Gravitational duality, topologically massive gravity and holographic fluids
aa r X i v : . [ h e p - t h ] J un Gravitational duality, topologically massivegravity and holographic fluids
P.M. Petropoulos ∗ Centre de Physique Théorique, CNRS † , Ecole Polytechnique,91128 Palaiseau Cedex, France August 22, 2018
CPHT-RR017.0414A
BSTRACT
Self-duality in Euclidean gravitational set ups is a tool for finding remarkable geometries infour dimensions. From a holographic perspective, self-duality sets an algebraic relationshipbetween two a priori independent boundary data: the boundary energy–momentum tensorand the boundary Cotton tensor. This relationship, which can be viewed as resulting froma topological mass term for gravity boundary dynamics, survives under the Lorentzian sig-nature and provides a tool for generating exact bulk Einstein spaces carrying, among others,nut charge. In turn, the holographic analysis exhibits perfect-fluid-like equilibrium statesand the presence of non-trivial vorticity allows to show that infinite number of transportcoefficients vanish. ∗ [email protected] † Unité mixte UMR7644. ontents
Gravitational duality is known to map the curvature form of a connection onto a dual cur-vature form. It allows for constructing self-dual, four-dimensional, Euclidean-signature ge-ometries, which are in particular Ricci-flat. Many exact solutions to Einstein’s vacuum equa-tions have been obtained in this manner, such as Taub–NUT [1], Eguchi–Hanson [2, 3], orAtiyah–Hitchin [4] gravitational instantons.The remarkable integrability properties underlying the above constructions have createdthe lore that in one way or another, integrability is related with self-duality, in a general andsomewhat loose sense. In particular, this statement applies to conformal self-duality condi-tions, either for Kähler or for Einstein spaces, which have delivered many exact geometries(LeBrun, Fubini–Study, Calderbank–Pedersen, Przanowski–Tod, Tod–Hitchin, . . . [6–19]). Quoting Ward (1985, [5]):. . . many (and perhaps all?) of the ordinary or partial differential equations that are regarded asbeing integrable or solvable may be obtained from the self-duality equations (or its generalizations)by reduction. i.e. on the boundary data,(ii) whether its underlying integrability properties extend to Lorentzian three-dimensionalboundaries and allow to obtain exact bulk Einstein spaces, and (iii) what the physical contentis for a boundary fluid emerging from such exact bulk solutions.The aim of these lecture notes is to provide a tentative answer to the above questions.They exhibit our present understanding of the subject, as it emerges from our works [20–23].The exact reconstruction of the bulk Einstein geometry or, equivalently, the resummabilityof the Fefferman–Graham expansion are achieved assuming a specific relationship amongthe two a priori independent boundary data, which are the boundary metric g µν and theboundary momentum F µν interpreted as the boundary field theory energy–momentum ten-sor expectation value T µν : wT µν + C µν =
0. (1.1)Here C µν is the Cotton–York tensor of the boundary geometry. In the Euclidean case, (anti-)self-duality corresponds precisely to the choice w = ± k / κ ( k is related to the cosmologicalconstant, Λ = − k , and κ to Newton’s constant, κ = k / π G N ). Equation (1.1) appears as thenatural extension of this duality – and integrability, in the spirit of the above discussion –requirement, irrespective of the signature of the metric, with arbitrary real w . This answersquestions (i) and (ii). Furthermore, the boundary condition (1.1) can be recast as δ S δ g µν = S = S matter + w Z ω ( γ ) , (1.3)where S matter is the action of the holographic boundary matter and ω ( γ ) the Chern–Simonsdensity ( γ is the boundary connection one-form). The reader will have recognized the dy-namics of matter coupled to a topological mass term for gravity [24]. Exact bulk Einsteinspaces satisfying this boundary dynamics turn out to provide laboratories for probing trans-port properties of three-dimensional holographic fluids, and this is an important spin-off ofthe present analysis that will answer question (iii). We will here review some basic facts about gravitational duality and their application to thefilling-in problem, which can be considered as the ancestor of holography. All this will be The relation is T µν = κ F µν , given in Eq. (3.1). The Cahen–Debever–Defrise decomposition, more commonly known as Atiyah–Hitchin–Singer [25, 26], is a convenient taming of the 20 independent components of the Riemanntensor. In Cartan’s formalism, these are captured by a set of curvature two-forms ( a , b , . . . =
0, . . . , 3) R ab = d ω ab + ω ac ∧ ω cb = R abcd θ c ∧ θ d , (2.1)where { θ a } are a basis of the cotangent space and ω ab = Γ abc θ c the set of connection one-forms. We will assume the basis { θ a } to be orthonormal with respect to the metricd s = δ ab θ a θ b , (2.2)and the connection to be torsionless and metric – this latter statement is equivalent to ω ab = − ω ba , where the connection satisfiesd θ a + ω ab ∧ θ b =
0. (2.3)The general holonomy group in four dimensions is SO ( ) , and (2.2) is invariant underlocal transfromations Λ ( x ) such that θ a ′ = Λ − ab θ b ,under which the curvature two-form transform as R a ′ b = Λ − ac R cd Λ db .Both ω ab and R ab are antisymmetric-matrix-valued forms, belonging to the representation of SO ( ) .Four dimensions is a special case as SO ( ) is factorized into SO ( ) × SO ( ) . Both con-nection and curvature forms are therefore reduced with respect to each SO ( ) factor as × + × , where and are respectively the vector and singlet representations. Theconnection and curvature decomposition leads to ( λ , µ , ν , . . . =
1, 2, 3 and ǫ = Σ λ = (cid:18) ω λ + ǫ λµν ω µν (cid:19) , A λ = (cid:18) ω λ − ǫ λµν ω µν (cid:19) , (2.4) S λ = (cid:18) R λ + ǫ λµν R µν (cid:19) , A λ = (cid:18) R λ − ǫ λµν R µν (cid:19) . (2.5) See also [27] for a review. Note the transformation of the connection: ω a ′ b = Λ − ac ω cd Λ db + Λ − ac d Λ cb . S λ = d Σ λ − ǫ λµν Σ µ ∧ Σ ν , A λ = d A λ + ǫ λµν A µ ∧ A ν . (2.6)Usually S and A are referred to as self-dual and anti-self-dual components of the Rie-mann curvature. This follows from the definition of the dual forms (supported by the fullyantisymmetric symbol ǫ abcd ) ˜ R ab = ǫ a dbc R cd ,borrowed from Yang–Mills. Under this involutive operation, S remains unaltered whereas A changes sign. Similar relations hold for the components ( Σ , A ) of the connection.Following the previous reduction pattern, the basis of 6 independent two-forms can bedecomposed in terms of two sets of singlets/vectors with respect to the two SO ( ) factors: φ λ = θ ∧ θ λ + ǫ λµν θ µ ∧ θ ν , χ λ = θ ∧ θ λ − ǫ λµν θ µ ∧ θ ν .In this basis, the 6 curvature two-forms S and A are decomposed as SA ! = r φχ ! ,where the 6 × r reads: r = A C + C − B ! = W + C + C − W − ! + s I . (2.7)The 20 independent components of the Riemann tensor are stored inside the symmetricmatrix r as follows: • s = Tr r = A = B = R / is the scalar curvature. • The 9 components of the traceless part of the Ricci tensor S ab = R ab − R g ab ( R ab = R cacb )are given in C + = ( C − ) t as S = Tr C + , S λ = ǫ µνλ C − µν , S λµ = C + λµ + C − λµ − Tr C + δ λµ . • The 5 entries of the symmetric and traceless W + are the components of the self-dualWeyl tensor, while W − provides the corresponding 5 anti-self-dual ones. A remark is in order here for D = ψ αβγ α , β , γ ∈ {
1, . . . , 7 } andthe dual G -invariant antisymmetric symbol ψ αβγδ allow to define a duality relation in 7 and 8 dimensions withrespect to an SO ( ) ⊃ G , and an SO ( ) ⊃ Spin respectively. Note, however, that neither SO ( ) nor SO ( ) isfactorized, as opposed to SO ( ) .
4n summary, S λ = W + λ + s φ λ + C + λµ χ µ , (2.8) A λ = W − λ + s χ λ + C − λµ φ µ , (2.9)where W + λ = W + λµ φ µ , W − λ = W − λµ χ µ are the self-dual and anti-self-dual Weyl two-forms respectively.Given the above decomposition, the following nomenclature is used (see e.g. [27] fordetails): Einstein C ± = ⇔ R ab = R g ab ) Ricci flat C ± = s = Self-dual A = ⇔ { W − = C ± = s = } Anti-self-dual S = ⇔ { W + = C ± = s = } Conformally self-dual W − = Conformally anti-self-dual W + = Conformally flat W + = W − = on-shell Weyltensor , defined as the antisymmetric-matrix-valued two-from:ˆ W ab = R ab + k θ a ∧ θ b . (2.10)Decomposing the latter à la Atiyah–Hitchin–Singer, we obtain:ˆ W + λ = S λ + k φ λ = W + λ + (cid:0) s + k (cid:1) φ λ + C + λµ χ µ , (2.11)ˆ W − λ = A λ + k χ λ = W − λ + (cid:0) s + k (cid:1) χ λ + C − λµ φ µ . (2.12)A quaternionic space is such that either ˆ W + or ˆ W − vanish. A round three-sphere is a positive-curvature, maximally symmetric Einstein space with SU ( ) × SU ( ) isometry. Its metric can be expressed using the Maurer–Cartan forms of5 U ( ) : d Ω = (cid:0) σ (cid:1) + (cid:0) σ (cid:1) + (cid:0) σ (cid:1) (2.13)with σ = sin ϑ sin ψ d ϕ + cos ψ d ϑσ = sin ϑ cos ψ d ϕ − sin ψ d ϑσ = cos ϑ d ϕ + d ψ ;0 ≤ ϑ ≤ π , 0 ≤ ϕ ≤ π , 0 ≤ ψ ≤ π are the Euler angles.A hyperbolic four-space H is a negative-curvature, maximally symmetric Einstein space.It is a foliation over three-spheres and its metric reads:d s H = d r + k r + k r d Ω .(we assumed R ab = − k g ab for H ). The conformal boundary of H is reached at r → ∞ asd s H −→ r → ∞ k r d Ω .In this sense, the round three-sphere is filled-in with H , the latter being the only regularmetric filling-in this three-dimensional space.The natural question to ask in view of the above is how to fill-in the more general Bergersphere S , which is a homogeneous but non-isotropic deformation of (2.13):d Ω S = (cid:0) σ (cid:1) + (cid:0) σ (cid:1) + n k (cid:0) σ (cid:1) (2.14)with nk constant. This metric is invariant under SU ( ) × U ( ) , respectively generated bythe Killings ξ = − sin ϕ cot ϑ ∂ ϕ + cos ϕ ∂ ϑ + sin ϕ sin ϑ ∂ ψ ξ = cos ϕ cot ϑ ∂ ϕ + sin ϕ ∂ ϑ − cos ϕ sin ϑ ∂ ψ ξ = ∂ ϕ ,and ∂ ψ .LeBrun studied the filling-in problem in general terms [8] and showed that an analyticthree-metric can be regularly filled-in by a four-dimensional Einstein space that has self-dual(or anti-self-dual) Weyl tensor, i.e. by a quaternionic space. In modern holographic words,LeBrun’s result states that requiring regularity makes the boundary metric a sufficient pieceof data for reconstructing the bulk. Regularity translates into conformal self-duality, whicheffectively reduces by half the independent Cauchy data of the problem, as we will see inSect. 3.2. 6 .3 A concrete example LeBrun’s analysis is very general. We can illustrate it in the specific example of the Bergersphere S . We search therefore a four-dimensional foliation over S , which is Einstein. Thisleads to the Bianchi IX Euclidean Schwarzschild–Taub–NUT family on hyperbolic space ( i.e. with Λ = − k ): d s = d r V ( r ) + (cid:0) r − n (cid:1) (cid:16)(cid:0) σ (cid:1) + (cid:0) σ (cid:1) (cid:17) + n V ( r ) (cid:0) σ (cid:1) (2.15)with V ( r ) = r − n h r + n − Mr + k (cid:16) r − n r − n (cid:17)i , (2.16)where M and n are the mass and nut charge. Clearly the metric fulfills the boundary require-ment since d s −→ r → ∞ r d Ω S ,where d Ω S is given in (2.14).The family of solutions at hand depends on 2 parameters, M and n , of which only thesecond remains visible on the conformal boundary. In that sense, the bulk is not fully deter-mined by the boundary metric. However, regularity is not always guaranteed either, as d s is potentially singular at r = + n or r = − n (depending on whether the range for r is chosenpositive or negative). Actually, this locus coincides with the fixed points of the Killing vector ∂ ψ , generating the extra U ( ) . In the present case, these are nuts and they are removableprovided the space surrounding them is locally flat.In order to make the above argument clear, let us focus for concreteness on r = n (as-suming thus r > r = n + ǫ and expand the metric using momentarily ǫ as radialcoordinate: d s ≈ d ǫ V ( n ) + ǫ V ′ ( n ) + n ǫ (cid:0) d ϑ + sin ϑ d ϕ (cid:1) + n (cid:0) V ( n ) + ǫ V ′ ( n ) (cid:1) ( d ψ + cos ϑ d ϕ ) . (2.17)Clearly to reconstruct locally flat space we must impose V ( n ) = V ′ ( n ) = / n . Thefirst of these requirements is equivalent to M = n (cid:0) − k n (cid:1) , (2.18)and makes the second automatically satisfied. Under (2.18) and with τ = √ n ǫ (proper In four dimensions, the fixed locus of an isometry is either a zero-dimensional or a two-dimensional space.The first case corresponds to a nut, the second to a bolt, and both can be removable singularities under appro-priate conditions (see [28] for a complete presentation). s ≈ d τ + τ (cid:0) d ψ + d ϕ + d ϑ + ϑ d ψ d ϕ (cid:1) ,which is indeed R .We can similarly analyze the behavior around r = − n . We then reach the same conclu-sion, with an overall change of sign in condition (2.18). These conditions are nothing butconformal (anti-)self-duality requirements, as we see by computing the Weyl components ofthe curvature, W ± , in the decomposition (2.7): W ± = M ∓ n ( − k n )( r ∓ n ) − − .The regularity requirement for the family of Einstein spaces (2.15) is thus equivalent to de-mand the space be quaternionic. In that case, the boundary metric contains enough infor-mation for determining the bulk and solving thereby the filling-in problem for the Bergersphere.For the quaternionic Schwarzschild–Taub–NUT geometries (2.15) with (2.18), the func-tion V ( r ) in (2.16) reads: V ( r ) = r − nr + n (cid:2) + k ( r − n )( r + n ) (cid:3) .These geometries belong to the general class of Calderbank–Pedersen [19], which is the fam-ily of quaternionic spaces with at least two commuting Killing fields. They belong to a wideweb of structures, and are in particular conformal to a family of spaces, which are Kähler andWeyl-anti-self-dual with vanishing scalar curvature, known as LeBrun geometries [29]. Thelimit n → ∞ deserves a particular attention, as it corresponds to the pseudo-Fubini–Study metric on g CP = SU ( ) U ( ) . Further holographic properties of these geometries can be foundin [30, 31]. The filling-in problem was presented as the ancestor of holography in the sense that (i) itposes the problem of reconstructing the bulk out of the boundary and (ii) it raises the issueof regularity as a mean to relate a priori independent boundary data. The bonus is that in the The metrics at hand are sometimes called spherical Calderbank–Pedersen , because they possess in total fourKillings, of which three form an SU ( ) algebra. This is the non-compact Fubini–Study. The ordinary Fubini–Study corresponds to the compact CP = SU ( ) U ( ) and has positive cosmological constant. The work of LeBrun [8], quoted previously in the framework of the filling-in problem, ledFefferman and Graham to set up a systematic expansion for Einstein metrics in powers of aradial coordinate [32, 33]. The infinite set of coefficients are data of the boundary, expressedin terms of two independent ones: g µν and F µν . From a Hamiltonian perspective, with theradial coordinate as evolution parameter, g µν and F µν are Cauchy data of “coordinate” and“momentum” type. The former is of geometric nature, the latter is not. In the holographiclanguage, g µν corresponds to a non-normalizable mode and is the boundary metric, whereas F µν is related to a normalizable operator and carries information on the energy–momentum-tensor expectation value of the boundary field theory: T µν = k π G N F µν , (3.1)where G N is four-dimensional Newton’s constant.The method of Fefferman–Graham is well suited for holography and has led to impor-tant developments (see e.g. [34–36]). It nicely fits the gravito-electric/gravito-magnetic splitHamiltonian formalism of four-dimensional gravity [37, 38]. In the Euclidean, this formal-ism is basically adapted to the self-dual/anti-self-dual splitting of the gravitational degreesof freedom presented in Sect. 2.1.Let us summarize here the basic facts, leaving aside the rigorous and complete exhibitionthat can be found in the above references. In Palatini formulation, the four-dimensional(bulk) Einstein–Hilbert action reads: I EH = − π G N Z M ǫ abcd (cid:18) R ab + k θ a ∧ θ b (cid:19) ∧ θ c ∧ θ d .As we already mentioned, θ a , a = r , λ are basis elements of a coframe, orthonormal withrespect to the signature (+ η + +) . The first direction r is the holographic one, and x ≡ ( t , x , x ) are the remaining coordinates, surviving on the conformal boundary – with t ≡ x in the Euclidean instance ( η = + ). 9he most general form for the coframe is θ r = N d rkr , θ λ = N λ d r + ˜ θ λ ,whereas the Levi–Civita connection generally reads: ω r λ = q r λ d r + K λ , ω µν = − ǫ µνλ (cid:18) Q λ d rkr + B λ (cid:19) .Without loss of generality, we can make the following gauge choice: N = N µ = q r µ = Q ρ = s = d r k r + η µν ˜ θ µ ˜ θ ν . (3.2)The connection is encapsulated in K µ and B λ . In Euclidean signature ( η = + ), these arevector-valued (with respect to the holonomy SO ( ) subgroups) connection one-forms, re-lated to the (anti-)self-dual ones introduced in (2.4): K λ = A λ + Σ λ , B λ = A λ − Σ λ . (3.3)The zero-torsion condition (2.3) translates in this language into K λ ∧ ˜ θ λ =
0d ˜ θ λ = kr K λ ∧ d r − ǫ λµν B µ ∧ ˜ θ ν . (3.4)With the present choice of gauge, all relevant information on the bulk geometry is storedinside (cid:8) ˜ θ λ , K λ , B λ (cid:9) . Assuming the metric be Einstein, leads to a very specific r -expansion ofthese vector-valued one-forms, in terms of the boundary data. This is the Fefferman–Grahamexpansion: ˜ θ λ ( r , x ) = kr E λ ( x ) + ∞ ∑ ℓ = ( kr ) ℓ + F λ [ ℓ + ] ( x ) , (3.5) K λ ( r , x ) = − k r E λ ( x ) + k ∞ ∑ ℓ = ℓ + ( kr ) ℓ + F λ [ ℓ + ] ( x ) , (3.6) B λ ( r , x ) = B λ ( x ) + ∞ ∑ ℓ = ( kr ) ℓ + B λ [ ℓ + ] ( x ) . (3.7)The boundary data are vector-valued one-forms. They are not all independent, and higher10rders are derivatives of lower orders (we will meet an example of this “horizontal” relation-ship in a short while). Furthermore, due to the zero-torsion condition (3.4), further “vertical”relations exist order by order amongst the three sets. This is manifest when comparing (3.5)and (3.6), where the relations are algebraic. The forms in (3.5) and (3.7) are also related, in adifferential manner, though.The form E λ is the boundary coframe. It is the first independent coefficient and it allowsto reconstruct the three-dimensional boundary metric:d s = lim r → ∞ d s k r = η µν E µ E ν .The one-form B µ appearing in the expansion of the magnetic component of the bulk connec-tion, Eq. (3.7), is the boundary Levi–Civita connection, differentially related to the coframe(boundary zero-torsion condition): d E λ = ǫ λµν B ν ∧ E µ .Other forms such as F µ [ ] = F µ [ ] ν E ν or B µ [ ] = B µ [ ] ν E ν are also geometric, respectively related tothe Schouten and Cotton–York tensors: S µν = − k F µν [ ] , C µν = k B µν [ ] . (3.8)There is again a differential relationship among the two, following basically from the bulkzero-torsion condition (3.4), since by definition C µν = η µρσ ∇ ρ S νσ (3.9)( η µρσ = ǫ µρσ / √ | g | ).Other curvature tensors of arbitrary order appear in the Fefferman–Graham expansion,all differentially related to the ones already described above. These tensors do not exhaust,however, all coefficients of the series (3.5), (3.6) and (3.7), as some infinite sequences of thoseare not of geometric nature, i.e. are not determined by the boundary metric itself (or bythe coframe E µ ). Instead, they follow differentially from the second independent piece ofdata, F µ ≡ F µ [ ] , related to the energy–momentum expectation value according to (3.1). Theinterested reader will find a more complete exhibition of the Fefferman–Graham expansionin the literature, and particularly in [37, 38] for the gravito-electric/gravito-magnetic splitformalism. In three dimensions, the Schouten tensor is defined as S µν = R µν − R g µν , whereas the Cotton–York tensoris the Hodge-dual of the Cotton tensor, defined in Eq. (3.9). The latter replaces the always vanishing three-dimensional Weyl tensor. In particular, conformally flat boundaries have zero Cotton tensor and vice versa. .2 Self-duality and its Lorentzian extension Riemann self-duality
A word on Riemann self-duality is in order at this stage, before ex-ploring the more subtle issue of Weyl self-duality.Demanding the Riemann tensor be (anti-)self-dual (see end of Sect. 2.1) guarantees Ricciflatness and Weyl (anti-)self-duality. Such a requirement on the curvature is easily trans-ported to the connection, using Eq. (2.6): the anti-self-dual connection K µ + B µ (see Eq.(3.3)) of a self-dual Riemann is either vanishing or a pure gauge (flat). This basically re-moves the corresponding degrees of freedom and gives an easy way to handle the problemvia first-order differential equations.The case of Bianchi foliations along the radial (holographic) direction, as the examplewe described in Sect. 2.3, has been largely analyzed in the literature (see [27] for a generaldiscussion, [39] for Bianchi IX, or [40–42] for a more recent general and exhaustive Bianchianalysis). The requirement of (anti-)self-dual Riemann leads to the following equation: K µ ± B µ = λ µν σ ν , (3.10)where σ ν are the Maurer–Cartan forms of the Bianchi group, and λ µν a constant matrix pa-rameterizing the homomorphisms mapping SO ( ) onto the Bianchi group. Expressing K µ and B µ in terms of the metric, (3.10) provides a set of first-order differential equations thathave usually remarkable integrability properties. For concreteness, in the case of BianchiIX ( SO ( ) ) foliations, λ µν = δ µν . The former case leads to the Lagrange equations,whereas the latter to the Darboux–Halphen system. Both systems are integrable, with cel-ebrated solutions such as Eguchi–Hanson or BGPP for the first [2, 3, 43], and Taub–NUT orAtiyah–Hitchin for the second [1, 4]. Weyl self-duality
Demanding Weyl (anti-)self-duality is not sufficient for setting K µ ± B µ as a pure gauge (flat connection). In the case of Bianchi foliations e.g. Eq. (3.10) is stillvalid but λ µν is a function of the radial coordinate r , and satisfies a first-order differentialequation. The general structure of this equation (independently of any ansatz such as aBianchi foliation) imposes a certain behavior and this is how Weyl (anti-)self-duality affectsboundary conditions in a way that becomes transparent in the Fefferman–Graham large- r expansion.We are specifically interested in quaternionic spaces, which are Einstein and conformally(anti-)self-dual. Thanks to the on-shell Weyl tensor (2.10), these requirements are simplyeither ˆ W + λ = W − λ = k (cid:2) ( ℓ + ) − (cid:3) F λ [ ℓ + ] ± ( ℓ + ) B λ [ ℓ + ] = ∀ ℓ ≥ + sign corresponds to the self-dual case), of which only the first is independent:3 kF λ [ ] ± B λ [ ] =
0. (3.11)The others follow from the already existing horizontal differential relationships. This al-gebraic equation between a priori independent boundary data is at the heart of conformalself-duality. In terms of the boundary energy–momentum and Cotton tensors (see (3.1) and(3.8)), Eq. (3.11) reads: 8 π G N k T µν ± C µν =
0. (3.12)Several important comments are in order at this stage. Firstly, referring to the originalproblem of Sect. 2.2, Eq. (3.11) provides the filling-in boundary condition for some a priori given boundary metric (not necessarily a three-sphere as originally studied in [8]). Thiscondition tunes algebraically the Cauchy data (“initial position” and “initial momentum”),in such a way that any boundary metric can be filled-in regularly . Following the intuitiondeveloped in the example of Sect. 2.3, we may slightly relax this condition and trade it for wT µν + C µν =
0, (3.13)where we now allow for any real w and not solely w = ± π G N k . The filling-in is stillexpected to occur, without guaranty for the regularity though.Secondly, as discussed in the introduction, duality is underlying integrability. This state-ment is clear in the case of Riemann self-duality, where the key is the reduction of the differ-ential order of the equations. For conformal self-duality it operates via an appropriate tuningof the boundary conditions, the effect of which would be better qualified as exactness ratherthan integrability: the equations of motion are not simplified, but the initial conditions selecta specific corner of the phase space, which enables for exact solutions to emerge, i.e. for theFefferman–Graham series to be resummable. Furthermore, even though self-duality (Rie-mann or Weyl) does not apply to the Lorentzian frame, condition (3.13) remains consistent When dealing with the Fefferman–Graham expansion together with Einstein dynamics, attention should bepayed to the underlying variational principle. This sometimes requires Gibbons–Hawking boundary terms tobe well posed. In the Hamiltonian language, these terms are generators of canonical transformations and inAdS/CFT their effect is known as holographic renormalization. These subtleties are discussed in [37, 38, 44–47],together with the specific role of the Chern–Simons boundary term, which produces the boundary Cotton tensor,and in conjunction with Dirichlet vs. Neumann boundary conditions. One should also quote the related works[48, 49], in the linearized version of gravitational duality though. In four-dimensional metrics with Lorentzian signature, self-duality leads either to complex solutions, or toMinkowski and AdS , which are both self-dual and anti-self-dual (they have vanishing Riemann and vanishingWeyl, respectively). S is the phenomenological holographicmatter action, whereas the second is the Chern–Simons term with ω the Lagrangian densitygiven in terms of the boundary connection one-form γ : ω ( γ ) =
12 Tr (cid:18) γ ∧ d γ + γ ∧ γ ∧ γ (cid:19) .Conceptually, this is a non-trivial step as holography is not supposed a priori to endow theboundary theory with gravitational dynamics. It raises three questions:1. What are the allowed boundary geometries, given certain assumptions on the energy–momentum tensor ?2. What are the bulk geometries that reproduce holographically the boundary data? Arethose exact Einstein spaces, i.e. is the corresponding Fefferman–Graham expansion re-summable in accordance with the above discussion ?3. Are there situations where gravitational degrees of freedom emerge?We will answer questions 1 and 2, at least in some specific framework, leaving open inter-esting extensions. As we will see, in some situations, the boundary geometry is really atopologically massive gravity vacuum – as if the three-dimensional Einstein–Hilbert termwere effectively present in (1.3). We will not delve into question 3, because this is a def-initely different direction of investigation. The interested reader may find Ref. [46] usefuland inspiring regarding that issue. The purpose of the present part is to answer questions 1 and 2 raised in Sect. 3.2. SolvingEq. (3.13) is possible, provided some assumptions are made both on the energy–momentumtensor, and on the boundary metric. These assumptions are motivated by our goal to probetransport coefficients for holographic fluids, without performing linear-response analysis.For that we must study equilibrium configurations of the fluid in various exact non-trivial14ackgrounds and design accordingly the boundary data. These satisfy Eq. (3.13) and areintegrable i.e. the corresponding Fefferman–Graham expansion is resummable.
Hydrodynamic description
A given bulk configuration (geometry possibly supplementedwith other fields) provides a boundary geometry, and a finite-temperature and finite-densitystate of the – generally unknown – microscopic boundary theory. It has expectation value T µν for the energy–momentum tensor, satisfying ∇ µ T µν =
0, (4.1)and possibly other conserved currents. This state may be close to a hydrodynamic configu-ration and is potentially described within the hydrodynamic approximation. This assumes,among others, local thermodynamic equilibrium. For this description to hold, it is necessarythat the scale of variation of the diverse quantities describing the fluid be large comparedto any microscopic scale (such as the mean free path). We will work in this framework andfurthermore suppose the fluid neutral, as the only bulk degrees of freedom are gravitationalin our case.The relativistic fluid is described in terms of a velocity field u ( x ) , as well as of localthermodynamic quantities like T ( x ) , p ( x ) , ε ( x ) , s ( x ) , obeying an equation of state and ther-modynamic identities sT = ε + p d ε = T d s .All these enter the energy–momentum tensor. The energy–momentum tensor of a neu-tral hydrodynamic system can be expanded in derivatives of the hydrodynamic variables,namely T µν = T µν ( ) + T µν ( ) + T µν ( ) + · · · , (4.2)where the subscript denotes the number of covariant derivatives. The validity of this deriva-tive expansion is subject to the above assumptions regarding the scale of variation. The ze-roth order energy–momentum tensor is the so called perfect-fluid energy–momentum ten-sor: T µν ( ) = ε u µ u ν + p ∆ µν , (4.3)where ∆ µν = u µ u ν + g µν is the projector onto the space orthogonal to u. This corresponds to afluid being locally in equilibrium, in its proper frame. The conservation of the perfect-fluid Defining the local proper frame, i.e. the velocity field u, is somewhat ambiguous in relativistic fluids. Apossible choice is the
Landau frame , where the non-transverse part of the energy–momentum tensor vanisheswhen the pressure is zero. This will be our choice. ∇ u ε + ( ε + p ) Θ = ∇ ⊥ p + ( ε + p ) a =
0, (4.4)where ∇ u = u · ∇ , Θ = ∇ · u, ∇ ⊥ µ = ∆ νµ ∇ ν , and a = u · ∇ u (more formulas on kinematicsof relativistic fluids are collected in App. A).The higher-order corrections to the energy–momentum tensor involve the transport co-efficients of the fluid. These are phenomenological parameters that encode the microscopicproperties of the underlying system. Listing them order by order requires to classify alltransverse tensors (possibly limited to traceless and Weyl-covariant if the microscopic the-ory is conformally invariant) and this depends on the space–time dimension. In the con-text of field theories, the transport coefficients can be determined from studying correlationfunctions of the energy–momentum tensor at finite temperature in the low-frequency andlow-momentum regime (see for example [54]).
Equilibrium and perfect equilibrium
Studying fluids at equilibrium on non-trivial back-grounds can provide information on their transport properties. A fluid in global thermo-dynamic equilibrium is described by a stationary solution of the relativistic equationsof motion (4.1), assuming that such solutions exist. Finding solutions to these equations isgenerally a hard task, in particular because most of the transport coefficients are unknown.As it will become clear in a short while, the concept of perfect equilibrium provides a naturalway out, giving access to non-trivial information about transport properties.The prototype example, where global thermodynamic description applies, is the one ofan inertial fluid in Minkowski background with globally defined constant temperature, en-ergy density and pressure. In this case, irrespective of whether the fluid itself is viscous, itsenergy–momentum tensor, evaluated at the solution, takes the zeroth-order (perfect) form(4.3) because all derivatives of the hydrodynamic variables vanish. On the one hand, thisequilibrium situation is easy to handle because the relevant equations are the zeroth-orderones, (4.4); on the other hand, it does not allow to learn anything about transport propertiesbecause the effect of transport is washed out by the geometry itself. If we insist keepingMinkowski as a background, the only way, which would give access to the transport coeffi- We recommend Refs. [50, 51] for a recent account of that subject. Insightful information was also madeavailable thanks to the developments on fluid/gravity correspondence [52, 53]. This should not be confused with a steady state, where we have stationarity due to a balance betweenexternal driving forces and internal dissipation. Such situations will not be discussed here. It is admitted that a non-relativistic fluid is stationary when its velocity field is time-independent. This is ofcourse an observer-dependent statement. For relativistic fluids, one could make this more intrinsic saying thatthe velocity field commutes with a globally defined time-like Killing vector, assuming that the later exists. Notealso that statements about global thermodynamic equilibrium in gravitational fields are subtle and the subjectstill attracts interest [55]. i.e. the energy–momentum tensor, in equilibrium, takesthe perfect form (4.3) solving Eqs. (4.4)?As anticipated, we call these special configurations perfect-equilibrium states . For theseconfigurations to exist, all terms in (4.2), except for the first one, must vanish, either becausethe transport coefficients are zero, or because the corresponding tensors vanish kinemat-ically – requiring in particular a special relationship between the fluid’s velocity and thebackground geometry. It should be stressed that the fluid in perfect equilibrium is not per-fect – the equilibrium is.At this stage of the presentation, the question to answer is whether fluids exist, whichcan exhibit, on certain backgrounds, perfect-equilibrium configurations. Holography andthe methods discussed in Sects. 2 and 3 for finding exact bulk solutions provide the tools forthis analysis. The strategy to follow is straightforward: • Choose a class of backgrounds possessing a time-like Killing vector ξ . • Assume perfect equilibrium and show that indeed perfect Euler Eqs. (4.4) are solvedfor a conformal fluid i.e. for a fluid such that ε = p . A hint for solving them is toimpose that the fluid velocity field u is aligned with ξ . • Impose the “self-duality” condition (3.13) and restrict the family of backgrounds athand. The three-dimensional geometries obtained in that way are called perfect geome-tries because their Cotton–York tensor is of the perfect-fluid form. • Use the Fefferman–Graham expansion to reconstruct the four-dimensional bulk geom-etry, hoping indeed that Eq. (3.13) acts as an integrability condition, allowing for re-summation of the series into an exact Einstein space. This is crucial for sustaining theclaim that we are describing a holographic conformal fluid behaving exactly as a perfectfluid.If this procedure goes through with genuinely non-trivial geometries, it enables us toprobe transport properties of the holographic fluid despite its global equilibrium state: alltransport coefficients coupled to Weyl-covariant, traceless and transverse tensors T µν that arenon-vanishing and whose divergence is also non-vanishing, when evaluated in the perfect-equilibrium solution, must be zero. We call such tensors dangerous tensors . Listing themrequires the knowledge of the specific perfect geometry and of the kinematic configurationof the fluid. Any fluid, which would have non-vanishing corresponding transport coeffi-cient, would not be in equilibrium in the configuration at hand. This may occur for transport More data are available on the dangerous tensors in certain classes of geometries in [23].
Perfect equilibrium in Papapetrou–Randers backgrounds
A stationary three-dimensionalmetric can be written in the generic form ( x = (cid:0) x , x (cid:1) and i , j , . . . =
1, 2)d s = B ( x ) (cid:16) − ( d t − b i ( x ) d x i ) + a ij ( x ) d x i d x j (cid:17) , (4.5)where B , b i , a ij are space-dependent but time-independent functions. These metrics wereintroduced by Papapetrou in [56]. They will be called hereafter Papapetrou–Randers becausethey are part of an interesting network of relationships involving the Randers form [57].These metrics admit a generically unique time-like Killing vector, ξ ≡ ∂ t , with norm k ξ k = − B ( x ) .At this stage of the analysis, we would like to restrict ourselves to the case where theKilling vector is normalized, i.e. where B is constant and can therefore be consistently set to1. This is a severe limitation, because it excludes equilibrium situations where the tempera-ture or the chemical potential are x -dependent. However, it illustrates the onset of perfectequilibrium configurations, and allows to establish a wide class of perfect geometries, inti-mately connected with holography.In the background (4.5) (with B = ξ = ∂ t , satisfies ∇ ( µ ξ ν ) = ξ µ ξ µ = − ξ have vanishing acceleration, Remember that inside a stationary gravitational field, under certain conditions, global thermodynamic equi-librium requires T √− g be constant [58]. Here √− g = B . Holographically, if the rescaling of the boundarymetric by B ( x ) (as in (4.5)) is accompanied with an appropriate rescaling of the energy–momentum tensor, thebulk geometry is unaffected, and B ( x ) is generated by a bulk diffeomorphism. ω = d ξ ⇔ ω µν = ∇ µ ξ ν . Then,it is easy to show that a solution of the perfect Euler equations (4.4), for a conformal fluid is:u = ξ , ε = p = constant, T = constant, s = constant. (4.6)Therefore a fluid in perfect equilibrium will align its velocity field u with the vector ξ = ∂ t ,while thermalize at everywhere-constant p and T . Fluid worldlines form a shearless andexpansionless geodesic congruence.The normalized three-velocity one-form of the fluid at perfect equilibrium isu = − d t + b, (4.7)where b = b i d x i . We will often write the metric (4.5) asd s = − u + d ℓ d ℓ = a ij d x i d x j . (4.8)A conformal fluid in perfect equilibrium on Papapetrou–Randers backgrounds has the energy–momentum tensor T ( ) µν d x µ d x ν = p (cid:0) + d ℓ (cid:1) (4.9)with the velocity form being given by (4.7) and p constant. We will adopt the conventionthat hatted quantities will be referring to the two-dimensional positive-definite metric a ij ,therefore ˆ ∇ for the covariant derivative and ˆ R ij d x i d x j = ˆ R d ℓ for the Ricci tensor built out of a ij . We collect in App. B some useful formulas regarding Papapetrou–Randers backgroundsand the kinematics of fluids at perfect equilibrium.Let us close this chapter by insisting once more on the meaning of the perfect-equilibriumconfiguration (4.6) for a conformal fluid that is not a priori perfect. For this configuration tobe effectively realized, all higher-derivative corrections in (4.2) must be absent. It is easy tocheck that this is indeed the case for the first corrections, which in the 2 + T µν ( ) = − ησ µν − ζ H η ρλ ( µ u ρ σ ν ) λ . (4.10)The first term in (4.10) involves the shear viscosity η , which is a dissipative transport coef-ficient. The second is present in systems that break parity and involves the non-dissipative Vorticity is inherited from the fact that ∂ t is not hypersurface-orthogonal. For this very same reason,Papapetrou–Randers geometries may in general suffer from global hyperbolicity breakdown. This occurs when-ever regions exist, where constant- t surfaces cease being space-like, and potentially exhibit closed time-likecurves. All these issues were discussed in detail in [20–22]. One important point to note is that in perfect equilibrium we have no frame ambiguity in defining thevelocity field. Since the velocity field is geodesic and is aligned with a Killing vector field of unit norm, itdescribes a unique local frame where all forces (like those induced by a temperature gradient) vanish. ζ H . Notice that the bulk-viscosity term ζ ∆ µν Θ or theanomalous term ˜ ζ ∆ µν η αβγ u α ∇ β u γ cannot appear in a conformal fluid because they are trace-full, namely for conformal fluids ζ = ˜ ζ =
0. Since the fluid congruence is shearless, the firstcorrections (4.10) vanish. Demanding that higher-order corrections also vanish, on the onehand, sets constraints on the transport coefficients coupled to the dangerous tensors that canbe constructed with the vorticity only; on the other hand, it leaves free many other coeffi-cients, which couple to tensors vanishing because of the actual kinematic state of the fluid. Ifthe transport coefficients coupled to the dangerous tensors are non-zero, the geodesic fluidcongruence with constant temperature is not a solution of the full Euler equations (4.1).The resolution of the latter alters the above perfect equilibrium state, leading in general tou = ξ + δ u ( x ) and T = T + δ T ( x ) . Such an excursion will be stationary or not depend-ing on whether the non-vanishing corrections to the perfect energy–momentum tensor arenon-dissipative or dissipative. The strategy
The analysis presented in Sect. 4.1 is useful if there exist conformal fluids,which are indeed in perfect equilibrium on a Papapetrou–Randers background. This is notguaranteed a priori since it requires infinite classes of transport coefficients to vanish. Holog-raphy provides the appropriate tools for addressing this problem. The strategy has alreadybeen described above, and the remaining two steps are the following:1. Impose condition (3.13) with perfect energy–momentum tensor and hence restrict thePapapetrou–Randers geometries to those which have a Cotton–York tensor of the perfect-fluid form (4.9): C µν = c ( u µ u ν + g µν ) , (4.11)where c is a constant with the dimension of an energy density. This form is known inthe literature as Petrov class D t . Notice that the existence of perfect geometries is anissue unrelated to holography.2. Sum the Fefferman–Graham series expansion. It turns out that the bulk geometries ob-tained in this way are exact solutions of Einstein’s equations: perfect-Cotton geometriesare boundaries of 3 + We recall that ε has dimensions of energy density or equivalently ( length ) − , therefore the energy–momentum tensor and the Cotton–York tensor have the same natural dimensions. The subscript t stands for time-like and refers to the nature of the vector u. For an exhaustive review onPetrov & Segre classification of three-dimensional geometries see [59] (useful references are also [60–62]). lassification of the perfect Papapetrou–Randers geometries Consider a metric of theform (4.5) with B ( x ) =
1. Requiring its Cotton–York tensor (B.3) to be of the form (4.11) isequivalent to impose the conditions: ˆ ∇ q + q ( δ − q ) = c , (4.12) a ij (cid:16) ˆ ∇ q + q ( δ − q ) − c (cid:17) = ˆ ∇ i ˆ ∇ j q , (4.13)ˆ R + q = δ (4.14)with δ being a constant relating the curvature of the two-dimensional base space, ˆ R , with thevorticity strength q (see App. B for definitions and formulas).It is remarkable that perfect-Cotton geometries always possess an extra space-like Killingvector. To prove this we rewrite (4.12) and (4.13) as (cid:18) ˆ ∇ i ˆ ∇ j − a ij ˆ ∇ (cid:19) q =
0. (4.15)Any two-dimensional metric can be locally written asd ℓ = Ω ( z , ¯ z ) d z d ¯ z , (4.16)where z and ¯ z are complex-conjugate coordinates. Plugging (4.16) in (4.15) we find that thenon-diagonal equations are always satisfied (tracelessness of the Cotton–York tensor), whilethe diagonal ones read: ∂ z q = ∂ z Ω ∂ z q , ∂ z q = ∂ ¯ z Ω ∂ ¯ z q .The latter can be integrated to obtain ∂ z q = e Ω − C ( ¯ z ) , ∂ ¯ z q = e Ω − C ( z ) (4.17)with C ( z ) an arbitrary holomorphic function and ¯ C ( ¯ z ) its complex conjugate. Trading thesefunctions for w ( z ) = Z e C ( z ) d z , ¯ w ( ¯ z ) = Z e C ( ¯ z ) d ¯ z ,and introducing new coordinates ( X , Y ) as X = w ( z ) + ¯ w ( ¯ z ) , Y = i ( ¯ w ( ¯ z ) − w ( z )) ,we find using (4.17) that the vorticity strength depends only on X : q = q ( X ) . Hence, (4.16)reads: d ℓ = ∂ X q (cid:0) d X + d Y (cid:1) . I thank Jakob Gath for clarifying this point. X .The presence of the space-like isometry actually simplifies the perfect-Cotton conditionsfor Papapetrou–Randers metrics. Without loss of generality, we take the space-like Killingvector to be ∂ y and write the metric asd s = − ( d t − b ( x ) d y ) + d x G ( x ) + G ( x ) d y . (4.18)Thus q = − ∂ x b ,and (4.12)–(4.14) can be solved in full generality. The solution is written in terms of 6 arbi-trary parameters c i : b ( x ) = c + c x + c x , (4.19) G ( x ) = c + c x + c x + c x ( c + c x ) . (4.20)It follows that the vorticity strength takes the linear form q ( x ) = − c − c x , (4.21)and the constants c and δ are given by: c = − c + c c − c c , (4.22) δ = c − c . (4.23)Finally, the Ricci scalar of the two-dimensional base space is given byˆ R = − ( c + c x ( c + c x )) ,and using (B.1) one can easily find the form of the three-dimensional scalar curvature aswell. Not all the six parameters c i correspond to physical quantities: some of them can bejust reabsorbed in a change of coordinates. In particular, we set here c = t → t + p y , with constant p , which does not change the form of themetric. The bulk duals of the perfect geometries
At this stage, the reader may wonder what theinterpretation of the parameters c i is. It is more convenient to answer that question afterunravelling the Einstein metrics that fit the boundary data (4.9), and (4.18) (with (4.19) and(4.20)). As already advertised, with these boundary data, the Fefferman–Graham series is22esummable because (4.9) and (4.11) satisfy the “self-duality” condition (3.13) with w = − c / ε .The resulting exact Einstein space reads, in Eddington–Finkelstein coordinates (where g rr = g r µ = − u µ ): d s = − (cid:18) d r − k G ( x ) ∂ x q d y (cid:19) + ρ k d ℓ − (cid:18) r k + δ k − q k − ρ (cid:16) Mr + qc k (cid:17)(cid:19) u (4.24)with u = − d t + b d y , (4.25) ρ = r + q k . (4.26)The various quantities appearing in (4.24)–(4.26), b ( x ) , G ( x ) , q ( x ) , c and δ , are reported inEqs. (4.19)–(4.23). Notice also that a coordinate transformation is needed in order to recast(4.24) in Boyer–Lindqvist coordinates, and a further one to move to the canonical Fefferman–Graham frame (3.2). Details can be found in [23], which we will not present here becausethey lie beyond the main scope of these lectures. Even though r is not the Fefferman–Grahamradial coordinate, in the limit r → ∞ , they both coincide. It is easy then to see that theboundary geometry is indeed the stationary Papapetrou–Randers metric (4.8), (4.18), andthat the boundary energy–momentum tensor is of the perfect-fluid form with ε = Mk π G N .Upon performing coordinate transformations and parameter redefinitions, one can showthat for c =
0, the bulk metrics at hand belong to the general class of Plebañski–Demia `nskitype D, analyzed in [63]. For vanishing c , depending on the other parameters, one findsthe flat-horizon solution of [64], or the rotating topological black hole of [65], or a set ofmetrics, which were found (but still not fully studied) in [23]. All these solutions are AdSblack holes, which have mass M , nut charge n and angular velocity a . The accelerationparameter, present in Plebañski–Demia `nski [63] is missing here. Actually, this parameter isan obstruction to perfect-Cotton boundary ( i.e. to D t Petrov–Segre class), and this is why itdoes not appear in our classification (see also [66]).For all these metrics, the horizon is spherical, flat or hyperbolic. The isometry groupcontains at least the time-like Killing vector ∂ t and the space-like Killing vector ∂ y . In theabsence of rotation, two extra Killing fields appear, which together with ∂ y generate SU ( ) , This is a local property. In the flat or hyperbolic cases, a quotient by a discrete subgroup of the isome-try group is possible and allows to reshape the global structure, making the horizon compact without conicalsingularities (a two-torus for example). SL ( R ) . The bulk metric is then a foliation over Bianchi IX, II or VIII homo-geneous geometries.From the explicit form of the bulk space–time metric (4.24), we observe that it can have acurvature singularity when ρ =
0. The locus of this singularity will then be at r = q ( x ) = ∂ t becomes null, at r ( x ) solution of r k + δ k − q k − ρ (cid:16) Mr + qc k (cid:17) = + + s = ρ ∆ r d r − ∆ r ρ ( d t + β d ϕ ) + ρ ∆ ϑ d ϑ + sin ϑ ∆ ϑ ρ ( a d t + α d ϕ ) (4.27)with ρ = r + ( n − a cos ϑ ) , ∆ r = k r + r ( + k a + k n ) − Mr + ( a − n )( + k n ) , ∆ ϑ = + k a cos ϑ ( n − a cos ϑ ) The Killing vector ∂ t is time-like and normalized at the boundary, where it coincides with the velocity fieldof the fluid, but its norm gets altered along the holographic coordinate, towards the horizon. β = − ( a − n + a cos ϑ ) Ξ sin ϑ / , α = − r + ( n − a ) Ξ , Ξ = − k a . Back to the boundaries and transport properties
The boundary physics depends on thesubset of those parameters among the c i s, which are non-trivial. The boundary metric is ingeneral a function of two parameters, n and a , whereas M appears in the boundary energy–momentum tensor. The bulk isometry group is conserved. Thus, in the absence of rotationparameter a =
0, the boundary is a homogeneous and stationary space–time: squashed S (including e.g. Gödel space), squashed Heisenberg or squashed AdS . The fluid undergoesa homogeneous rotation ( i.e. without center, monopolar ) with constant vorticity strength q .For non-vanishing a , the boundary space–time is stationary but has only spatial axialsymmetry. The vorticity is a superposition of a monopole and a dipole , and the fluid has nowa cyclonic rotation around the poles on top of the uncentered one.We give for illustration the boundary metric of the Kerr–Taub–NUT space–time withspherical horizon (4.27):d s = − ( d t + β d ϕ ) + k ∆ ϑ (cid:18) d ϑ + ∆ ϑ Ξ sin ϑ d ϕ (cid:19) . (4.28)For vanishing a , d ℓ is an ordinary two-sphere and b = − β d ϕ is a Dirac-monopole-likepotential. Switching-on a deforms axially the base space d ℓ , while it adds a dipole contri-bution to b. From the perspective of transport in holographic fluids, the purpose is to list thedangerous tensors carried by this kind of boundaries. The more tensors we have, the moreinformation we gain on vanishing transport coefficients: since the energy–momentum ten-sor that emerges holographically is perfect, any transport coefficient coupled to a dangeroustensor is necessarily zero.For the boundary metric (4.28), the vorticity strength, the Cotton prefactor and the scalarcurvature read: q = k ( n − a cos ϑ ) , c = k n (cid:0) + k (cid:0) n − a (cid:1)(cid:1) , R = k (cid:0) + k n + k na cos ϑ + k a (cid:0) − ϑ (cid:1)(cid:1) .We observe that, on the one hand, the nut charge n is responsible for the 2 + a , on the other hand,25ntroduces a ϑ -dependence in q and R . This betrays the breaking of homogeneity due to a : when a vanishes, the boundary is an squashed S with SU ( ) × R isometry, which is ahomogeneous space–time, and all of its scalars are constants. Coming back to the discussion on the dangerous tensors, we expect them to be morenumerous when less symmetry is present. Indeed, for vanishing a , all scalars are constantand both the Riemann and the Cotton are combinations of u µ u ν and g µν with constant co-efficients. Any covariant derivative acting on those will be algebrised in a similar fashion.Thus • all hydrodynamic scalars are constants, • all hydrodynamic vectors are of the form Au µ with constant A , and • all hydrodynamic tensors are of the form Bu µ u ν + Cg µν with constant B and C .Hence there exists no traceless transverse tensor that can correct the hydrodynamic energy–momentum tensor in perfect equilibrium. In other words, there is no dangerous tensor.Therefore, in the case of monopolar geometries, the symmetry is too rich and in such ahighly symmetric kinematical configuration, the fluid dynamics cannot be sensitive to anydissipative or non-dissipative coefficient. As soon as a dipole component is added ( a = Thus, thisprogramme lies outside of the present framework, as it requires to work with perturbedbulk Einstein spaces, and handle fluid perturbations potentially bringing the fluid awayfrom perfect equilibrium.
Monopolar geometries have been mentioned in Sect. 4.2 around the example (4.28), whichappears as the boundary of Taub–NUT Schwarzschild AdS black hole with spherical hori-zon. This terminology is justified by the fact that the vorticity strength q is constant (like This family includes Gödel space–time (see [67, 68] for more information). The important issue of closedtime-like curves emerges as a consequence of the lack of global hyperbolicity. This was discussed in Refs. [20–22], in relation with holographic fluids. When the bulk geometry has hyperbolic horizon, this caveat can becircumvented. In 1919, Weyl exhibited multipolar Ricci-flat solutions, which do not seem extendible to the Einstein case(see [69] for details). c = q , using the general equations(B.1) and (B.2) for Papapetrou–Randers, as well as (4.11)–(4.14) for perfect-Cotton geome-tries, we find: R = δ − q R µν d x µ d x ν = δ − q + (cid:18) δ − q (cid:19) d s , C µν d x µ d x ν = q (cid:0) δ − q (cid:1) (cid:0) + d s (cid:1) .These expressions can be combined into R µν − R g µν + λ g µν = µ C µν (5.1)with λ = δ − q
12 , µ = q λ and µ . This is not surprising, as it is a knownfact that, for example, squashed anti-de-Sitter or squashed three-spheres solve topologicallymassive gravity equations [59–62]. However, what is worth stressing here is that reversingthe argument and requiring a generic Papapetrou–Randers background (4.5) to solve (5.1)leads necessarily to a monopolar geometry. We leave as an exercise to set that result. As already advertised, the topological mass term (resulting from the Chern–Simons ac-tion in (1.3)) appears explicitly, in the cases under consideration, as part of topologicallymassive gravity equations. The reader might be puzzled by this connection. The 2 + a priori be considered as a sign ofdynamics. Nevertheless, as for the general “self-dual” case (Eqs. (3.13) obtained by varying(1.3)), we should leave open the option of introducing some topologically massive gravitondynamics on the boundary. This approach should not be confused with that of some recentworks [70, 71], where topologically massive gravity and its homogeneous solutions play therole of bulk geometries. Investigating the interplay between these two viewpoints might be Use the expression for the Ricci tensor for Papapetrou–Randers geometries (B.2), impose tracelessness andextract λ . Then use (B.3) and (4.20) and conclude that q must be constant and related to µ . Combine theseresults and reach the conclusion that all solutions are fibrations over a two-dimensional space with metric d ℓ of constant curvature ˆ R = λ − µ / . They are thus homogeneous spaces of either positive ( S ), null ( R ) ornegative curvature ( H ).
27f some relevance.
Modified versions of Einstein’s gravity are of interest primarily in cosmology. The aim ofthe present lectures is to set a bridge with a somewhat less expected area of applications,namely holography. Prior to holography we actually find, in four-dimensional Euclideanframework, quaternionic spaces. These, from the Fefferman–Graham viewpoint, require aboundary condition, which is obtained holographically as the extremization of S = S holographic matter + S Chern–Simons . (6.1)Assuming homogeneity for the boundary metric, further restricts (6.1) to the topologicallymassive gravity action, as shown in the last paragraph of these notes. Although, at this stage,only the extremum of this action is relevant, investigating boundary graviton dynamics inholographic set-ups might prove interesting in the future.Translating the bulk Weyl self-duality condition into boundary data opens up the pos-sibility to make it applicable for Lorentzian-signature bulk and boundary geometries. Thissort of integrability requirement is not necessary, however, and many Einstein spaces exist,which do not satisfy (3.13). Investigating further the relationships amongst the boundaryenergy–momentum tensor and the boundary Cotton tensor may be instructive in the caseof exact Einstein spaces, which fall outside of the class studied here. This could be usefulboth for understanding the underlying gravitational structure and for studying transportproperties in conformal holographic fluids.Besides potential generalizations of (3.13), appears also here the issue of the form of theboundary metric and of the energy–momentum tensor. Our analysis has been limited to (i)stationary Papapetrou–Randers boundary geometries (4.5) with B =
1, and (ii) perfect-fluid-like boundary energy–momentum tensors. These options make operational the determina-tion of vanishing transport coefficients by imposing perfect equilibrium, which turns out toexist holographically. We may however scan more general situations as many more exactEinstein spaces exist that deserve to be analyzed. We have already quoted in Sect. 4.2 thePlebañski–Demia `nski Einstein stationary solutions [63], for which the acceleration param-eter is a source of deviation from the perfect-Cotton boundary geometry. Non-stationaryspaces provide equally interesting laboratories for further investigation (see footnote 29).Finally, on the Euclidean side, a great deal of techniques (isomonodromic deformations,twistors, . . . ) have been developed for finding the families of quaternionic spaces quotedin Refs. [9, 10, 13, 15, 16, 18, 29] (see also [73] for a review). Among these, the Calderbank– As usual with instantons, self-duality selects ground states, but exact excited states can also exist. Recently this was discussed for a non-stationary solution of Einstein’s equations [72]. of the Kerr–Taub–NUT (4.27). Since this family contains more self-dual metrics than our exhaustive analysis of Sect. 4.2 has revealed, these metrics must neces-sarily lead to a non-perfect boundary energy–momentum tensor, potentially combined witha Papapetrou–Randers boundary geometry with non-constant B . Although this discussionis valid in the Euclidean and not all Euclidean solutions admit a real-time continuation, itshould help clarifying the landscape of self-duality holographic properties, and possibly beuseful for Lorentzian extensions.Last, but very intriguing, comes the limitation in the dimension. We have been analyz-ing four dimensional bulk geometries because our guideline was self-duality, which indeedexists in this (Euclidean) framework. It can however be generalized in eight-dimensionalspaces. There, it is known that the octonionic symbols Ψ ABCD allow to define a duality map:˜ R AB = Ψ ABCD R CD . Reducing the Riemann two-form R AB , which belongs to the of SO ( ) , with respect to Spin ⊂ SO ( ) leads to a self-dual component S and an anti-self-dual one A . Equations (2.8) and (2.9) are now traded for S = W φ + s φ + W χ , A = W χ + s χ + S φ ,where the singlet s is the scalar curvature, S is the traceless Ricci, and the W I are the threeirreducible components of the Weyl tensor. Riemann self-dual gravitational instantons, ob-tained by setting A =
0, are known to exist [75–79]. Those are Ricci flat. The question is stillopen to find Weyl self-dual Einstein spaces, by demanding S = W =
0. From theboundary perspective, W = Acknowledgements
The present notes will appear in the proceedings of the 7th Aegean summer school BeyondEinstein’s theoryof gravity, held in Paros, Greece, September 23 – 28, 2013. I wish to thankthe organizers of this school, where the lectures were delivered. The material presented hereis borrowed from recent or on-going works realized in collaboration with M. Caldarelli, C.Charmousis, J.–P. Derendinger, J. Gath, R. Leigh, A. Mukhopadhyay, A. Petkou, V. Pozzoli,K. Sfetsos, K. Siampos and P. Vanhove. I also benefited from interesting discussions with I.Bakas, D. Klemm, N. Obers and Ph. Spindel. The feedback from the Southampton Universitygroup was also valuable during a recent presentation of this work in their seminar. This In this case, (2.18) is traded for M = n (cid:0) − k (cid:0) n − a (cid:1)(cid:1) (see also [74]). A On vector-field congruences
We consider a manifold endowed with a space–time metric of the generic formd s = g µν dx µ dx ν = η µν E µ E ν (to avoid inflation of indices we do not distinguish between flat and curved ones). Con-sider now an arbitrary time-like vector field u, normalised as u µ u µ = −
1, later identifiedwith the fluid velocity. Its integral curves define a congruence which is characterised by itsacceleration, shear, expansion and vorticity (see e.g. [81, 82]): ∇ µ u ν = − u µ a ν + D − Θ∆ µν + σ µν + ω µν with a µ = u ν ∇ ν u µ , Θ = ∇ µ u µ , σ µν = ∆ ρµ ∆ σν (cid:0) ∇ ρ u σ + ∇ σ u ρ (cid:1) − D − ∆ µν ∆ ρσ ∇ ρ u σ = ∇ ( µ u ν ) + a ( µ u ν ) − D − ∆ µν ∇ ρ u ρ , ω µν = ∆ ρµ ∆ σν (cid:0) ∇ ρ u σ − ∇ σ u ρ (cid:1) = ∇ [ µ u ν ] + u [ µ a ν ] .The latter allows to define the vorticity form as2 ω = ω µν dx µ ∧ dx ν = du + u ∧ a. (A.1)The time-like vector field u has been used to decompose any tensor field on the manifold intransverse and longitudinal components. The decomposition is performed by introducingthe longitudinal and transverse projectors: U µν = − u µ u ν , ∆ µν = u µ u ν + δ µν , (A.2)where ∆ µν is also the induced metric on the surface orthogonal to u. The projectors satisfythe usual identities: U µρ U ρν = U µν , U µρ ∆ ρν = ∆ µρ ∆ ρν = ∆ µν , U µµ = ∆ µµ = D − Our conventions are: A ( µν ) = / (cid:0) A µν + A νµ (cid:1) and A [ µν ] = / (cid:0) A µν − A νµ (cid:1) . u µ a µ = u µ σ µν = u µ ω µν = u µ ∇ ν u µ = ∆ ρµ ∇ ν u ρ = ∇ ν u µ . B Papapetrou–Randers backgrounds and aligned fluids
In this appendix, we collect a number of useful expressions for stationary Papapetrou–Randers three-dimensional geometries (4.5) with B =
1, and for fluids in perfect equilibriumon these backgrounds. The latter follow geodesic congruences, aligned with the normalizedKilling vector ∂ t , with velocity one-form given in (4.7).We introduce the inverse two-dimensional metric a ij , and b i such that a ij a jk = δ ik , b i = a ij b j .The three-dimensional metric components read: g = − g i = b i , g ij = a ij − b i b j ,and those of the inverse metric: g = a ij b i b j − g i = b i , g ij = a ij .Finally, q | g | = √ a ,where a is the determinant of the symmetric matrix with entries a ij .Using (4.7) and (A.1) we find that the vorticity of the aligned fluid can be written as thefollowing two-form (the acceleration term is absent here) ω = ω µν d x µ ∧ d x ν =
12 db.The Hodge-dual of ω µν is ψ µ = η µνρ ω νρ ⇔ ω νρ = − η νρµ ψ µ .In 2 + ψ µ = qu µ ,31here, in our set-up, q ( x ) = − ǫ ij ∂ i b j √ a .It is a static scalar field that we call the vorticity strength , carrying dimensions of inverselength. Together with ˆ R ( x ) – the curvature of the two-dimensional metric d ℓ introduced in(4.8), the above scalar carries all relevant information for the curvature of the Papapetrou–Randers geometry. We quote for latter use the three-dimensional curvature scalar: R = ˆ R + q R µν d x µ d x ν = q + ˆ R + q ℓ − u d x ρ u σ η ρσµ ∇ µ q , (B.2)as well as the three-dimensional Cotton–York tensor: C µν d x µ d x ν = (cid:16) ˆ ∇ q + q ( ˆ R + q ) (cid:17) (cid:0) + d ℓ (cid:1) − (cid:16) ˆ ∇ i ˆ ∇ j q d x i d x j + ˆ ∇ q u (cid:17) − u2 d x ρ u σ η ρσµ ∇ µ ( ˆ R + q ) . (B.3) References [1] E.T. Newman, L. Tamburino and T.J. Unti,
Empty-space generalization of the Schwarzschildmetric , Journ. Math. Phys. (1963) 915.[2] T. Eguchi and A.J. Hanson, Self-dual solutions to Euclidean gravity , Annals Phys. (1979) 82.[3] T. Eguchi and A.J. Hanson,
Gravitational instantons , Gen. Rel. Grav. (1979) 315.[4] M.F. Atiyah and N.J. Hitchin, Low-energy scattering of non-abelian monopoles , Phys. Lett. (1985) 21.[5] R.S. Ward,
Integrable and solvable systems, and relations among them , Philos. Trans. R. Soc.London,
A315 (1985) 451.[6] G.W. Gibbons and C.N. Pope, CP as a gravitational instanton , Comm. Math. Phys. (1978) 239.[7] R.S. Ward, Self-dual space–times with cosmological constant , Commun. Math. Phys. (1980) 1.[8] C.R. Lebrun, H -space with a cosmological constant , Proc. R. Soc. Lond. A380 (1982) 171.329] H. Pedersen,
Eguchi–Hanson metrics with cosmological constant , Class. Quantum Grav. (1985) 579.[10] H. Pedersen, Einstein metrics, spinning top motions and monopoles , Math. Ann. (1986)35.[11] H. Pedersen and Y.S. Poon,
Hyper-Kähler metrics and a generalization of the Bogomolnyequations , Comm. Math. Phys. (1988) 569.[12] H. Pedersen and Y.S. Poon,
Kähler surfaces with zero scalar curvature , Class. Quant. Grav. (1990) 1707.[13] M. Przanowski, Killing vector fields in self-dual, Euclidean Einstein spaces with Λ =
0, J.Math. Phys. (1991), 1004.[14] K.P. Tod, A comment on a paper of Pedersen and Poon , Class. Quant. Grav. (1991) 1049.[15] K.P. Tod, Self-dual Einstein metrics from the Painlevé VI equation , Phys. Lett.
A190 (1994)221.[16] N.J. Hitchin,
Twistor spaces, Einstein metrics and isomonodromic deformations , J. Diff.Geom. (1995) 30.[17] R. Maszczyk, L.J. Mason and N.M.J. Woodhouse, Self-dual Bianchi metric and Painlevétranscendents , Class. Quant. Grav. (1994) 65.[18] D.M.J. Calderbank and H. Pedersen, Self-dual spaces with complex structures, Einstein–Weyl geometry and geodesics , Ann. Inst. Fourier (2000) 921.[19] D.M.J. Calderbank and H. Pedersen, Self-dual Einstein metrics with torus symmetry , J.Diff. Geom. (2002) 485.[20] R.G. Leigh, A.C. Petkou and P.M. Petropoulos, Holographic three-dimensional fluids withnon-trivial vorticity , Phys. Rev.
D85 (2012) 086010 [arXiv:1108.1393 [hep-th]].[21] R.G. Leigh, A.C. Petkou and P.M. Petropoulos,
Holographic fluids with vorticity and ana-logue gravity systems , JHEP (2012) 121 [arXiv:1205.6140 [hep-th]].[22] M.M. Caldarelli, R.G. Leigh, A.C. Petkou, P.M. Petropoulos, V. Pozzoli and K. Siampos,
Vorticity in holographic fluids , Proc. of Science
Corfu11 (2012) 076 [arXiv:1206.4351 [hep-th]].[23] A. Mukhopadhyay, A.C. Petkou, P.M. Petropoulos, V. Pozzoli and K. Siampos,
Holo-graphic perfect fluidity, Cotton energy–momentum duality and transport properties , JHEP (2014) 136 [arXiv:1309.2310 [hep-th]].[24] S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories , Ann. Phys. (1982) 372; Erratum-ibid. (1988) 406;
Three-dimensional massive gauge theories , Phys.Rev. Lett. (1982) 975.[25] M. Cahen, R. Debever and L. Defrise, A complex vectorial formalism in general relativity ,Journal of Mathematics and Mechanics, (1967) 761.3326] M.F. Atiyah, N.J. Hitchin and I.M. Singer, Self-duality in four dimensional Riemanniangeometry , Proc. Roy. Soc. London
A362 (1978) 425.[27] T. Eguchi, P.B. Gilkey and A.J. Hanson,
Gravitation, gauge theories and differential geome-try , Phys. Rept. (1980) 213.[28] G.W. Gibbons and S.W. Hawking, Classification of gravitational instanton symmetries ,Commun. Math. Phys. (1979) 291.[29] C.R. Lebrun, Counter-examples to the generalized positive action conjecture , Commun.Math. Phys. (1988) 591.[30] K. Zoubos,
Holography and quaternionic Taub–NUT , JHEP (2002) 037[arXiv:hep-th/0209235].[31] K. Zoubos,
A Conformally invariant holographic two point function on the Berger sphere ,JHEP (2005) 031 [arXiv:hep-th/0403292].[32] C. Fefferman and C.R. Graham,
Conformal invariants , in Elie Cartan et les mathéma-tiquesd’aujourd’hui, Astérisque, 1985, numéro hors série Soc. Math. France, Paris, 95.[33] C. Fefferman and C.R. Graham,
The ambient metric , arXiv:0710.0919 [math.DG].[34] K. Skenderis and S.N. Solodukhin,
Quantum effective action from the AdS/CFT correspon-dence , Phys. Lett.
B472 (2000) 316 [arXiv:hep-th/9910023].[35] S. de Haro, K. Skenderis and S.N. Solodukhin,
Holographic reconstruction of spacetimeand renormalization in the AdS/CFT correspondance , Commun. Math. Phys. (2001) 595[arXiv:hep-th/0002230].[36] I. Papadimitriou and K. Skenderis,
Thermodynamics of asymptotically locally AdS space-times , JHEP (2005) 004 [arXiv:hep-th/0505190].[37] D.S. Mansi, A.C. Petkou and G. Tagliabue,
Gravity in the + -split formalism I: holog-raphy as an initial value problem , Class. Quant. Grav. (2009) 045008 [arXiv:0808.1212[hep-th]].[38] D.S. Mansi, A.C. Petkou and G. Tagliabue, Gravity in the + -split formalism II: self-duality and the emergence of the gravitational Chern–Simons in the boundary , Class. Quant.Grav. , 045009 (2009) [arXiv:0808.1213 [hep-th]].[39] G.W. Gibbons and C.N. Pope, The positive action conjecture and asymptotically Euclideanmetrics in quantum gravity , Commun. Math. Phys. (1979) 267.[40] F. Bourliot, J. Estes, P.M. Petropoulos and Ph. Spindel, Gravitational instantons, self-duality and geometric flows , Phys. Rev.
D81 (2010) 104001 [arXiv:0906.4558 [hep-th]].[41] F. Bourliot, J. Estes, P.M. Petropoulos and Ph. Spindel,
G3-homogeneous gravitational in-stantons , Class. Quant. Grav. (2010) 105007 [arXiv:0912.4848 [hep-th]].3442] P.M. Petropoulos, V. Pozzoli and K. Siampos, Self-dual gravitational instantons and ge-ometric flows of all Bianchi types , Class. Quant. Grav. (2011) 245004 [arXiv:1108.0003[hep-th]].[43] V.A. Belinsky, G.W. Gibbons, D.N. Page and C.N. Pope, Asymptotically Euclidean BianchiIX metrics in quantum gravity , Phys. Lett. (1978) 433.[44] R.G. Leigh and A.C. Petkou,
Gravitational duality transformations on (A)dS , JHEP ,079 (2007) [arXiv:0704.0531 [hep-th]].[45] S. de Haro and A.C. Petkou, Holographic aspects of electric–magnetic dualities , J. Phys.Conf. Ser. (2008) 102003 [arXiv:0710.0965 [hep-th]].[46] S. de Haro,
Dual gravitons in AdS / CFT and the holographic Cotton tensor , JHEP (2009) 042 [arXiv:0808.2054 [hep-th]].[47] O. Miskovic and R. Olea, Topological regularization and self-duality in four-dimensionalanti-de Sitter gravity , Phys. Rev.
D79 (2009) 124020 [arXiv:0902.2082 [hep-th]].[48] I. Bakas,
Energy-momentum/Cotton tensor duality for AdS black holes , JHEP (2009)003 [arXiv:0809.4852 [hep-th]].[49] I. Bakas, Duality in linearized gravity and holography , Class. Quant. Grav. (2009) 065013[arXiv:0812.0152 [hep-th]].[50] P. Romatschke, New developments in relativistic viscous hydrodynamics , Int. J. Mod. Phys.
E19 (2010) 1 [arXiv:0902.3663 [hep-ph]].[51] P. Kovtun,
Lectures on hydrodynamic fluctuations in relativistic theories , J. Phys.
A45 (2012)473001 [arXiv:1205.5040 [hep-th]].[52] V.E. Hubeny, S. Minwalla and M. Rangamani,
The fluid/gravity correspondence ,arXiv:1107.5780 [hep-th].[53] M. Rangamani,
Gravity and hydrodynamics: lectures on the fluid-gravity correspondence ,Class. Quant. Grav. , 224003 (2009) [arXiv:0905.4352 [hep-th]].[54] G.D. Moore and K.A. Sohrabi, Kubo formulæ for second-order hydrodynamic coefficients ,Phys. Rev. Lett. (2011) 122302 [arXiv:1007.5333 [hep-ph]].[55] N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla and T. Sharma,
Constraints on fluid dynamics from equilibrium partition functions , JHEP (2012) 046[arXiv:1203.3544 [hep-th]].[56] A. Papapetrou,
Champs gravitationnels stationnaires à symétrie axiale , Ann. Inst. H.Poincaré A4 (1966) 83.[57] G. Randers, On an asymmetrical metric in the four-space of general relativity , Phys. Rev. (1941) 195.[58] L.D. Landau and E.M. Lifchitz, PhysiqueThéorique, Vol. 5 Physique statistique §27, MIR1969. 3559] D.D.K. Chow, C.N. Pope and E. Sezgin,
Classification of solutions in topologically massivegravity , Class. Quant. Grav. (2010) 105001 [arXiv:0906.3559 [hep-th]].[60] G. Guralnik, A. Iorio, R. Jackiw and S.Y. Pi, Dimensionally reduced gravitational Chern–Simons term and its kink , Annals Phys. (2003) 222 [arXiv:hep-th/0305117].[61] D. Grumiller and W. Kummer,
The Classical solutions of the dimensionally reduced gravita-tional Chern–Simons theory , Annals Phys. (2003) 211 [arXiv:hep-th/0306036].[62] G. Moutsopoulos and P. Ritter,
An exact conformal symmetry ansatz on Kaluza–Klein re-duced TMG , Gen. Rel. Grav. (2011) 3047 [arXiv:1103.0152 [hep-th]].[63] J.F. Plebañski and M. Demia `nski, Rotating, charged, and uniformly accelerating mass ingeneral relativity , Ann. Phys. (NY) (1976) 98.[64] N. Alonso-Alberca, P. Meessen and T. Ortin, Supersymmetry of topological Kerr–Newman–Taub–NUT–AdS space–times , Class. Quant. Grav. (2000) 2783 [arXiv:0003071 [hep-th]].[65] D. Klemm, V. Moretti and L. Vanzo, Rotating topological black holes , Phys. Rev.
D57 (1998)6127 [Erratum-ibid.
D60 (1999) 109902] [arXiv:gr-qc/9710123].[66] D. Klemm and A. Maiorana,
Fluid dynamics on ultrastatic spacetimes and dual black holes ,arXiv:1404.0176 [hep-th].[67] A.K. Raychaudhuri and S.N. Guha Thakurta,
Homogeneous space–times of the Gödel type ,Phys. Rev.
D22 (1980) 802.[68] M.J. Rebouças and J. Tiomno,
Homogeneity of Riemannian space–times of Gödel type , Phys.Rev.
D28 (1983) 1251.[69] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, Exact solutionsto Einstein’s field equations, Cambridge Monographs on Mathematical Physics, CUP2003.[70] D. Anninos, W. Li, M. Padi, W. Song and A. Strominger,
Warped AdS black holes , JHEP (2009) 130 [arXiv:0807.3040 [hep-th]].[71] D. Anninos, S. de Buyl and S. Detournay, Holography for a de Sitter–Esque geometry , JHEP (2011) 003 [arXiv:1102.3178 [hep-th]].[72] G.B. de Freitas and H.S. Reall,
Algebraically special solutions in AdS/CFT , arXiv:1403.3537[hep-th].[73] P.M. Petropoulos and P. Vanhove,
Gravity, strings, modular and quasimodular forms , An-nales Mathématiques Blaise Pascal (2012) 379 [arXiv:1206.0571 [math-ph]].[74] K. Behrndt, G. Dall’Agata, D. Lüst and S. Mahapatra, Intersecting six-branes from newseven manifolds with G holonomy , JHEP (2002) 027 [arXiv:hep-th/0207117].[75] B.S. Acharya and M. O’Loughlin, Self-duality in D ≤ eight-dimensional Euclidean gravity ,Phys. Rev. D55 (1997) 4521 [arXiv:hep-th/9612182].3676] E.G. Floratos and A. Kehagias,
Eight-dimensional self-dual spaces , Phys. Lett.
B427 (1998)283 [arXiv:hep-th/9802107].[77] I. Bakas, E.G. Floratos and A. Kehagias,
Octonionic gravitational instantons , Phys. Lett.
B445 (1998) 69 [arXiv:hep-th/9810042].[78] A. Bilal, J.-P. Derendinger and K. Sfetsos, (Weak) G holonomy from self-duality, flux andsupersymmetry , Nucl. Phys. B628 (2002) 112 [arXiv:hep-th/0111274].[79] R. Hernandez and K. Sfetsos,
An eight-dimensional approach to G manifolds , Phys. Lett. B536 (2002) 294 [arXiv:hep-th/0202135].[80] J. Zanelli,
Introduction to Chern–Simons theories , contribution to the 7th Aegean summerschool, Paros (GR), September 2013.[81] J. Ehlers,
Contributions to the relativistic mechanics of continuous media , Gen. Rel. Grav. (1993) 1225.[82] H. van Elst and C. Uggla, General relativistic + -orthonormal frame approach revisited ,Class. Quant. Grav.14