ∂ ξ H − ∂ η H ) ( E ∧ E − E ∧ E ) , (3.10)where we introduced the useful combination H ( η, ξ ) = g F (cid:48) ( ξ ) + G (cid:48) ( η ) η − ξ . (3.11)We recall that G − = gR (cid:80) I =1 λ I X I , and we have to compute the functions λ , λ , λ .Eq. (2.12) gives λ = 12 ∇ R − ∂ ξ H ∂ η H , (3.12)8here ∇ R was expressed in terms of orthotoric data above. In order to solve for λ , λ ,we have to analyse the constraint (2.18) on the K¨ahler metric. Plugging our ansatz in,we obtain the equation ∂ ξ (cid:2) F ∂ ξ H ∂ ξ ( ∂ ξ H + ∂ η H ) + F ∂ ξ (cid:0) ∇ R − ∂ ξ H ∂ η H (cid:1)(cid:3) + ∂ η (cid:2) G ∂ η H ∂ η ( ∂ ξ H + ∂ η H ) + G ∂ η (cid:0) ∇ R − ∂ ξ H ∂ η H (cid:1)(cid:3) = 0 . (3.13)This is a complicated sixth-order equation for the two functions F ( ξ ) and G ( η ), that wehave not been able to solve in general. However, we have found the cubic polynomialsolution G ( η ) = g ( η − g )( η − g )( η − g ) , F ( ξ ) = −G ( ξ ) + f ( ξ + f ) , (3.14)comprising six arbitrary parameters g , . . . , g , f , f . We thus continue assuming that F and G take the form (3.14). We can then solve eq. (2.14) for λ , λ . Assuming adependence on η, ξ only, the solution is λ + iλ = i g F (cid:48)(cid:48)(cid:48) + G (cid:48)(cid:48)(cid:48) ( η − ξ ) (cid:112) F ( ξ ) G ( η ) + g c + ic (cid:112) F ( ξ ) G ( η ) , (3.15)with c , c real integration constants. One can promote c + ic to an arbitrary anti-holomorphic function, however we will not discuss such generalisation in this paper (see [9]for an example where this has been done explicitly).We now have all the ingredients to solve eq. (2.9) and determine ω . The solution is ω = F (cid:48)(cid:48)(cid:48) + G (cid:48)(cid:48)(cid:48) g ( η − ξ ) (cid:110) (cid:2) F ( ξ ) + ( η − ξ ) (cid:0) F (cid:48) ( ξ ) − F (cid:48)(cid:48)(cid:48) ( ξ )( f + ξ ) (cid:1)(cid:3) (dΦ + η dΨ)+ G ( η )(dΦ + ξ dΨ) (cid:111) − F (cid:48)(cid:48)(cid:48) G (cid:48)(cid:48)(cid:48) g [( η + ξ )dΦ + ηξ dΨ] − c g (cid:16) I ξ d ξ F ( ξ ) + I η d η G ( η ) + ΦdΨ (cid:17) − c g [( I − I )dΦ + ( I − I )dΨ] + d χ , (3.16)where I = (cid:90) d η G ( η ) , I = (cid:90) d ξ F ( ξ ) , I = (cid:90) η d η G ( η ) , I = (cid:90) ξ d ξ F ( ξ ) . (3.17) This includes the case where g → G degeneratesto a polynomial of lower degree. Same for F . χ is an arbitrary locally exact one-form, which in the five-dimensional metriccan be reabsorbed by a change of the y coordinate. For F and G as in (3.14), the integrals I , . . . , I can be expressed in terms of the roots of the polynomials. We have: I = log( η − g ) g ( g − g )( g − g ) + cycl(1 , , , I = g log( η − g ) g ( g − g )( g − g ) + cycl(1 , , , (3.18)and similarly for I and I (although the roots of F in (3.14) expressed in terms of the pa-rameters g , . . . , g , f , f are less simple). Here, cycl(1 , ,
3) denotes cyclic permutationsof the roots.Note that if c (cid:54) = 0 then ω explicitly depends on one of the angular coordinates Φ , Ψ,hence the U (1) × U (1) symmetry of the orthotoric base is broken to a single U (1) in thefive-dimensional metric.Also note that when F (cid:48)(cid:48)(cid:48) + G (cid:48)(cid:48)(cid:48) = 0, namely the coefficients of the cubic terms in thepolynomials are opposite, expression (3.16) simplifies drastically. Then we see that theterm in d ω independent of c , c is proportional to the K¨ahler form X . Moreover thebase becomes K¨ahler-Einstein. This class of solutions was pointed out in [1], where it wasexplored for the case the base is the space SU (2 , /S ( U (2) × U (1)) endowed with theBergmann metric. This is the non-compact analog of C P with the Fubini-Study metric,and the corresponding solution with c = c = 0 is pure AdS .To summarise, we started from the orthotoric ansatz (3.1) for the four-dimensionalK¨ahler metric, studied the sixth-order constraint (2.18) and found a solution in terms ofcubic polynomials F , G containing six arbitrary parameters, cf. (3.14). We also providedexplicit expressions for P , f and ω ( cf. (3.6), (3.9), (3.16)), with the solution for ω containing the additional parameters c , c . Plugging these expressions in the metric (2.4)and Maxwell field (2.8), we thus obtain a supersymmetric solution to minimal gaugedsupergravity controlled by eight parameters. We now show that three of the six parametersin the polynomials are actually trivial in the five-dimensional solution. As a first thing, we observe that one is always free to rescale the four-dimensional K¨ahlerbase by a constant factor. This is because the spinor solving the supersymmetry equa-tion (2.3) is defined up to a multiplicative constant, and the spinor bilinears inherit such10escaling freedom. This leads to the transformation X I → κ X I , f → κ f , y → κ − y , d s B → κ d s B , P → P , ω → κ − ω , (3.19)where κ is a non-zero constant. Clearly this leaves the five-dimensional metric (2.4) andthe gauge field (2.8) invariant.Let us now consider a supersymmetric solution whose K¨ahler base metric d s B is inthe orthotoric form (3.1), with some given functions F ( ξ ) and G ( η ). Then we can usethe symmetry above to rescale these two functions. Indeed after performing the transfor-mation we have (d s B ) old = κ (d s B ) new , and the new K¨ahler metric is again in orthotoricform, with the redefinitions F old = κ − F new , G old = κ − G new , Φ old = κ Φ new , Ψ old = κ Ψ new . (3.20)Hence the overall scale of F and G is irrelevant as far as the five-dimensional solution isconcerned. A slightly more complicated transformation that we can perform is ξ old = κ ξ new + κ , η old = κ η new + κ , Ψ old = κ κ Ψ new , Φ old = κ ( κ Φ new − κ κ Ψ new ) , F old ( ξ old ) = κ − F new ( ξ new ) , G old ( η old ) = κ − G new ( η new ) . (3.21)with arbitrary constants κ (cid:54) = 0, κ (cid:54) = 0 and κ , such that κ κ = κ . It is easy to see thatthe new metric (d s B ) new is again orthotoric, though with different cubic functions F and G compared to the old ones.We conclude that a supersymmetric solution with orthotoric K¨ahler base is locallyequivalent to another orthotoric solution, with functions F new ( ξ ) = κ F old ( κ ξ + κ ) , G new ( η ) = κ G old ( κ η + κ ) . (3.22)Using this freedom, we can argue that three of the six parameters in our orthotoric solutionare trivial. In the next section we will show that the remaining ones are not trivial byrelating our solution with c = c = 0 to the solution of [6]. The authors of [6] provide a four-parameter family of AAdS solutions to minimal five-dimensional gauged supergravity. The generic solution preserves U (1) × U (1) × R sym-metry (where R is the time direction) and is non-supersymmetric. By fixing one of the11arameters, one obtains a family of supersymmetric solutions, controlled by the threeremaining parameters a, b, m . This includes the most general supersymmetric black holefree of closed timelike curves (CTC’s) known in minimal gauged supergravity, as well asa family of topological solitons. Generically, the supersymmetric solutions are 1/4 BPSin the five-dimensional theory, namely they preserve two real supercharges. For b = a or b = − a , the symmetry is enhanced to SU (2) × U (1) × R .We find that upon a change of coordinates the supersymmetric solution of [6] fits inour orthotoric solution, with polynomial functions F , G of the type discussed above. Indetail, the five-dimensional metric and gauge field strength of [6] match (2.4), (2.8), withthe data given in the previous section and c = c = 0. The change of coordinates is t CCLP = yθ CCLP = 12 arccos ηr = 12 ( a − b ) ˜ m ξ + 1 g [( a + b ) ˜ m + a + b + abg ] + 12 ( a + b ) ˜ m ,φ CCLP = g y − − a g ( a − b ) g ˜ m (Φ − Ψ) ,ψ CCLP = g y − − b g ( a − b ) g ˜ m (Φ + Ψ) , (3.23)where “CCLP” labels the coordinates of [6]. Here, we found convenient to trade m for˜ m = m g ( a + b )(1 + ag )(1 + bg )(1 + ag + bg ) − , (3.24)which is defined so that the black hole solution of [6] corresponds to ˜ m = 0. The cubicpolynomials F ( ξ ) and G ( η ) read G ( η ) = − a − b ) g ˜ m (1 − η ) (cid:2) (1 − a g )(1 + η ) + (1 − b g )(1 − η ) (cid:3) , F ( ξ ) = −G ( ξ ) − m ˜ m (cid:18) ag + bg ( a − b ) g + ξ (cid:19) , (3.25)and are clearly of the form (3.14). The function χ in (3.16) is χ = − g ˜ m . The Killingvector arising as a bilinear of the spinor (cid:15) solving the supersymmetry equation (2.3) is V = ∂∂y = ∂∂t CCLP + g ∂∂φ CCLP + g ∂∂ψ CCLP . (3.26) Note that the present orthotoric form of the solution in [6], which is adapted to supersymmetry, doesnot use the same coordinates of the Pleba´nski-Demia´nski-like form appearing in [24]. c = c = 0, the family of supersymmetric solutions we have constructed is (at leastlocally) equivalent to the supersymmetric solutions of [6].We checked that when either c or c (or both) are switched on, the boundary metric isno more conformally flat, hence the solution becomes AlAdS and is not diffeomorphic tothe c = c = 0 case. We have thus obtained a new two-parameter AlAdS deformation ofthe AAdS solutions of [6]. Choosing c (cid:54) = 0 , c = 0 and χ in (3.16) as χ = − g ˜ m + c g I I ,the boundary metric appears to be regular and of type Petrov III like that of [1, 8]. Itsexplicit expression in the coordinates of [6] is (below we drop the label “CCLP” on thecoordinates): d s = d s , CCLP + d s c , (3.27)where the undeformed boundary metric of [6], obtained sending gr → ∞ , isd s , CCLP = − ∆ θ Ξ a Ξ b d t + 1 g (cid:18) d θ ∆ θ + sin θ Ξ a d φ + cos θ Ξ b d ψ (cid:19) , (3.28)with Ξ a = 1 − a g , Ξ b = 1 − b g and∆ θ = 1 − a g cos θ − b g sin θ , (3.29)while the deformation is linear in c and readsd s c = c g ˜ m ( a − b ) a Ξ b (cid:0) gt (Ξ a + Ξ b ) − Ξ b φ − Ξ a ψ (cid:1)(cid:0) − g d t (Ξ a − Ξ b ) − Ξ b d φ + Ξ a d ψ (cid:1) × (cid:0) − (Ξ a cos θ + Ξ b sin θ ) g d t + Ξ a cos θ d ψ + Ξ b sin θ d φ (cid:1) . (3.30)It would be interesting to study further the regularity properties of these deformationsand see if they generalise the similar solutions of [1, 8, 9].Note that both the change of coordinates (3.23) and the polynomials (3.25) are singularin the limits ˜ m → b → a , while they remain finite when b → − a . (When we take b → ± a , it is understood that we keep m , and not ˜ m , fixed). As already mentioned, theseare physically relevant limits: ˜ m → b → ± a leads to solutions with enhanced symmetry. We clarify the singular limits in thenext section. See [25] for a discussion of the Petrov type of supersymmetric boundaries. A scaling limit and two special cases
In the following we show that a simple scaling limit of the orthotoric metric yields certain non -orthotoric K¨ahler metrics, that have previously been employed to construct super-symmetric solutions. We recover on the one hand the base metric considered in [9], and onthe other hand an SU (2) × U (1) invariant K¨ahler metric. This proves that our orthotoricansatz captures all known supersymmetric solutions to minimal five-dimensional gaugedsupergravity belonging to the timelike class. The procedure will also clarify the singularlimits pointed out in the previous section.We start by redefining three of the four orthotoric coordinates { η, ξ, Φ , Ψ } asΦ = ε φ , Ψ = ε ψ , ξ = − ε − ρ , (4.1)where ε is a parameter that we will send to zero. For the metric to be well-behaved inthe limit, we also assume that the functions F , G satisfy G ( η ) = ε − (cid:101) G ( η ) + O (1) , F ( ξ ) = ε − (cid:101) F ( ρ ) + O ( ε − ) , (4.2)where (cid:101) G ( η ), (cid:101) F ( ρ ) are independent of ε and thus remain finite in the limit. Plugging thesein the orthotoric metric (3.1) and sending ε → g d s B = g lim ε → d s = ρ (cid:101) F ( ρ ) d ρ + (cid:101) F ( ρ ) ρ (d φ + η d ψ ) + ρ (cid:18) d η (cid:101) G ( η ) + (cid:101) G ( η )d ψ (cid:19) . (4.3)This is a K¨ahler metric of Calabi type (see e.g. [20]), with associated K¨ahler form X = − g d [ ρ (d φ + η d ψ )] . (4.4)At this stage the functions (cid:101) F ( ρ ) and (cid:101) G ( η ) are arbitrary. Of course, for (4.3) to be thebase of a supersymmetric solution we still need to impose on (cid:101) F ( ρ ), (cid:101) G ( η ) the equationfollowing from the constraint (2.18).We next consider two subcases: in the former we fix (cid:101) F and recover the metric studiedin [9], while in the latter we fix (cid:101) G and obtain an SU (2) × U (1) invariant metric. Case 1.
We take (cid:101) F ( ρ ) = 4 ρ + ρ and subsequently redefine ρ = sinh ( gσ ). Then(4.3) becomesd s B = d σ + 14 g sinh ( gσ ) (cid:18) d η (cid:101) G ( η ) + (cid:101) G ( η )d ψ + cosh ( gσ )(d φ + η d ψ ) (cid:19) , (4.5)14hich is precisely the metric appearing in eq. (7.8) of [9] (upon identifying η = x and (cid:101) G ( η ) = H ( x )). In this case our equation (2.18) becomes (cid:0) (cid:101) G (cid:101) G (cid:48)(cid:48)(cid:48)(cid:48) (cid:1) (cid:48)(cid:48) = 0 , (4.6)that coincides with the constraint found in [9]. As discussed in [9], this K¨ahler basemetric supports the most general supersymmetric black hole solution free of CTC’s thatis known within minimal five-dimensional gauged supergravity. This is obtained fromthe supersymmetric solutions of [6] by setting ˜ m = 0. In fact, the limit ˜ m → ε → (cid:101) G ( η ) isa cubic polynomial [7, 9]. Particular non-polynomial solutions to eq. (4.6) were foundin [9], however in the same paper these were shown to yield unacceptable singularities inthe five-dimensional metric.
Case 2.
If instead we take (cid:101) G ( η ) = 1 − η and redefine η = cos θ , then the metric (4.3)becomes g d s B = ρ (cid:101) F ( ρ ) d ρ + (cid:101) F ( ρ ) ρ (d φ + cos θ d ψ ) + ρ (cid:0) d θ + sin θ d ψ (cid:1) , (4.7)with K¨ahler form X = − g d [ ρ (d φ + cos θ d ψ )] . (4.8)This has enhanced SU (2) × U (1) symmetry compared to the U (1) × U (1) invariant or-thotoric metric. It is in fact the most general K¨ahler metric with such symmetry andis equivalent, by a simple change of variable, to the metric ansatz employed in [2] toconstruct the first supersymmetric AAdS black hole free of CTC’s. The constraint (2.18)becomes a sixth-order equation for (cid:101) F ( ρ ). This is explicitly solved if (cid:101) F ( ρ ) satisfies thefifth-order equation16( (cid:101) F (cid:48) ) + 4 ρ (cid:16) (cid:101) F (cid:48)(cid:48) + ( (cid:101) F (cid:48)(cid:48) ) − ρ (cid:101) F (3) (cid:17) + 2 ρ (cid:101) F (cid:48) (cid:16) − − (cid:101) F (cid:48)(cid:48) − ρ (cid:101) F (3) + 3 ρ (cid:101) F (4) (cid:17) − (cid:101) F (cid:16) −
16 + 8 (cid:101) F (cid:48)(cid:48) − ρ (cid:101) F (3) + 4 ρ (cid:101) F (4) − ρ (cid:101) F (5) (cid:17) = 0 . (4.9) This can be seen starting from (3.23), (3.25) and redefining ˜ m = − α ( a − b ) ε and r = r +4 α ρ , wherewe are denoting α = r + (1+ ag + bg ) g and r = a + b + abgg . It follows that ξ = ε − ρ + O (1). Then implement-ing the scaling limit described above we get (cid:101) F ( ρ ) = 4 ρ + ρ and (cid:101) G ( η ) = (1 − η ) (cid:2) A + A + ( A − A ) η (cid:3) with A = − a g g α and A = − b g g α . This makes contact with the description of the supersymmetric blackholes of [6] given in [7, 9]. SU (2) × U (1)invariant five-dimensional metric and graviphoton. We find that a simple solution to (4.9)is provided by a cubic polynomial (cid:101) F ( ρ ) = f + f ρ + f ρ + f ρ , such that f + 3 f (1 − f ) = 0 . (4.10)Supersymmetric AAdS solutions with SU (2) × U (1) symmetry were also found in [5] andfurther discussed in [15]. It is easy to check that after scaling away a trivial parameter,the five-dimensional solution determined by (4.10) in fact reproduces the two-parameter“case B” solution given in [15, sect. 3.4]. In turn, the latter includes the black hole of [2],and a family of topological solitons for particular values of the parameters.The special case f = 0, f = 1 yields the most general K¨ahler-Einstein metricwith SU (2) × U (1) isometry; this has curvature R = − g f and is diffeomorphic tothe Bergmann metric only for f = 0. The corresponding SU (2) × U (1) invariant five-dimensional solution is “Lorentzian Sasaki-Einstein”: for f = 0 this is just AdS , whilefor f (cid:54) = 0 it features a curvature singularity at ρ = 0.In [10], a different solution of equation (4.9) was put forward, leading to a smoothAlAdS five-dimensional metric. The non-conformally flat boundary is given by a squashed S × R , where the squashing is along the Hopf fibre and thus preserves SU (2) × U (1)symmetry.A particular example of this ε → b → a limit in the map (3.23),(3.25) relating the solution of [6] and the one based on our orthotoric ansatz. In fact,taking b = a in the solutions of [6] yields precisely the solutions presented in [15, sect. 3.4].Note that since the black hole of [2] is obtained from the general solution of [6] bytaking ˜ m = 0 and b = a , it belongs both to our cases 1 and 2.In figure 1 we summarise the relation between different K¨ahler metrics and the corre-sponding AAdS solutions in five dimensions. In the case the charges are set equal, so that the two vector multiplets of the U (1) gauged theorycan be truncated away and the solution exists within minimal gauged supergravity. This can be seen starting from (3.23), (3.25), redefining b = a + 8(1 − a g ) (cid:2) g m (1+2 ag )(1+ ag ) − ag (cid:3) − ε after having re-expressed ˜ m as in (3.24), and implementing the scaling limit. This gives (cid:101) G ( η ) = 1 − η and a cubic polynomial (cid:101) F ( ρ ) satisfying (4.10). rthotoric SU(2)xU(1)invariant
Kahler base metrics (above) and corresponding known AAdS solutions (below), with therelevant references ←~ m b → a scalinglimit 1 scalinglimit 2 SU ( )× U ( ) invariant [2] Gutowski-Reall; [4] Klemm-Sabra; [6] CCLP; [9] Herdeiro; [15] CGLP Kählermetric of [9] b →− a solution of [6], generic a , b , m : Case A of [15]black hole of [6] b → a : black hole of [2] includes black hole of [2]Case B of [15] Figure 1: K¨ahler base metrics (above) and corresponding known AAdS solutions (below),with relevant references.
In this section we focus on a sub-family of the solution of [6], comprising “topologicalsolitons” with non-trivial geometry but no horizon.
A priori , these may be consideredas candidate gravity dual to pure states of SCFT’s defined on S × R . In section 5.1 weconsider the non-vanishing vacuum expectation values of the energy and R -charge of suchtheories, and we look for a possible gravity dual. The constraints from the superalgebranaturally lead us to consider a 1/2 BPS topological soliton, however a direct comparisonof the charges with the SCFT vacuum expectation values shows that these do not match.In section 5.2 we argue that in the dual SCFT certain background R -symmetry field mustbe turned on, implying a constraint on the R -charges and suggesting that the state dual tothe topological soliton is different from the vacuum. Finally, in section 5.3 we show thatthe one-parameter family of 1/4 BPS topological solitons presented in [6] is genericallyplagued with conical singularities. In this section we assess the possible relevance of the supergravity solutions discussedabove to account for the vacuum state of dual four-dimensional N = 1 SCFT’s definedon the cylinder S × R [26, 27, 14, 28, 29]. The field theory background is specified by around metric on S with radius r , and by a flat connection for a non-dynamical gaugefield A cs coupling to the R -current. Crucially, A cs is chosen in such a way that halfof the eight supercharges in the superconformal algebra commute with the Hamiltoniangenerating time translations on the cylinder. In this way we ensure that this half of the The label “cs” refers to the fact that this is the gauge field of the four-dimensional conformal super-gravity that determines how the SCFT is coupled to curved space. H susy , is related to the operator ∆generating dilatations in flat space as H susy = ∆ − r R , where R is the R -charge operator(see [14] for more details). We will call Q α , Q † α , α = 1 , , ) representation of the SU (2) left × SU (2) right group acting on S , and their anti-commutator is [30]12 {Q α , Q † β } = δ βα (cid:16) H susy − r R (cid:17) − r σ i βα J i left , (5.1)where the J i left , i = 1 , ,
3, generate the SU (2) left angular momentum and σ i are the Paulimatrices. The input from [14] is that the vacuum preserves all four Q supercharges, andthat the bosonic charges evaluate to (cid:104) H susy (cid:105) ≡ (cid:104) ∆ (cid:105) − r (cid:104) R (cid:105) = 1 r (cid:104) R (cid:105) = 427 r ( a + 3 c ) , (cid:104) J left (cid:105) = 0 , (5.2)where a , c are the SCFT central charges. In [31, 14], these a priori divergent quantitieswere proved free of ambiguities as long as their regularisation does not break supersym-metry. For this proof to hold, it is important that the supercharges are preserved whenthe Euclidean time is compactified.Guided by the information above, we infer that the dual supergravity solution shouldbe AAdS and preserve (at least) four supercharges. Moreover, it should allow for agraviphoton behaving as A → c d t at the boundary, where c is a constant chosen so thatthe asymptotic Killing spinors generating (5.1) do not depend on time; in our conventions,this must be c = − √ . Indeed, the general Killing spinor of AdS that solves (2.3) asymp-totically reduces to a Weyl spinor on the boundary which, in standard two-componentnotation, may be written as (cid:15) −−−→ r →∞ (cid:15) α = ( gr ) / (cid:16) e i ( √ c +1) gt ζ α + e i ( √ c − gt ( σ ¯ η ) α (cid:17) , (5.3)where ζ α and ¯ η ˙ α are arbitrary Weyl spinors on the S × R boundary, independent of t andtransforming as ( , ) and ( , ), respectively, under the action of SU (2) left × SU (2) right .We see that choosing c = − √ , half of the spinors become independent of time. These arethe spinors that should be preserved by the solution: asymptotically, they generate thesuperalgebra (5.1). Note that if we Wick rotate t → − iτ and compactify the Euclideantime τ , then the other half of the Killing spinors is not well-defined. Hence we should18egard Euclidean AAdS spaces (including pure AdS) with compact S × S boundary aspreserving at most four supercharges.Assuming to work in the context of type IIB supergravity on Sasaki-Einstein five-manifolds, we can translate in gravity units the value of the vacuum energy and R -chargegiven in (5.2) using the standard dictionary a = c = π g κ . We shall also fix the radiusof the boundary S to r = 1 /g for simplicity. Finally, we map the field theory vevsinto supergravity charges as (cid:104) ∆ (cid:105) = E , (cid:104) R (cid:105) = − √ g Q and (cid:104) J left (cid:105) = J left , where E is thetotal gravitational energy, Q the electric charge under the graviphoton and J left the leftangular momentum. We thus obtain the following expected values for the charges of thedual gravity solution: E = − √ Q = 89 π g κ , J left = 0 . (5.4)Note that the relation between E and Q and the vanishing of J left also follow from the factthat the subset of the AdS supercharges respected by the solution anticommute into [15] {Q sugra , Q † sugra } = E + √ Q + 2 gσ i J i left . (5.5)( J right instead appears in the anti-commutator of broken supercharges.) This clearlyreflects the field theory considerations above. While there exist different prescriptionsfor the computation of the energy in asymptotically AdS spacetimes, here we will baseour statements on the fact that this must be related to the charge Q as dictated by thesuperalgebra. Regarding the evaluation of Q , we will rely on the standard formula Q = 1 κ (cid:90) S ∗ F , (5.6)where the integration is performed over the three-sphere at the boundary. In general thisformula would contain an additional A ∧ F term, however our boundary conditions impose F → ;indeed the boundary is S × R and we are free to switch on a graviphoton component A t = c without introducing any pathology. However since F = 0 everywhere, the charge Q computed as in (5.6) obviously vanishes. A possible solution to this mismatch with (5.4)may come from a careful analysis of the compatibility between supersymmetry and the Recall that E is not the same as E susy = (cid:104) H susy (cid:105) , but the two quantities are related as E = (cid:104) ∆ (cid:105) = (cid:104) H susy (cid:105) = E susy . We hope this notation will not cause confusion in the reader. c = c = 0, hence we areleft precisely with the supersymmetric solutions of [6], controlled by the three parameters a, b, m . In addition, the need to preserve four supercharges imposes a + b = 0. Thisidentifies a two-parameter family of solutions with SU (2) × U (1) invariance, originallyfound in [4] and further studied in [15]. As already observed, the limit b → − a (at fixed m ) is smooth in the transformation (3.23), (3.25) between the coordinates of [6] and ourorthotoric coordinates, and yields t CCLP = y , θ CCLP = 12 arccos ηr = (cid:18) − α g q (cid:19) ( αgξ + q ) − α q ,φ CCLP = g y − αg (Φ − Ψ) , ψ CCLP = g y − αg (Φ + Ψ) , (5.7)where we renamed the surviving parameters as a = αq , m = ( q − α g ) q . (5.8)The polynomials F ( ξ ) and G ( η ) in the orthotoric metric now become G ( η ) = − αg (1 − η ) , F ( ξ ) = −G ( ξ ) − (cid:18) qαg + ξ (cid:19) . (5.9)The coordinates { t, θ, φ, ψ, r } used in [15] are reached by the further transformation y = t , η = cos θ , ξ = r αg , Φ = αg φ + 2 gt ) , Ψ = αg ψ . (5.10) See the “case A” solutions in section 3.3 of [15], with all charges set equal, q = q = q = q , so thatthe solution fits in minimal gauged supergravity. Refs. [6, 15] base their statements on the amount ofsupersymmetry of the solutions on a study of the eigenvalues of the Bogomolnyi matrix arising from theAdS superalgebra. We have done a check based on the integrability condition (given in (A.17) below) ofthe Killing spinor equation, and found agreement.
20n these coordinates, the five-dimensional metric readsd s = − r V B d t + d r V + B (d ψ + cos θ d φ + f d t ) + 14 ( r + q )(d θ + sin θ d φ ) , (5.11)where V = r + g ( r + q ) − g α r ( r + q ) , B = ( r + q ) − α r + q ) , f = 2 αr α − ( r + q ) , (5.12)and the graviphoton is A = √ r + q (cid:18) q d t − α (d ψ + cos θ d φ ) (cid:19) + c d t . (5.13)Here, θ ∈ [0 , π ], φ ∈ [0 , π ), ψ ∈ [0 , π ) are the standard Euler angles parameterising thethree-sphere in the S × R boundary at r = ∞ .We observe that although the metric (5.11) and the gauge field (5.13) are invariantunder the SU (2) left × U (1) right subgroup of the SU (2) left × SU (2) right acting on the S atinfinity, the solution does not fall in the ansatz of [2], and hence in case 2 of section 4,because the bilinears of the Killing spinors and the K¨ahler base metric (3.1) do not sharethe same symmetry. In particular, in the coordinates of [15] the supersymmetric Killingvector (3.26) reads V = ∂∂y = ∂∂t − g ∂∂φ , (5.14)which is invariant under SU (2) right and transforms under SU (2) left , while the metric isinvariant under SU (2) left × U (1) right . In fact, the bilinears of the two independent Killingspinors of this 1/2 BPS solution give rise to three Killing vectors, which generate SU (2) left and are SU (2) right invariant. We can now discuss the charges, computed using the method of [15]. From (5.6), thecharge under the graviphoton is found to be Q = − √ q π κ . (5.15)The angular momentum conjugate to a rotational Killing vector K µ is given by the Komarintegral J = κ (cid:82) S ∗ d K , where K = K µ d x µ . For the angular momentum J left conjugateto ∂∂φ we get J left = 0 , (5.16) It follows that the Killing vector ∂∂t + 2 g ∂∂ψ put forward in [15] does not arise as a bilinear of theKilling spinors of the solution. J right , conjugate to ∂∂ψ , is controlled by α and reads J right = 2 α π κ . (5.17)Finally, the energy was computed in [15] by integrating the first law of thermodynamics,with the result E = − √ Q = 6 q π κ . (5.18)These values of the charges are in agreement with the superalgebra (5.5). It thus remainsto check the numerical value of Q against the expected one in (5.4). Whether these matchor not depends on the value of the parameter q . In order to see how this must be fixed,we need to discuss the global structure of the solution.Let us first observe that by setting the rotational parameter α = 0, the SU (2) × U (1)symmetry of (5.11), (5.13) is enhanced to SO (4). This solution was originally found in [3]and contains a naked singularity for any value of q (cid:54) = 0. So while the α = 0 limit providesthe natural symmetries to describe the vacuum of an SCFT on S × R , it yields a solutionthat for any q (cid:54) = 0 is pathological, at least in supergravity. In appendix A we prove thatthere are no other supersymmetric solutions with SO (4) × R symmetry within minimalgauged supergravity.It was shown in [15] that the two-parameter family of solutions given by (5.11), (5.13)contains a regular topological soliton (while there are no black holes free of CTC’s). Thisis obtained by tuning the rotational parameter α to the critical value α = q . (5.19)Then the metric (5.11) has no horizon, is free of CTC’s, and extends from r = 0 to ∞ .In addition, for the r, ψ part of the metric to avoid a conical singularity while it shrinksas r →
0, one has to impose q = 19 g . (5.20)In this way one obtains a spin c manifold with topology R × ( O ( − → S ), where thefirst factor is the time direction, and the second has the topology of Taub-Bolt space [15].Since √ gA is a connection on a spin c bundle, as it can be seen from (2.3), one must alsocheck the quantisation condition for the flux threading the two-cycle at r = 0. This reads12 π √ g (cid:90) S F ∈ Z + 12 , (5.21)22here the quantisation in half-integer units arises because the manifold is spin c ratherthan spin. One can check that12 π √ g (cid:90) S F = 32 g q / = 12 , (5.22)hence the condition is satisfied.We can then proceed to plug (5.20) into (5.15). This gives E = − √ Q = 23 π g κ , (5.23)which is different from (5.4). In field theory units, this reads (cid:104) R (cid:105) = a (cid:54) = a , where thelatter is the vev of the R -charge in a supersymmetric vacuum [14] (recall footnote 14). Weconclude that although this 1/2 BPS topological soliton is smooth and seemingly fullfillsthe requirements imposed by the field theory superalgebra, it is not dual to the vacuumstate of an SCFT on the S × R background. Below we will give further evidence that thissolution cannot describe the supersymmetric vacuum state of a generic SCFT on S × R . Firstly, we note that the non-trivial topology of the solution entails an obstruction to itsembedding into string theory, precisely analogous to the situation of the “bolt solutions”found in [32]. For instance, although the solution cannot be uplifted to type IIB super-gravity on S [15], there is a viable embedding if the orbifold S / Z is chosen instead. Wediscuss this issue in some detail in appendix B, where we also allow for a more generalLens space S / Z p topology for the spatial part of the boundary geometry.Further information comes from studying regularity of the graviphoton A in (5.13). Itwas noted in [15] that this is not well-defined as r →
0. Indeed, although F µν F µν remainsfinite, A µ A µ diverges as A µ A µ = qr + O (1) , (5.24)where we have used the critical value α = q . In order to cure this, one can introducetwo new gauge potentials, A (cid:48) and A (cid:48)(cid:48) , the first being well-defined around r = 0 , θ = 0,and the second being well-defined around r = 0, θ = π . These are related to the original23 by the gauge transformations A −→ A (cid:48) = A + √ q / (d ψ + d φ ) ,A −→ A (cid:48)(cid:48) = A + √ q / (d ψ − d φ ) . (5.25)It was claimed in [15] that the new gauge fields are not well-defined near to r = ∞ due toa singular term at order O (1 /r ), and for this reason a third gauge patch was introduced.However, we obtain a behavior different from the one displayed in eq. (3.37) of [15]. Wefind g µν A (cid:48) µ A (cid:48) ν = 6 q ( r + q )(1 + cos θ ) + . . . ,g µν A (cid:48)(cid:48) µ A (cid:48)(cid:48) ν = 6 q ( r + q )(1 − cos θ ) + . . . , (5.26)where the ellipsis denote a regular function of r only. The expressions are regular in therespective gauge patches. Hence it is not necessary to introduce a third gauge patch.Extending to infinity the two gauge patches introduced above, we obtain the boundaryvalues A (cid:48)∞ = c d t + √ q / (d ψ + d φ ) near θ = 0 ,A (cid:48)(cid:48)∞ = c d t + √ q / (d ψ − d φ ) near θ = π , (5.27)where A ∞ is the graviphoton evaluated at r = ∞ .Let us close this section with some comments on the interpretation of these flat fields inthe putative field theory duals. A ∞ is related to the background gauge field A cs couplingcanonically to the R -current of the dual SCFT by the conversion factor A cs = √ gA ∞ .Therefore, also using q / = g and c = − √ , A cs reads A cs = − g t + 14 (d ψ + d φ ) near θ = 0 ,A cs = − g t + 14 (d ψ − d φ ) near θ = π . (5.28)We see that in passing from the patch including θ = 0 to the one including θ = π , thegauge transformation A cs → A cs − d φ is performed. Correspondingly, the dynamical These gauge shifts have an opposite sign compared to those appearing in eq. (3.36) of [15]. To seethis, one has to recall that α = q / and take into account the different normalisation A here = −√ A there . e − i q R φ , where q R is their R -charge. Since allbosonic gauge invariant operators in the SCFT should be well-defined in both patchesas φ → φ + 2 π , we conclude that their R -charges must satisfy q R ∈ Z . We willmake further comments in appendix B, where we will show that for a number of concreteexamples this condition is automatically satisfied, after taking into account the constraintson the internal Sasaki-Einstein manifolds which follow from the conditions for upliftingthe topological soliton to type IIB supergravity.
We now consider the 1/4 BPS topological solitons mentioned in [6]. We will show thatunder the assumption that the boundary has the topology of S × R , there are no regulartopological solitons among the supersymmetric solutions of [6] apart for the 1/2 BPS onediscussed above.Let us start from the boundary of the solution in [6]. The boundary metric is obtainedby sending ( gr ) → + ∞ , and is given in (3.28). Requiring that this is (conformal to) thestandard metric on S × R fixes the range of the coordinates as θ ∈ [0 , π ], φ ∼ φ + 2 π and ψ ∼ ψ + 2 π . Moreover, requiring positivity of the spatial part of the boundary metric, cf. (3.28), we have that the parameters a , b should satisfy | ag | < , | bg | < . (5.29)As discussed in [6], the condition that there is no horizon fixes the parameter m as m = − (1 + ag )(1 + bg )(1 + ag + bg )(2 a + b + abg )( a + 2 b + abg ) . (5.30)Then the five-dimensional metric degenerates at r = − ( a + b + abg ) , (5.31)(since the solution only depends on even powers of r , it can be continued to negative r ).This is best seen by introducing a new radial coordinate( r (cid:48) ) = r − r , (5.32) This condition is reminiscent of the quantisation of the R -charges which is imposed on supersymmetricfield theories on S × T by an R -symmetry monopole through S , see e.g. [33]. Note that if we imposethe much more restrictive condition that the basic (scalar) fields of the gauge theory should have R -charge q R ∈ Z then all known dual field theories would be ruled out because these have a superpotential with R -charge 2, implying all scalar fields in the theory have R -charges < In the present subsection 5.3, we drop the label “CCLP” previously used to denote the coordinatesof [6]. One should recall anyway that these are not the same as the coordinates { r, θ, φ, ψ } of [15]appearing in subsections 5.1, 5.2. r (cid:48) = 0 to ∞ , and making the change of angular coordinates φ = φ (cid:48) ,ψ = ψ (cid:48) + (1 − bg )(2 a + b + abg )(1 − ag )( a + 2 b + abg ) φ (cid:48) . (5.33)As r (cid:48) →
0, the orbit of ∂∂φ (cid:48) shrinks to zero size, while the orbit generated by ∂∂ψ (cid:48) remainsfinite. Regularity requires that the orbit of ∂∂φ (cid:48) is closed, hence the fraction in (5.33)must be a rational number . In this case, it follows that the angles φ (cid:48) , ψ (cid:48) have the sameperiodicities as φ , ψ , namely 2 π . Then one can see that absence of conical singularitiesin the ( r (cid:48) , φ (cid:48) ) plane as r (cid:48) → (cid:20) ( a + b + abg )(3 + 5 ag + 5 bg + 3 abg )(1 − ag )( a + 2 b + abg ) (cid:21) = 1 . (5.34)However, this is not the only condition needed for regularity. Let us consider the boltsurface at r (cid:48) = 0 and t = const, parameterised by θ, ψ (cid:48) . One can see that as θ →
0, theleading terms of the metric on this surface ared s = b (2 a + b + abg ) ag − θ + b ( ag − a + 2 b + abg ) (1 − bg ) (2 a + b + abg ) θ (d ψ (cid:48) ) + . . . . (5.35)Therefore, the bolt has a conical singularity at θ = 0 unless the additional condition (cid:20) (1 − ag )( a + 2 b + abg )(1 − bg )(2 a + b + abg ) (cid:21) = 1 (5.36)is satisfied. Notice that this implies ψ = ψ (cid:48) ± φ (cid:48) , which is consistent with the assumptionwe made that the two angular variables be related by a linear rational transformation in(5.33). We have checked that the behaviour close to θ = π/ a = − b = ± g (5.37)(corresponding to ψ = ψ (cid:48) − φ (cid:48) in (5.33)) and a = b = − √ g (5.38)(corresponding to ψ = ψ (cid:48) + φ (cid:48) in (5.33)). Since a = ± b , the corresponding solutions have SU (2) × U (1) invariance and were discussed in [15]. The solution following from (5.37)corresponds to the 1/2 BPS topological soliton already discussed above, contained in the26case A” of [15, section 3.3]. Case (5.38) is contained in the “case B” solution of [15,section 3.4]. In both cases, the metric on the bolt is the one of a round two-sphere.Closer inspection reveals that only (5.37) is acceptable. Indeed, an additional con-straint on the parameters comes from considering the g θθ component of the five-dimensionalmetric in [6], that reads g θθ = ( r (cid:48) ) − ( a + b + abg ) + a cos θ + b sin θ − a g cos θ − b g sin θ . (5.39)This should remain positive all the way from r (cid:48) = ∞ down to r (cid:48) = 0. It is easy to checkthat while (5.37) does satisfy the condition, (5.38) does not, implying that the signatureof the metric changes while one moves towards small r (cid:48) .Therefore we conclude that among the supersymmetric solutions of [6], the only onecorresponding to a completely regular topological soliton with an S × R boundary is the1/2 BPS soliton of [15] that we considered above. In this paper, we studied supersymmetric solutions to minimal gauged supergravity in fivedimensions, building on the approach of [1]. We derived a general expression for the sixth-order constraint that must be satisfied by the K¨ahler base metric in the timelike class of [1], cf. (2.18). We then considered a general ansatz comprising an orthotoric K¨ahler base, forwhich the constraint reduced to a single sixth-order equation for two functions, each of onevariable, cf. (3.13). We succeded in finding an analytic solution to this equation, yieldinga family of AlAdS solutions with five non-trivial parameters. We showed that after settingtwo of the parameters to zero, this reproduces the solution of [6] and hence encompasses(taking into account scaling limits) all AAdS solutions of the timelike class that areknown within minimal gauged supergravity. This highlights the role of orthotoric K¨ahlermetrics in providing supersymmetric solutions to five-dimensional gauged supergravity.For general values of the parameters, we obtained an AlAdS generalisation of the solutionsof [6], of the type previously presented in [1, 8, 9] in more restricted setups. There exists afurther generalisation by an arbitrary anti-holomorphic function [1]; it would be interestingto study regularity and global properties of these AlAdS solutions.It would also be interesting to investigate further the existence of solutions to our“master equation” (3.13), perhaps aided by numerical analysis. In particular, our ortho-toric setup could be used as the starting point for constructing a supersymmetric AlAdS27olution dual to SCFT’s on a squashed S × R background, where the squashing of thethree-sphere preserves just U (1) × U (1) symmetry. This would generalise the SU (2) × U (1)invariant solution of [10].Finally, we have discussed the possible relevance of the solutions above to accountfor the non-vanishing supersymmetric vacuum energy and R -charge of a four-dimensional N = 1 SCFT defined on the cylinder S × R . The most obvious candidate for the gravitydual to the vacuum of an SCFT on S × R is AdS in global coordinates; however thiscomes with a vanishing R -charge. In appendix A we have performed a complete analysisof supersymmetric solutions with SO (4) × R symmetry, proving that there exists a uniquesingular solution, where the charge is an arbitrary parameter [3]. Imposing regularity ofthe solution together with some basic requirements from the supersymmetry algebra ledus to focus on the 1/2 BPS smooth topological soliton of [15]. A direct evaluation of theenergy and electric charge however showed that these do not match the SCFT vacuumexpectation values.We cannot exclude that there exist other solutions, possibly within our orthotoricansatz, or perhaps in the null class of [1], that match the supersymmetric Casimir energyof a four-dimensional N = 1 SCFT defined on the cylinder S × R . It would also be worthrevisiting the evaluation of the charges of empty AdS space, and see if suitable boundaryterms can shift the values of both the energy and electric charge, in a way compatiblewith supersymmetry.Although we reached a negative conclusion about the relevance of the 1/2 BPS topo-logical soliton to provide the gravity dual of the vacuum of a generic SCFT, our analysisclarified its properties, and may be useful for finding a holographic interpretation of thissolution. In appendix B we showed that the embedding of this solution into string theoryincludes simple internal Sasaki-Einstein manifolds such as S and T , . In particular, itshould be possible to match the supergravity solution to a field theory calculation in thecontext of well-known theories such as N = 4 super Yang-Mills placed on S / Z m × R and the Klebanov-Witten theory placed on S / Z m × R . Acknowledgments
We would like to thank Zohar Komargodski and Alberto Zaffaroni for discussions. D.C.is supported by an European Commission Marie Curie Fellowship under the contractPIEF-GA-2013-627243. J.L. and D.M. acknowledge support by the ERC Starting GrantN. 304806, “The Gauge/Gravity Duality and Geometry in String Theory.”28 SO (4) -symmetric solutions In this appendix, we present an analysis of solutions to minimal gauged supergravitypossessing SO (4) × R symmetry. In particular, we prove that the only supersymmetry-preserving solution of this type is the singular one found long ago in [3]. To our knowledge,a proof of uniqueness had not appeared in the literature before.This appendix is somewhat independent of the rest of the paper, and the notationadopted here is not necessarily related to that.The most general ansatz for a metric and a gauge field with SO (4) × R symmetry isd s = − U ( r )d t + W ( r )d r + 2 X ( r )d t d r + Y ( r )dΩ , (A.1) A = A t ( r )d t , (A.2)where dΩ is the metric on the round S of unit radius,dΩ = 14 (cid:0) σ + σ + σ (cid:1) ,σ = − sin ψ d θ + cos ψ sin θ d φ ,σ = cos ψ d θ + sin ψ sin θ d φ ,σ = d ψ + cos θ d φ . (A.3)The crossed term X ( r )d t d r in the metric can be removed by changing the t coordinate,so we continue assuming X ( r ) = 0. We will make use of the frame e = √ U d t , e , , = 12 √ Y σ , , , e = √ W d r . (A.4) Equations of motion
We proceed by first solving the equations of motion and then examining the additionalconstraint imposed by supersymmetry. The action and equations of motion are given byequations (2.1) and (2.2). With the ansatz (A.2), the Maxwell equation is0 = ∇ ν F νµ ⇔ A (cid:48)(cid:48) t + 12 A (cid:48) t (cid:18) log Y U W (cid:19) (cid:48) . (A.5)This can be integrated to A (cid:48) t = c (cid:114) U WY , (A.6)29ith c a constant of integration. The Einstein equations read (using frame indices) R = − R = 4 g + ( A (cid:48) t ) U W ,R = R = R = − g + ( A (cid:48) t ) U W , (A.7)where the Ricci tensor components are R = U (cid:48)(cid:48) U W − U (cid:48) W (cid:48) U W + 3 U (cid:48) Y (cid:48) U W Y − U (cid:48) U W ,R = R = R = − U (cid:48) Y (cid:48) U W Y + W (cid:48) Y (cid:48) W Y − Y (cid:48)(cid:48) W Y − Y (cid:48) W Y + 2 Y ,R = − U (cid:48)(cid:48) U W + U (cid:48) W (cid:48) U W + U (cid:48) U W + 3 W (cid:48) Y (cid:48) W Y − Y (cid:48)(cid:48) W Y + 3 Y (cid:48) W Y . (A.8)To solve these, let us define T ( r ) = U ( r ) W ( r ) Y ( r ) . (A.9)Combining two of the Einstein equations yields,0 = R + R = 3 U T ( T (cid:48) Y (cid:48) − T Y (cid:48)(cid:48) ) , (A.10)which can be integrated to T ( r ) = c Y (cid:48) ( r ) , (A.11)with c (cid:54) = 0 a constant of integration. Using this, the angular components of the Einsteinequations can be integrated, yielding U ( r ) = 4 c + 4 c g Y + 1 Y c + c c Y , (A.12)with a third constant of integration c . This solves all the equations of motion.We can now use the freedom to redefine the radial coordinate to choose one of thefunctions. In particular, we can choose the function W ( r ) so that W U = 4 s , where wetake s >
0. From (A.9) and (A.11) we then obtain (cid:18) d Y d r (cid:19) = 4 s c Y ⇒ Y ( r ) = s c r , (A.13)where we used the freedom to shift r to set to zero an integration constant. Finally, afterperforming the trivial redefinitions r old = √ c s r new , U old = 4 c U new , t new = 2 √ c t old , wearrive at the solution d s = − U ( r )d t + 1 U ( r ) d r + r dΩ , (A.14) A = (cid:16) c − c r (cid:17) d t , (A.15)30ith U ( r ) = 1 + g r + c c r + c r , (A.16)and c another arbitrary constant. Hence, the solution depends on three constants: c ,which is essentially the charge, the ratio c /c , and c which is quite trivial but may playa role in global considerations. Supersymmetry
The integrability condition of the Killing spinor equation (2.3) is0 = I µν (cid:15) ≡ R µνκλ γ κλ (cid:15) + i √ (cid:0) γ [ µκλ + 4 γ κ δ λ [ µ (cid:1) ∇ ν ] F κλ (cid:15) + 148 (cid:0) F κλ F κλ γ µν + 4 F κλ F κ [ µ γ ν ] λ − F µκ F νλ γ κλ + 4 F κλ F ρ [ µ γ ν ] κλρ (cid:1) (cid:15) + ig √ (cid:0) F κλ γ κλµν − F κ [ µ γ ν ] κ − F µν (cid:1) (cid:15) + g γ µν (cid:15) , (A.17)where we used [ ∇ µ , ∇ ν ] (cid:15) = R µνκλ γ κλ (cid:15) . A necessary condition for the solution to preservesupersymmetry is that det Cliff I µν = 0 for all µ, ν , (A.18)where the determinant is taken over the spinor indices. This gives for the SO (4) × R invariant solution (in flat indices a, b ):det Cliff I ab = 9 (16 c c − c ) c r ab . (A.19)Hence, the supersymmetry condition is c c = − √ c , (A.20)where we fixed a sign without loss of generality. Plugging this back into (A.16), we have U ( r ) = (cid:18) − c √ r (cid:19) + g r . (A.21)This recovers a solution first found in [3]. It is also obtained from (5.11)–(5.13) by setting α = 0 and changing the radial coordinate. 31herefore we conclude that in the context of minimal gauged supergravity, the mostgeneral supersymmetric solution possessing SO (4) × R symmetry is the one-parameterfamily found in [3]. This preserves four supercharges and has a naked singularity. Itwould be interesting to determine if this can be acceptable in a string theory framework.
B Uplifting topological solitons to type IIB
In this appendix we discuss the uplift to type IIB supergravity of the 1/2 BPS topologicalsoliton of [15]. Recall that this is obtained from the solution in (5.11)–(5.13) by choosingthe rotational parameter as α = q and fixing the remaining parameter q so that conicalsingularities are avoided. Compared to section 5.1, we will consider the slightly moregeneral case where the spatial part of the boundary has the topology of S / Z p ratherthan just S . Then the periodicity of ψ is πp and the condition (5.20) on q becomes q = p g . (B.1)The four-dimensional hypersurfaces at constant t have the topology of O ( − p ) → S .These manifolds are spin for p even, while they are spin c for p odd.Locally, all solutions to five-dimensional minimal gauged supergravity can be embed-ded into type IIB supergravity on a Sasaki-Einstein five-manifold [34]. However, whenthe external spacetime has non-trivial topology one may encounter global obstructions.In particular, it was pointed out in [15] that the 1/2 BPS topological soliton cannot beuplifted when the internal manifold is S . Here we identify the Sasaki-Einstein manifoldsthat make the uplift of that solution viable.The truncation ansatz for the ten-dimensional metric reads [34]d s = g µν d x µ d x ν + 1 g (cid:18) d s ( M ) + 19 (d ζ + 3 σ − √ gA µ d x µ ) (cid:19) (B.2)Following the presentation in e.g. [35], the metricd s ( SE ) = d s ( M ) + (cid:18)
13 d ζ + σ (cid:19) (B.3)is Sasaki-Einstein, where d s ( M ) is an a priori local K¨ahler-Einstein metric, with K¨ahlertwo-form J = d σ . The contact one-form is d ζ + σ and the dual Reeb vector field is3 ∂∂ζ . The graviphoton A gauges the space-time dependent reparameterisations of ζ .32or eq. (B.2) to provide a good ten-dimensional metric, the one-form13 d ζ + σ − g √ A (B.4)must be globally defined, and this imposes a constraint on the choice of Sasaki-Einsteinmanifold. In particular, as discussed in [32] in a closely related scenario, the one-form(B.4) can be globally defined only if ζ is periodically identified, thus one can never upliftto irregular Sasaki-Einstein manifolds.Let us then assume for simplicity to have a regular Sasaki-Einstein manifold, that isa circle bundle over a Fano K¨ahler-Einstein manifold M , with Fano index I ( M ). Forexample, for M = C P the Fano index is I ( C P ) = 3, while for M = C P × C P it is I ( C P × C P ) = 2 (see e.g. [36]). Then, for any integer k that divides I , the period of ζ can be taken 2 πI/k , with the Sasaki-Einstein five-manifold being simply connected ifand only if k = 1. Thus, for M = C P , taking ζ to have period 6 π gives S , while for M = C P × C P , a ζ with period 4 π gives T , . If k divides I but is larger than one,a ζ with period 2 πI/k yields a Sasaki-Einstein manifold that is still regular, albeit notsimply connected. Defining ˜ ζ = kI ζ , so that ˜ ζ has canonical period 2 π , we see that for theten-dimensional metric (B.2) to be globally defined, the termd ˜ ζ − √ gkI A (B.5)must be a bona fide connection on a circle bundle, implying the quantisation condition √ gkI (cid:90) S F π ∈ Z . (B.6)Using the computation in (5.22) with q chosen as in (B.1), we obtain k pI ∈ Z . (B.7)This condition relates the topology of the boundary manifold R × S / Z p to the topologyof the internal manifold.Let us now provide some examples of choices that obey (B.7), together with somebrief comments on the field theory duals. We begin considering the case p = 1 as in themain body of the paper. For M = C P the condition (B.7) implies that k = 3. Thismeans that the Sasaki-Einstein manifold is S / Z , and we can put the dual field theoryon S × R . This a quiver gauge theory with three nodes and nine bi-fundamental fields,arising from D3 branes placed at the O ( − → C P singularity (see e.g. [37]). According33o the discussion at the end of section 5.2, the R -charges of gauge-invariant operators ofthis theory, that may be constructed as closed loops of bi-fundamental fields in the quiver,must be even integers. This is in fact automatic, since the shortest loops are superpotentialterms, that have R -charge precisely equal to 2. Similarly, for M = C P × C P condition(B.7) with p = 1 implies k = 2. Then the Sasaki-Einstein manifold is T , / Z , and we canput the dual field theory on S × R . This is a quiver with four nodes. In this case thereare two possibilities for the bi-fundamental fields and superpotential, known as “toricphases” related by Seiberg duality [38, 37]. Again, one can check that all loops in thequiver are superpotential terms, and therefore have R -charge 2. One can go through allremaining regular Sasaki-Einstein cases, where the base is a del Pezzo surface M = dP i ,with 3 ≤ i ≤
9, which all have Fano index I ( dP i ) = 1, implying k = 1. In fact, accordingto (B.7) these theories can be placed on S / Z p × R for any p ≥
1. For general p , theconstraint on R -charges of gauge invariant operators derived in section 5.2 is that thesemust be q R ∈ p Z . We have verified that for all four toric phases of the quivers dual tothe third del Pezzo surface M = dP , the shortest loops are again superpotential terms[38], and therefore satisfy this condition (for any p ).There is in fact a more geometric way of understanding the restriction on the choice ofinternal manifold Y , that is directly related to the field theory dual description. Assumingthat it is a regular Sasaki-Einstein, Y can be identified with the unit circle bundle in L = K k/I , where K denotes the canonical line bundle of the K¨ahler-Einstein manifold M . Then scalar BPS operators in the dual field theory are in 1-1 correspondence withholomorphic functions on the Calabi-Yau cone over Y . These correspond to holomorphicsections of L − n , with n ∈ N a positive integer. Converting into field theory background R -symmetry gauge field the connection term in (B.5) this reads − kI A cs , showing that the R -charge of the holomorphic functions is given by q R = kI n .Let us conclude illustrating these general comments in two concrete examples with I > p >
1. In particular, take M = C P , and consider placing the theory on S / Z m × R , thus picking p = 3 m . Then (B.7) can be solved for either k = 1 or k = 3.Choosing k = 1 we can consider the theory dual to S , namely N = 4 super Yang-Mills.The gauge invariant operators in this theory are constructed with the three adjoints asTr(Φ n I I Φ n J J Φ n K K ), I, J, K = 1 , ,
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