Separability in Consistent Truncations
aa r X i v : . [ h e p - t h ] F e b Separability in Consistent Truncations
Krzysztof Pilch , Robert Walker and Nicholas P. Warner , , Department of Physics and Astronomy,University of Southern California,Los Angeles, CA 90089-0484, USA Instituut voor Theoretische Fysica, KU Leuven,Celestijnenlaan 200D, B-3001 Leuven, Belgium Institut de Physique Th´eorique,Universit´e Paris Saclay, CEA, CNRS,Orme des Merisiers, F-91191 Gif sur Yvette, France Department of Mathematics,University of Southern California,Los Angeles, CA 90089, USA pilch @ usc.edu , robert.walker @ kuleuven.be, warner @ usc.edu
Abstract
The separability of the Hamilton-Jacobi equation has a well-known connection to the existenceof Killing vectors and rank-two Killing tensors. This paper combines this connection with thedetailed knowledge of the compactification metrics of consistent truncations on spheres. The factthat both the inverse metric of such compactifications, as well as the rank-two Killing tensors canbe written in terms of bilinears of Killing vectors on the underlying “round metric,” enables usto perform a detailed analyses of the separability of the Hamilton-Jacobi equation for consistenttruncations. We introduce the idea of a separating isometry and show that when a consistenttruncation, without reduction gauge vectors, has such an isometry, then the Hamilton-Jacobiequation is always separable. When gauge vectors are present, the gauge group is required tobe an abelian subgroup of the separating isometry to not impede separability. We classify theseparating isometries for consistent truncations on spheres, S n , for n = 2 , . . . ,
7, and exhibitall the corresponding Killing tensors. These results may be of practical use in both identifyingwhen supergravity solutions belong to consistent truncations and generating separable solutionsamenable to scalar probe calculations. Finally, while our primary focus is the Hamilton-Jacobiequation, we also make some remarks about separability of the wave equation. ontents S n , n = 1 , . . . , S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.4 Examples of non-separating isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 , m, n ) superstrata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.3 Separable U(1) × U(1) subsectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
C.1 The N = 1 flow in Section 6.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46C.2 The N = 2 flow in Section 6.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47C.3 The N = 1 flow in Section 6.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49C.4 The solution in Section 6.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50C.5 The (1 , , n ) superstata solution in Section 7.3.1 . . . . . . . . . . . . . . . . . . . . . . . 51C.6 The (1 , , n ) superstata solution in Section 7.3.2 . . . . . . . . . . . . . . . . . . . . . . . 532 Introduction
Geodesics and scalar waves represent the two simplest, and perhaps most useful geometric probes. Thegeometric optics approximation relates these probes in that, at high frequencies, or short wavelengths,the normals to wave fronts become tangents to geodesics. The wave equation ∇ M ∇ M Ψ = 1 p | G | ∂ M (cid:16)p | G | G MN ∂ N Ψ (cid:17) = − m Ψ , (1.1)reduces then to the Hamilton-Jacobi equation G MN ∂S∂x M ∂S∂x N = − m , (1.2)for the principal function S , where Ψ( x ) ∼ exp S . The analysis of these equations is, of course, greatlysimplified if there are isometries. For every Killing vector, K , generated by an isometry, there is aconserved momentum, p K ≡ K M ∂ M S , for the geodesics, and the equations (1.1) and (1.2) may be, atleast partially, separated by introducing coordinates along the symmetry directions,More generally, the wave equation and the corresponding Hamilton-Jacobi equation might haveadditional separation constants that go beyond those that are a consequence of the isometries. For thegeodesic problem, such separation constants correspond to non-trivial rank two Killing tensors, K MN ,satisfying ∇ ( M K NP ) = 0 , (1.3)that define non-trivial conserved quantities, which are quadratic in velocities: K MN dx M dλ dx N dλ . (1.4)Perhaps the most celebrated example of this was the discovery of the non-trivial Killing tensor in theKerr metric [1, 2].There is a further generalization of this story to the conformal structure of the manifold. There arecircumstances in which only the massless Hamilton-Jacobi equation is separable, then there are onlyconformal Killing vectors, ζ M , or conformal Killing tensors, ξ MN , defined by ∇ ( M ζ N ) = γ G MN , ∇ ( P ξ MN ) = η ( P G MN ) , (1.5)where G MN is the metric and γ = 1 n ∇ M ζ M , η M = 1 n + 2 (cid:0) ∇ M ( ξ N N ) + 2 ∇ P ξ P M (cid:1) . (1.6)Since the right-hand side of (1.5) vanishes when all the indices are contracted with a null tangent vector,such conformal Killing vectors and tensors provide conserved quantities along null geodesics. This stillprovides invaluable insight into the geometry of the background.The purpose of this paper is to highlight the fact that consistent truncations provide a fertile groundfor the discovery of non-trivial Killing and conformal Killing tensors. Indeed, we will exhibit severalconsistent truncations whose metrics have insufficient isometries to make separability manifest, and3et have fully separable massless Hamilton-Jacobi equations. Such metrics therefore have at least onenon-trivial conformal Killing tensor, as was discussed recently in [3].Conversely, holography and the study of microstate geometries has led to physically interestingmetrics that are generated by non-trivial solutions of consistent truncations. Such metrics are oftenvery complicated and seem rather intractable. The fact that some of them have conformal Killingvectors, or conformal Killing tensors, means that these geometries can be probed using scalar wavesand geodesics far more easily than one would expect.A consistent truncation in supergravity is the embedding of a lower-dimensional supergravity theoryinto a higher-dimensional one. To say that the truncation is “consistent,” means that if one solves theequations of motion in the lower-dimensional theory, then the result also leads to a solution of thehigher-dimensional theory, via an “uplift.” For this to occur, the space-time manifold, M , of the D -dimensional theory must be a fibration with a compact n -dimensional Riemannian manifold, F , overthe spacetime manifold, B , of the d -dimensional theory (and so D = d + n ).The simplest examples arise when F has a transitive symmetry group, G , and one can easily obtaina consistent truncation by restricting to the fields that are singlets under the action of G . This includesthe “trivial” torus compactifications. However, there are now well-known consistent truncations usingspheres in which one allows fields to have very specific, non-trivial (and non-singlet) dependence on thespherical harmonics. Such consistent truncations include the M-theory on S [4, 5] and on S [6, 7],respectively, type IIB supergravity on S [8–12], massive type IIA on S [13, 14], certain other spherereductions [15–17], and particular S compactifications of six-dimensional supergravities [18]. Indeed,it was the surprising separability of the massless wave equation for the “superstrata” solutions of six-dimensional supergravity [19,20], which were subsequently shown to be part of an S truncation in [21],that stimulated this more detailed study of separability in consistent truncations.A consistent truncation requires “uplift” formulae that precisely define how the dynamics of thefields on B are encoded into the dynamics of the fields on M . Once a solution of the theory on B is found, inserting it into the uplift ansatz must then, necessarily, give a solution to the equations ofmotion on M . Since we are interested in the geodesics and the Hamilton-Jacobi equation (1.2), we areonly going to need the metric uplift formula, that is, how the vectors, scalars and metric on B upliftto the complete metric on M . Fortunately there is a universal approach to obtaining such an upliftformula based on the Kaluza-Klein Ansatz using the techniques first developed in [4] and then used toconstruct the consistent truncations above.The general uplift formula for the metric will be discussed in Section 3.1. For now, we note that, inthe absence of Kaluza-Klein (KK) vector fields, the full metric takes the form of a warped product: ds M ≡ G MN dx M dx N = ∆ − d − ds B + ds F . (1.7)where ds B = g µν ( x ) dx µ dx ν is the metric on the d -dimensional base, B . A mixing between the baseand the fiber in the metric (1.7) arises from a nontrivial dependence of the internal metric, ds F = g mn ( x, y ) dy m dy n , and of the warp factor, ∆( x, y ). From the perspective of the lower-dimensional theory,these deformations of the internal metric, and warp factor, encode non-trivial scalar fields. Typically, ifall these scalar fields vanish then the internal metric, d ◦ s F = ◦ g mn ( y ) dy m dy n , is “round” in that it has atransitive isometry, and then the scalar fields on B are viewed as inducing a deformation of the “round4etric,” on F .More precisely, for a consistent truncation on a sphere S n , the internal metric is given by theuniversal formula for its inverse:∆ − d − g mn = 14 M ABCD ( x ) K m AB ( y ) K n CD ( y ) , (1.8)where K AB are the Killing vectors corresponding to the so ( n + 1) isometry of the round sphere metricand M ABCD is an so ( n + 1) tensor determined by the scalar fields. In particular, for the vanishingscalars, M ABCD = δ ABCD , ∆ = 1, and (1.8) reproduces the round sphere metric, ◦ g mn . Requiring thisround sphere metric to solve the supergravity equations defines the vacuum of the theory, and thechoices M ABCD and ∆ for this vacuum sets the normalization of the Killing vectors, K AB .We will show in this paper that there is a remarkable synergy between consistent truncations onspheres and the separability of the Hamilton-Jacobi equation. At its core, this synergy works becauseof the theorem that all rank two Killing tensors on a “round” sphere, or more generally any constantcurvature manifold, are linear combinations of bilinears in the Killing vectors [22–25]. This means that(1.8) can be decomposed into Killing tensors and thus the Hamilton-Jacobi equation can be separated tosome degree, depending on the details of those Killing tensors. We will show how this works in detail,establishing when a complete separation can be achieved and when the Hamilton-Jacobi equation isonly partially separable. There is a lot of literature on separability of the Hamilton-Jacobi equationsin general, but the new element in our work is the symbiosis with the form of the internal metrics inconsistent truncations.We will also introduce the idea of “separating isometries” as a powerful tool as well as a classificationtechnique. The idea is to consider subgroups, G , of the isometry group ◦ G of the “round” metric andconsider the most general consistent truncations that have G as an isometry. We find that separabilityof such a generic consistent truncation (in the absence of KK vector fields) is entirely determined by G , and if the massless Hamilton-Jacobi equation is fully separable we will refer to G as a separatingisometry.In the body of this paper, we will focus on the separation of the Hamilton-Jacobi equation, andreturn to the wave equation in Appendix A. We simply note here that if one broadens the notion ofseparability to encompass finite dimensional representations of gauge symmetries, then such a separationof the wave equation can be extended to include non-Abelian KK fields, and this may be importantto probing non-trivial supergravity solutions. However, outside Appendix A, we will use the standarddefinition of separability and apply it to the Hamilton-Jacobi equation. As we will discuss, this will limitus to Abelian KK fields. The separation of the Hamilton-Jacobi equation is, of course, also importantto probing non-supergravity solutions because it is directly related to the integrability of the geodesicproblem.In Section 2 we give a brief survey of the relevant ideas, and literature, on separability, and inSection 3 we review the relevant parts of consistent truncations. In particular, we recall the universaluplift formula for the metric of consistent truncations on spheres, including vector fields, and introducethe notion of “partial” separability of the Hamilton-Jacobi equation. The corresponding discussion ofpartial separability of massless scalar wave equations can be found in Appendix A. This is followed, inSection 4, by a pedagogical discussion of an explicit example coming from an SO(3) × SO(3) invariant5runcation of type IIB supergravity, which captures most of the generic features of our problem. Then inSection 5 we summarize some pertinent technical results for separability of Hamilton-Jacobi equationson Riemannian manifolds, introduce the idea of separable isometries and then use this to classifyisometries that lead to separable Hamilton-Jacobi equation in generic consistent truncations on S , S , . . . , S . This is illustrated with several explicit examples arising from known solutions of M-theory andtype IIB supergravity in Section 6. In Section 7 we go back to six-dimensional microstate geometriesand demonstrate separability in a large family of superstrata. A distinct feature here is that non-trivial Kaluza-Klein vector fields participate in the separation of variables. Finally, we conclude with adiscussion of our results and their significance in Section 8. Conditions for separability of the geodesic Hamilton-Jacobi equation, (1.2), on Riemannian andLorentzian manifolds have been well studied with the fundamental results going back to St¨ackel [26],Levi-Civita [27] and Eisenhart [28, 29]. Separability here means that (1.2) admits a complete solutionof the form: S ( x , . . . , x D ; c , . . . , c D ) = D X M =1 S M ( x M ; c , . . . , c D ) , det (cid:18) ∂ S∂z M ∂c N (cid:19) = 0 . (2.1)One can also consider multiplicative separability of the wave equation, (1.1), of the form: Ψ( z ) = D Y M =1 Ψ M ( x M , ˜ c , ˜ c , · · · ˜ c D ) , det ∂∂ ˜ c N (cid:16) Ψ ′ M Ψ M (cid:17) ∂∂ ˜ c N (cid:16) Ψ ′′ M Ψ M (cid:17) = 0 , (2.2)for a separate set of constants ˜ c M . It turns out that separability of the wave equation of the form(2.2) implies that the Hamilton-Jacobi equation (1.2) will separate additively as in (2.1). However, theconverse is not necessarily true. For orthogonal metrics it can be shown to occur when the Ricci-tensorhas vanishing off diagonal terms, while a full consideration for non-orthogonal systems are far moreinvolved. Following on the earlier work by Levi-Civita and Eisenhart, more recently, Kalnins and Miller(see, the monograph [31] and the references therein), Benenti (see, e.g., the review [32]), and othersgave a variety of methods for establishing separability of both the Hamilton-Jacobi equations and thecorresponding wave equations in a geometric and coordinate independent way with a central role playedby non-trivial (conformal) Killing tensors.A convenient framework for analyzing symmetries of the Hamilton-Jacobi equation (1.2) is to thinkof symmetric tensors on M as functions on the phase space, T ∗ M , using the map: K M ...M k ←→ K ≡ K M ...M k p M . . . p M k . (2.3) There is no sum over the index M in the completeness condition here, so that the determinant is of a 2 D × D matrix,see (4.1) of [30]. See [22] and the references therein for a discussion of these issues. G MN , is then defined as H M = 12 G MN p M p N , (2.4)and the (conformal) Killing vector/tensor equations, (1.3) and (1.5), are equivalent to the followingequations for the Poisson brackets, { H M , K} = 0 , { H M , ζ } = − γ H M , { H M , ξ } = − η H M . (2.5) We now assume that the metric has the form given in (1.7) and (1.8). It follows that the geodesicHamiltonian, H M , on M can be written as H M = Ω( x, y ) [ H B ( x ; π ) + H F ( x, y ; p ) ] , (2.6)where H B ≡ g µν π µ π ν is the geodesic Hamiltonian on the base and H F ≡
12 ∆ − d − g mn p m p n = 18 M ABCD K AB K CD , (2.7)is the Hamiltonian on the fiber. It is rather remarkable that the uplift formula for the metric (1.8)leads to a simple, factorized dependence of H M on the warp factor Ω = ∆ d − . This turns out crucialfor our analysis in two respects.First, when considering the massless Hamilton-Jacobi equation (1.2) with m = 0, the dependenceon the warp factor drops out and one is left with the dynamics described by a much simpler Hamiltonian H M + H F . In many physically relevant examples that we will discuss, such as holographic RG-flows ormicrostate geometries, the Hamilton-Jacobi equation for H B trivially separates in natural coordinateson B , and so the separability reduces to considering the fiber, F . The Hamiltonian H F can be expanded H F = X ω g ω ( x ) K ω ( y ; p ) , (2.8)where g ω ( x ) are linearly independent functions on the base and K ω ( y ; p ) are Killing tensors for theround metric on S n . In order to separate the massless Hamilton-Jacobi equation between the base andthe fiber, one must be able to set those Killing tensors to constants. This can be done consistently onlywhen { H F , K ω } = 0, and, for generic functions, g ω ( x ), this means that the Killing tensors in (2.8) mustbe in involution on S n : {K ω , K ω ′ } = 0 . (2.9)The full separation of the massless Hamilton-Jacobi equation reduces then to a well studied problem [33],but with an interesting twist. Since the geodesic Hamiltonian for the round metric, ◦ g mn , arises in theexpansion (2.8), one is looking for separating coordinates for the Hamilton-Jacobi equation on the roundsphere, S n , that simultanously provide consistent separation of all other terms in (2.8). Note that we define the fiber Hamiltonian using the rescaled metric ∆ − / ( d − g mn on F . G . In the higherdimensional theory, this symmetry becomes an isometry group of the uplifted metric (1.7) along thefiber, and restricts the tensors, K ω , in (2.8) to those invariant under G . This leads to a subtle interplaybetween the symmetry of a truncation and separability of the massless Hamilton-Jacobi equation. Itturns out that in all examples where the massless Hamilton-Jacobi equation separates, it does so becauseof isometries of the uplifted metric, and one can classify those symmetries by rather straightforwardgroup theoretic arguments that we will present in Section 5.Secondly, the factorized warp factor in (2.6) determines symmetries of the full Hamilton-Jacobiequation. Note that, given a Killing tensor, K ( y ; p ), with respect to the fiber Hamiltonian, it satisfies { H M , K} = { ln Ω , K} H M , (2.10)and hence typically becomes a conformal Killing tensor for the full metric. Reversing the argument,conformal Killing tensors for a warped metric may be a telltale of a Killing tensor lurking in thebackground and a separable massless Hamilton-Jacobi equation.The bottom line is that the structure of the base manifold depends on the details of the low-dimensional physics. In many situations the metric on B does not present an obstacle to separabilityand so we will focus on the role of the compactification manifold, F . We therefore begin by reviewingsome of the pertinent literature. Let M be an n -dimensional Riemannian manifold. A classic result of Levi-Civita [27] gives the necessaryand sufficient conditions for separability of the Hamilton-Jacobi equation for a “natural” Hamiltonian H = 12 g MN p M p N + U ( x ) , (2.11)in a given coordinate system, x M . Those Levi-Civita separability conditions require that: ∂H∂x M ∂H∂x N ∂ H∂p M ∂p N + ∂H∂p M ∂H∂p N ∂ H∂x M ∂x N − ∂H∂x M ∂H∂p N ∂ H∂p M ∂x N − ∂H∂p M ∂H∂x N ∂ H∂x M ∂p N = 0 , (2.12)where there is no sum on M and N . Thus (2.12) must me identically satisfied at all points on T ∗ M for all distinct values of M and N , M = N . In particular, the vanishing of the quartic terms in themomenta in (2.12) implies that the Hamilton-Jacobi equation for the geodesic Hamiltionian, given bythe first term in (2.11), must be separable irrespective of the potential, U ( x ).Separability is not simply a property of the Hamiltonian, but also a property of coordinates. Inthis respect, the Levi-Civita conditions (2.12) provide a direct calculation by which one can verifyseparability of the Hamilton-Jacobi equation in given coordinates without constructing the resultingsystem of ODEs. What is less obvious is that, starting with the Hamiltonian (2.11) written in somearbitrary coordinates, one can use (2.12) to obtain coordinate-independent conditions on H under whichthe Hamilton-Jacobi equation is separable and eventually find the system of separating coordinates. Much of this discussion also carries over to pseudo-Riemannian manifolds.
C.1 There must exist ≤ r ≤ n commuting Killing vectors, K α , and m = n − r Killing tensors, K ω , ofrank two such that the system ( K α , K ω ) is integrable. This means that the Killing tensors, K ω , areinvariant under K α ’s and have vanishing Poisson brackets among each other. Using the metric to lower/raise indices of the Killing tensors, one can view them as endomorphism of
T M or T ∗ M . Let ∆ ⊂ T M be the distribution spanned by the Killing vectors and ∆ ⊥ its orthogonalcomplement with respect to the scalar product defined by the metric. C.2 As endomorphisms of
T M , the Killing tensors, K ω , must preserve ∆ ⊥ and, as linear operators on ∆ ⊥ , must be linearly independent and commute at each point on M . Condition C.2 implies that at each point on M there are m linearly independent mutually orthogonaleigenvectors, V a , a = 1 , . . . , m , of the Killing tensors acting on ∆ ⊥ . Then, using (i), one proves thatthose eigenvectors form a completely integrable distribution, which, by the Frobenius theorem, impliesthat there exists a coordinate system, ( x M ) = ( x a , ξ α ), such that V a = v a ( x ) ∂ a (no sum) , K α = ∂ α . (2.13)Note that the eigenvectors, V a , do not necessarily commute and the functions, v a ( x ) can depend on allthe coordinates x a . The important point is that the commutators of the V a close into the V a , and sodefine a foliation, `a la Frobenius, that can be coordinatized by the x a . The system ( x M ) = ( x α , ξ α )then defines a set of separable coordinates.The set of the Killing tensors, K ω , ω = 1 , . . . , m , trivially includes the metric itself, which weidentify with K m . In the separable coordinates, ( x a , ξ α ), the metric and other Killing tensors have thesemi-diagonal standard form, K ω ≡ K MNω p M p N = m X a =1 K ωaa p a + r X αβ =1 K αβω p α p β , (2.14)where K aaω = ϕ ωa , K αβω = ϕ ωa φ αβa . (2.15)9he m × m matrix ( ϕ ωa ) = ( ϕ aω ) − is the inverse of what is known as the St¨ackel matrix. Both ϕ aω = ϕ aω ( x a ) and φ αβa = φ αβa ( x a ) depend only on the coordinate, x a , corresponding to the lowerindex, and all quantities in (2.15) are independent of the ξ α .Finally, the Killing vectors and tensors provide n independent integrals of motion K α = c α , K ω = c ω , (2.16)and the corresponding separated solution of the geodesic Hamilton-Jacobi equation is found from thegeneralized St¨ackel system of ODEs, which we write using the momenta, p M = ∂ M S , p α = c α , p a + φ αβa c α c β = ϕ aω c ω . (2.17)This makes the relation between Killing vectors/tensors, conserved quantities and separability of theHamilton-Jacobi equation precise.Finally, for completness, let us note that for the natural Hamilton-Jacobi equation (2.11) with anontrivial potential, U , additional conditions for separability are K α U = 0 and d ( K ω dU ) = 0 , (2.18)for all Killing vectors and tensors. In the second equation, one should recall that, by raising an indexwith the metric, K ω acts as an endomorphism on T ∗ M and so the term in the parentheses is to beviewed as a one-form. The corresponding modification of (2.17) can be found in [30]. In this section we summarize the results for the uplift of the metric for generic Kaluza-Klein reductions.We then define the notion of “partial” separability of the Hamilton-Jacobi equation between the compactmanifold, F , and the manifold of the reduced theory, B , and derive necessary conditions for when itholds. We then specialize to consistent truncations on spheres and discuss the structure of the metricon the fiber. The uplift formulae for consistent truncations can be extremely complicated, especially for the tensorgauge fields. However, the uplift formula for the metric can be deduced using some relatively simpleinsights.To make a long story short, we start with the standard Kaluza-Klein ansatz for the metric on M , G MN = G µν ( x, y ) + B µm ( x ) B νn ( x, y ) g mn ( x, y ) B µm ( x, y ) g mn ( x, y ) g mn ( x, y ) B ν n ( x, y ) g mn ( x, y ) , (3.1) See, however, more recent progress in, for example, [35–37, 12]. x µ , µ = 1 , . . . , d , on thebase space-time, B , and y m , m = 1 , . . . , n , on the internal compact manifold, F . The metric, G µν ( x, y ),is related to the actual metric, g µν ( x ), in the lower-dimensional theory on B by the rescaling G µν = ∆ − d − g µν , (3.2)where the warp factor, ∆( x, y ), is given by∆ = s det( g mn )det( ◦ g mn ) . (3.3)With this conformal rescaling, the Einstein action on M reduces to the Einstein action on B .Next, one implements the main premise of the Kaluza-Klein program that gauge invariance of thevector fields must descend from general coordinate invariance of the fibration, which implies: B µm = A I µ ( x ) K I m ( y ) , (3.4)where A I µ ( x ) are the Kaluza-Klein gauge fields, K I m ( y ) are the Killing vectors for the round metric , ◦ g mn ( y ), on F and I , J , . . . are adjoint gauge indices.Finally, the correct formula for the inverse of the internal metric on F in supergravity theories canbe deduced by using (3.4) to compare the supersymmetry transformations of the metric on M withsupersymmetry transformations of the gauge fields in the theory on B [4, 5, 9]. One finds that:∆ − d − g mn = U IJ ( x ) K I m ( y ) K J n ( y ) , (3.5)where U IJ ( x ) is a specific, and known, matrix constructed from the scalar fields of the theory on B .The normalization of this matrix depends on the normalization of the Killing vectors, but can be setusing the fact that when the scalars vanish one has U IJ ∼ δ IJ and the metric must be that of theround compact manifold.Putting this all together, consistent truncation for a background involving scalar fields and Kaluza-Klein vectors on B , requires that the inverse metric on M must be given by: G MN = ∆ d − g µν ( x ) − A I µ ( x ) K I n ( y ) − A I ν ( x ) K I m ( y ) (cid:16) U IJ ( x ) + A I σ ( x ) A J σ ( x ) (cid:17) K I m ( y ) K J n ( y ) . (3.6)The inverse metric following from ds F , is then given by (3.5), and gives a part of the lower right entryof (3.6) with the warp factor factored out. The Hamilton-Jacobi equation (1.2) for the inverse metric (3.6) reads∆ d − " g µν ∂S∂x µ ∂S∂x ν − A I µ K I m ∂S∂x µ ∂S∂y m + (cid:16) U IJ + A I σ A J σ (cid:17) K I m K J n ∂S∂y m ∂S∂y n = − m . (3.7) For a careful check of the relation between diffeomorphism and gauge symmetry for the Kaluza-Klein Ansatz, see [6]. Note that µ, ν, . . . indices will always be raised and lowered with the metric g µν in (3.2).
11e will say that (3.7) is “partially separable” if the principal function S ( x, y ) can be taken as a sum S ( x, y ) = S B ( x ) + S F ( y ) , (3.8)of functions that depend on the coordinates along B and F , respectively. The question is under whatconditions the equation (3.7) can be then reduced to a Hamilton-Jacobi equation along B of the form, g µν ( x ) ∂S B ∂x µ ∂S B ∂x ν + Λ µ ( x ) ∂S B ∂x µ + V ( x ) = 0 . (3.9)It is clear that for a warp factor, ∆( x, y ), that is a generic function on M , we must set m = 0, thatis, the separation will hold only for the massless Hamilton-Jacobi equation given by the expression in thesquare bracket in (3.7) set to zero. Next consider the dependence on the vector fields, A µ I . Althoughthe sum over the indices I and J runs over the entire adjoint representation, the Kaluza-Klein vectorfields may gauge only a subgroup of the isometries of the round metric on F . Let K i be the vector fieldson F corresponding to generators of the gauged symmetry, which are some linear combinations of theKilling vectors, K I . Hence, to be more precise, we should use A µ I K I m = A µi K im , (3.10)and rewrite (3.7) in terms of the sum on the right hand side. The rest of the analysis is easier to carryout by looking at the corresponding Hamiltonian obtained by setting π µ = ∂S∂x µ , p m = ∂S∂y m . (3.11)The resulting Hamiltonian for the massless Hamilton-Jacobi equation can be recast into the followingform H = H B ( x ; π ) − A µ i ( x ) π µ K i + X ω g ω ( x ) K ω ( y ; p ) , (3.12)where the functions g ω ( x ) are linearly independent combinations of the functions U IJ and A I σ A J σ while K ω are linear combinations of bilinears in the Killing vectors, K I . Observe that the entire dependenceof H on the coordinates and momenta along the fiber, F , is through K i and K ω . Separation of theHamilton-Jacobi equation between the base and the fiber, resulting in the effective equation (3.9),amounts to setting consistently both K i and K ω to constants. Consistency means that K i and K ω mustbe actual constants of motion, that is { H, K i } = 0 , { H, K ω } = 0 . (3.13)Assuming that the Kaluza-Klein vector fields are suitably generic, and assuming the linear independenceof the functions g ω ( x ) in the construction, we deduce from (3.13) that for a consistent partial separationwe must have { K i , K j } = 0 , { K i , K ω } = 0 , {K ω , K ω ′ } = 0 . (3.14) The exception being when there is only a single “breathing mode,” so that U IJ ∝ δ IJ and ∆ = ∆( x ). In this instancethe full massive Hamilton-Jacobi equation may be separable. H , that is the metric (3.6) without thewarp factor, (ii) K ω are Killing tensors for that metric and are invariant under the gauged isometries,and (iii) the Killing tensors, K ω , are in involution.The conditions (3.13) and (3.14) for the partial separation of the massless Hamilton-Jacobi equationdo not in general imply separability along the fiber. As was observed in Section 2.2, this requires somefurther conditions on the consistent truncation, and we will return to these in Section 5.Before concluding we note that the wave equation affords a broader and interesting notion of partialseparability. In particular, one can allow non-abelian KK gauge symmetries if one also allows the wavefunctions to lie in finite-dimensional representations of the gauge group. More details may be found inAppendix A. We now set the Kaluza-Klein vector fields to zero and specialize the metric to that of the sphere, S n ,embedded in R n +1 as the hypersurface, Y A Y A = 1, where Y A , A = 1 , . . . , n + 1, are the Cartesiancoordinates. We will not use the induced metric from R n +1 but the uplifted fiber metric (3.5). In termsof the Cartesian coordinates, the so ( n + 1) Killing vectors are given by: K AB = Y A P B − Y B P A , (3.15)where P A is the momentum conjugate to Y A in T ∗ R n +1 . Since the Killing vectors (3.15) are tangent tothe sphere, a Poisson bracket of any function of those vectors on T ∗ S n can be evaluated directly in theambient space. This is clearly much simpler than working in some explicit coordinates on the sphere,which will be defined only locally.The Hamiltonian, H F , in (2.6) is defined by the internal metric given by the left hand side in (3.5).It is convenient to normalize it such that H F = 18 M ABCD K AB K CD , (3.16)were M ABCD is an so ( n +1) tensor that depends on the position on the base, B , but is constant along S n .The geodesic Hamiltonian for the round metric on the unit radius sphere is then given by H S n = 14 K AB K AB . (3.17)It is clear that each term in the Hamiltonian (3.16) has the vanishing Poisson bracket with H S n andthus is a Killing tensor for the round metric. The tensor, M ABCD , has obvious symmetries M ABCD = − M BACD = − M ABDC and M ABCD = M CDAB , (3.18) The Killing tensors given by (3.16), with constant M ABCD , have been studied extensively in the mathematical literature(see, e.g., [38] and the references therein). M A [ BCD ] = 0 , (3.19)which together with (3.18) defines M ABCD as an algebraic Riemann tensor in R n +1 . When combinedwith the fact that all rank two Killing tensors on a sphere, or more generally any constant curva-ture manifold, are linear combinations of bilinears in the Killing vectors [22–25], one obtains the 1-1correspondence between the Killing tensors on S n and the algebraic Riemann tensors on the ambientspace [39].We also note, that, like the Riemann tensor, if one assumes the symmetries (3.18), then (3.19) isequivalent to: M [ ABCD ] = 0 . (3.20)As an so ( n + 1) tensor, M ABCD decomposes into three components, M ABCD = M • ABCD + M ABCD + M ABCD , (3.21)corresponding to the irreducible representations with dimensions dim • = 1 , dim = 12 n ( n + 3) , dim = 112 ( n − n + 1)( n + 3) . (3.22)Adding these dimensions, we obtain the dimension of the space of rank two Killing tensors on S n givenby the Delong [40], Takeuchi [24], Thompson [25] formula,dim K ( S n ) = 1 n (cid:18) n + 23 (cid:19)(cid:18) n + 12 (cid:19) . (3.23)The singlet representation in (3.21) corresponds, modulo the base dependent conformal factor, tothe geodesic Hamiltonian, H S n , for the round metric in (3.17). Using (3.15) and expanding the righthand side in (3.17) we get H S n = 12 (cid:2) ( Y · Y )( P · P ) − ( Y · P )( Y · P ) (cid:3) , (3.24)where the dot denotes the contraction in R n +1 . Let ( y m , p n ) be some coordinates and their conjugatemomenta on S n . Then p m = Y Am P A , where Y Am ≡ ∂ m Y A . Using standard identities for the harmonics, Y A and Y Am , on a unit sphere: Y A Y A = 1 , Y Am Y An = ◦ g mn , ◦ g mn Y Am Y Bn = δ AB − Y A Y B , (3.25)one can show that the momenta p m and P A satisfy P A = ◦ g mn Y Am p n + Y A ( Y · P ) . (3.26) One may also note that the three representations of so ( n + 1) descend from a single irreducible representation of gl ( n + 1 , R ) with the Young tableaux [39]. H S n = 12 ◦ g mn p m p n . (3.27)The Hamiltonian, H F , corresponds to a deformation of the round metric by a linear combination ofthe Killing tensors. As we have already discussed in Section 2, it is natural to expand H F into linearlyindependent functions on B , cf. (2.8) and (3.12), H F = X ω g ω ( x ) K ω ( y ; p ) , (3.28)where, for a separable Hamilton-Jacobi equation, the round sphere Killing tensors K ω must be ininvolution. This severely restricts the possible terms in (3.28).One method of imposing such a restriction in a controllable manner is to assume that the internalmetric is invariant under some isometry G ⊂ SO( n + 1). This is quite natural from the point of viewof consistent truncations where such isometries correspond to the scalar fields in the lower-dimensionalsupergravity being invariant under a subgroup of the gauge group. Let K i , i = 1 , . . . , dim g be theKilling vectors corresponding to the generators, T i , of the Lie algebra, g ⊂ so ( n + 1), of G . By the sameargument that led to (2.9), we must have { K i , K ω } = 0 , (3.29)that is the allowed K ω ’s in (3.28) must be invariant under g . As we will show in the next two sections,this restriction based on symmetry is quite powerful and leads to a large class of separable Hamiltonians. In this section we illustrate some of the general discussion above with a simple example of an uplift ofthe SO(3) × SO(3) invariant sector of the maximal gauged supergravity in five dimensions to type IIBsupergravity on S . The uplifted metric in this truncation was obtained in [41] using (3.5) and shownto reproduce the one for half-BPS Janus solution derived in [42] directly in type IIB supergravity. Analmost identical analysis holds for the uplift of the SO(4) × SO(4) invariant sector of N = 8, four-dimensional supergravity to M-theory. See, [43–45] for details of solutions constructed using upliftformulae. The ten-dimensional metric derived using (3.5) is given by ds = ( X X ) / (cid:20) ds , + 4 g (cid:18) dθ + cos θX d Ω + sin θX d e Ω (cid:19)(cid:21) . (4.1)The first term in the square bracket is the metric ds , = g µν dx µ dx ν for a solution in five-dimensionsand the second term is the metric along the fiber, S . This metric has the same structure as in (1.7), The coupling constant g is related to the radius of the internal manifold. In the following we set g = 1. − / = ( X X ) / , in both terms. The S here is a fibration of two unit two-spheres with the metrics d Ω = dφ + sin φ dφ , d e Ω = dξ + sin ξ dξ , (4.2)over the interval 0 ≤ θ ≤ π/
2. The deformations of the round metric on S are parametrized by twoscalar fields, α ( x ) and χ ( x ), in the five-dimensional theory through the functions X = sin θ + e α ( x ) cosh 4 χ ( x ) cos θ and X = e − α ( x ) cosh 4 χ ( x ) sin θ + cos θ . (4.3)From the form of the metric (4.1), separability of the Hamilton-Jacobi equation is by no meansobvious. Yet, it is quite straightforward to check that by taking S ( x, y ) = S x ( x ) + S θ ( θ ) + S φ ( φ ) + S φ ( φ ) + S ξ ( ξ ) + S ξ ( ξ ) , (4.4)the Hamilton-Jacobi equation along S separates into the following system of ODEs: S ′ φ ( φ ) = c φ , S ′ ξ ( ξ ) = c ξ , (4.5)for the cyclic coordinates, φ and ξ , respectively, and S ′ φ ( φ ) + csc φ c φ − c φ = 0 ,S ′ ξ ( ξ ) + csc ξ c ξ − c ξ = 0 ,S ′ θ ( θ ) + sec θ c φ + csc θ c ξ − c θ = 0 . (4.6)The resulting effective Hamilton-Jacobi equation in five-dimensions is g µν ∂S x ∂x µ ∂S x ∂x ν + 14 h c θ − c φ − c ξ + cosh 4 χ (cid:0) c φ e α + c ξ e − α (cid:1) i = 0 , (4.7)which proves partial separability in this example. Along the way we have also shown that the masslessHamilton-Jacobi equation on M fully separates along the S . For the solutions in [41, 42], the five-dimensional metric is a deformation of the AdS metric with the scalar fields and the components ofthe metric tensor depending only on a radial coordinate, r . The full separability of (4.7) is then trivial.From the ODEs (4.5) and (4.6), we also see that the separation involves two Killing vectors, whichcan be identified with the Cartan generators of the so (3) × so (3) isometry, two Killing tensors anda conformal Killing tensor, corresponding to the ODEs in (4.6) respectively. Those five conservedvectors/tensors are in involution, something that is much easier to see by working in the ambient space. The two SO(3)’s of the isometry act in R as rotations on ( Y , Y , Y ) and ( Y , Y , Y ), respectively,generated by the Killing vectors( K , K , K ) and ( K , K , K ) . (4.8)16sing this embedding of the so (3) × so (3) isometry in so (6) given by → ( , ) + ( , ) , (4.9)the resulting branchings of the so (6) irreps in (3.21) are: → ( , ) , ′ → ( , ) + ( , ) + ( , ) + ( , ) , → ( , ) + 2( , ) + ( , ) + ( , ) + ( , ) + ( , ) + ( , ) . (4.10)The uplifted metric (3.5) is a linear combination of three invariant Killing tensors on S correspondingto the singlets in (4.10). A convenient basis for those invariants consists of the “Casimir invariant” of so (6) C so (6)2 = 14 X A,B =1 K AB K AB , (4.11)which is the geodesic Hamiltonian for the round metric on S in (3.17), and the Casimir invariants ofthe two so (3)’s: C so (3) = 14 X A,B =1 K AB K AB , C so (3) = 14 X A,B =4 K AB K AB . (4.12)The most general Hamiltonian, H F , along the fiber invariant under the SO(3) × SO(3) isometry isgiven by a linear combination of these Casimirs, H F = g ( x ) C so (6)2 + g ( x ) C so (3) + g ( x ) C so (3) , (4.13)where g ω ( x ) are arbitrary functions on the base. The three Casimir tensors are clearly in involution,together with the two Cartan generators, K = K and K = K , of so (3) × so (3).As we have discussed in Section 2.2, the existence of this number of Killing vectors/tensors in invo-lution is a necessary condition for separability of the Hamiltonian (4.13). To demonstrate separabilityof H F , let us introduce explicit coordinates, ( θ, φ , , ξ , ), on S , such that Y = cos θ sin φ cos φ ,Y = cos θ sin φ sin φ ,Y = cos θ cos φ , Y = sin θ sin ξ cos ξ ,Y = sin θ sin ξ sin ξ ,Y = sin θ cos ξ . (4.14)In terms of these coordinates and canonically conjugate momenta, we find K = p φ , K = p ξ , (4.15) C so (3) = p φ φ p φ , C so (3) = p ξ ξ p ξ , (4.16) Our group theory conventions are the same as in [46]. To compute branching rules here and in Section 5, we havemade an extensive use of the Mathematica package LieArt [47]. C so (6)2 = p θ θ C so (3) + csc θ C so (3) . (4.17)The nested structure of (4.15)-(4.17) implies that by setting K = c φ , K = c ξ , C so (3) = c φ , C so (3) = c ξ , C so (6)2 = c θ , (4.18)the massless Hamilton-Jacobi equation with the fiber Hamiltonian (4.13) is partially separable and fullyseparable along the fiber, with the separation constants c θ , . . . , c ξ .The fiber metric in (4.1) is recovered by setting g = 14 , g = 14 ( e α cosh 4 χ − , g = 14 ( e − α cosh 4 χ − , (4.19)and is just a representative of a whole family of uplifted metrics for which the Hamilton-Jacobi equationis separable due to the so (3) × so (3) isometry.This simple calculation illustrates how an isometry can be used to restrict the allowed terms in (3.28)and makes it more clear how the complicated functions X and X given in (4.3) arise upon inversionof the metric from a simple internal Hamiltonian (4.13). It is also quite instructive to explore, in this explicit example, whether the isometry of the metricalone can account for separability. To this end consider (4.1) but with the most general SO(3) × SO(3)invariant metric on the S : ds = ds , + 4 g (cid:16) dθ + A ( x, θ ) d Ω + A ( x, θ ) d e Ω (cid:17) . (4.20)where A , ( x, θ ) are arbitrary functions of the base manifold and θ . We have also dropped the warpfactor for convenience.Proceeding with the separation of the massless Hamilton-Jacobi equation as in Section 4.1, we endup with the partially separated equation, g µν ∂S B ∂x µ ∂S B ∂x ν + 12 (cid:20) c θ + c φ (cid:18) A ( x, θ ) − θ (cid:19) + c ξ (cid:18) A ( x, θ ) − θ (cid:19)(cid:21) = 0 . (4.21)So the non-trivial identities required for separation are:1 A ( x, θ ) − θ = f ( x ) , A ( x, θ ) − θ = f ( x ) , (4.22)where f , ( x ) are some functions on B . For the specific SO(3) × SO(3) invariant solution considered inSection 4.1: A ( x, θ ) = sec θ X ( x, θ ) = e α ( x ) cosh 4 χ ( x ) + tan θ , (4.23) A ( x, θ ) = csc θ X ( x, θ ) = e − α ( x ) cosh 4 χ ( x ) + cot θ , (4.24)18nd the identities (4.22) are indeed satisfied. A priori there is no reason to expect A , ( x, θ ) to beconstrained in such a way to allow this non-trivial separation.We see from this example that separability of the massless Hamilton-Jacobi equation is not just aconsequence of the underlying symmetry, but results from an interplay between the symmetry and thestructure of the uplifted metric in (3.16). When looking at a consistent truncations on the n -sphere, S n , one sometimes tries to simplify theproblem by looking at sectors that are invariant under a particular sub-algebra, g , of the full algebraof isometries, so ( n + 1). This approach also turns out to be a powerful tool in the investigation ofseparability within consistent truncations, and, as we will discuss, the separability of the Hamilton-Jacobi equation for a generic g -invariant sector depends solely upon the choice of g . Indeed, we define g to be a separating isometry precisely when the generic g -invariant sector of the consistent truncationon S n has a separable Hamilton-Jacobi equation in the absence of reduction gauge vectors.It is important to note why we are insisting on the separability of generic g -invariant sectors. This isbecause, in Section 3.2 we crucially needed generic functions, g ω ( x ), in (2.8) in order to arrive at (3.14),and, in particular, show that partial separability requires that the Killing tensors, K ω , in (3.28) mustbe in involution. If some of the g ω ’s vanish, or these functions are linearly dependent, then one coulddiscard, or combine, some of the K ω ’s into a new basis so that the new g ω ’s become linearly independent.The resulting new K ω ’s might then be in involution with one another and lead to separability of theHamilton-Jacobi equation. This situation would arise if we started with a Lie algebra, g , that was nota separating isometry, but we chose a point in the configuration space at which g becomes enhanced toa larger symmetry, ˆ g , that is a separating isometry. It is also quite possible that there are examples inwhich there are separable loci in the configuration space of a non-separable isometries, g , and yet theseloci do not involve some enhancement of the isometry.To avoid the complications of such special cases, we insist upon generic g -invariant sectors andcan therefor invoke (3.14). Having made this step, we will systematically examine sub-algebras, g ⊂ so ( n + 1), for n = 2 , . . . ,
7, and determine which algebras lead to separating isometries, or not. This willautomatically lead to a partially separable massless Hamilton-Jacobi equation. We start by explainingour approach and we tabulate the group theoretic details in Appendix B. We then catalog the range ofpossibilities, giving the Killing tensors and the separable coordinates.
Our task is to classify “separating isometries,” g ⊂ so ( n + 1), of the uplifted metric (3.28) on S n forwhich the massless Hamilton-Jacobi equation for the Hamiltonian (3.12), with no Kaluza-Klein vectorfields, is fully separable along the fiber and partially separable between the fiber and the base.By separability along the fiber we mean that K ω have common separating coordinates as in (2.14).Since the Hamiltonian, H S n , for the round metric is obviously invariant under g , the results summarizedin Section 2.2 are directly applicable to our problem, which may now be restated as follows:19 or a given g ⊂ so ( n + 1) , determine whether the set of Killing tensors, K ω , invariant under g canbe developed into to a set of n Killing vectors/tensors that satisfy conditions C.1 and C.2 for separabilityof the geodesic Hamilton-Jacobi equation for the round metric on S n . Let K g be the set of Killing tensors on S n invariant under g . Since we are looking at the g -invariantsector, the K ω appearing in (3.28) must be a linear combinations of the K g .An isometry g of the uplifted metric may fail to be separating for one of three reasons:(i) The set K g is too large and the invariant Killing tensors are not all in involution.(ii) There is no extension of K g to a set of n Killing vectors/tensors in involution.(iii) Some of the Killing tensors are not integrable as endomorphisms of the tangent space.Intuitively one can understand the first two options as follows. If g is “too small,” then the K ω ’sappearing in a generic expansion (3.28) may simply be too numerous for all of the K ω ’s to be in involutionand so they cannot all be set to independent constants, which means that separation fails. If g is “toolarge,” then K g is small and there will need to be additional, non- g -invariant Killing tensors for theHamilton-Jacobi equation to separate. Those may or may not exist. The third possibility is a failure ofthe Killing tensors being integrable in the sense of endomorphisms of the tangent space as defined bycondition C.2 in Section 2.2. They may not preserve the subspace orthogonal to the Killing vectors ormay not commute. In the last instance, there can be an integrable system of geodesics and yet thereare no separating coordinates for the Hamilton-Jacobi equation. We will give an example of this inSection 5.4.3.Given g , the invariant Killing tensors, K ω ∈ K g , are in one-one correspondence with the singlets inthe branching of the three representation of so ( n + 1) in (3.21) under g . Obviously, there will be alwaysat least one such tensor, which is the round metric on S n . Other trivially invariant Killing tensors comefrom various Casimir invariants, which are constructed as follows.For a subalgebra h ⊂ so ( n + 1) with the generators, t i = ( t iAB ), in R n +1 , the Killing vectors, K ( t i ),satisfying { K ( t i ) , Y A } = t iAB Y B , (5.1)are K ( t i ) = 12 t iAB K AB . (5.2)The Casimir invariant of h is defined as C h ≡ − κ ij K ( t i ) K ( t j ) , (5.3)where κ ij is the Cartan-Killing form on h . In particular, we have H S n = C so ( n +1)2 . (5.4)Suppose that g is a direct sum algebra g = h ⊕ . . . ⊕ h s ⊕ u (1) ⊕ . . . ⊕ u (1) ν , (5.5)where h i are simple, and is embedded in so ( n + 1) through a chain of maximal subalgebras, so ( n + 1) ≡ g ⊃ g ⊃ . . . ⊃ g k ≡ g , (5.6)20f length k . Then the Casimir invariants and the bilinears in the Killing vectors of the u (1)’s in (5.5), C h i , i = 1 , . . . , s ; C g a , a = 1 , . . . , k ; K α K β , α, β = 1 , . . . , ν , (5.7)are all invariant under g and are in involution. However, note the actual number of the independentinvariants in (5.7) may be less than s + k + ν ( ν + 1) / g exceeds the numberof the obvious invariants (5.7), not all Killing tensors in K g will be in involution. This means that aseparating isometry cannot be too small. Another observation is that if one of the algebras in the chain(5.6) is a separating isometry then all other algebras higher in the chain are separating isometries aswell. Conversely, if some algebra in not separating, all algebras lower in the chain cannot be separating.This leads to a systematic procedure by which we have classified all chains of separating isometries forthe consistent truncations on n -spheres, S n , for n = 2 , . . . ,
7, listed in the tables in Appendix B.We thus have an effective method of enumerating many of the elements of K g and testing whetherthey can be in involution with one another. At the other extreme, when g is “large,” K g is “small,”we are going to need to find non- g -invariant Killing tensors so as to complete the separation of theHamilton-Jacobi equation. There is also a useful systematic way to achieve this. If one has a chain ofinclusions: g ⊂ g ⊂ . . . ⊂ g k ≡ g , (5.8)then the Casimirs C g a are all in involution with one another. As we will see, this can prove to be aneffective way of generating the missing Killing tensors. S n , n = 1 , . . . , In this, and the next, section, we will check explicitly separability for the isometries listed in Tables B.1and B.2. This is most easily done by simply constructing explicitly separating coordinates, which istypically quite straightforward given the isometry. We will also discuss some interesting examples whereseparability fails when the isometry is further decreased. We start with the families of separable chainsthat are present on all spheres. g = so ( n + 1) For the maximal isometry, we have only one invariant Killing tensor, K g = D C so ( n +1)2 E . (5.9)Separating coordinates for the geodesic Hamiltonian on spheres have been fully classified by Kalninsand Miller [33]. All those coordinates, up to an equivalence, are orthogonal, which means that themetric tensor is diagonal. The equivalence is a linear change of cyclic variables. An earlier, partialclassification in terms of polyspherical coordinates was given by Vilenkin [48]. In fact, these are theonly coordinates that we will need in our examples. For an ambient space characterization of separatingcoordinates on spheres, see [38]. 21 .2.2 g = so ( n ) The so ( n ) isometry acts on Y , . . . , Y n , and K g = D C so ( n +1)2 , C so ( n )2 E . (5.10)Setting Y I = cos θ y I , I = 1 , . . . , n Y n +1 = sin θ , (5.11)where y I y I = 1 define S n − , one finds that C so ( n +1)2 = p θ θ C so ( n )2 . (5.12)We can now use any separating coordinates for the geodesic Hamiltonian on S n − to separate simul-tanously both Killing tensors. g = so ( p ) ⊕ so ( q ), p + q = n + 1 The so ( p ) and so ( q ) isometries act on Y , . . . , , Y p and Y p +1 , . . . , Y p + q , respectively. There are threeindependent invariant Killing tensors, K g = D C so ( n +1)2 , C so ( p )2 , C so ( q )2 E . (5.13)The two sets of ambient coordinates parametrize S p × S q with radii cos θ and sin θ , respectively, fiberedover an interval parametrized by θ . Then C so ( n +1)2 = p θ θ C so ( p )2 + csc θ C so ( q )2 , (5.14)and all three invariants separate using any separating coordinates on S p and S q . g = su ( m ) ⊕ u (1), 2 m = n + 1 We take as the generator of u (1) the block diagonal, antisymmetric matrix with J = . . . = J m − m =1, which defines a complex structure in R m . Its Killing vector is J m = m X A =1 K A − A . (5.15)The su ( m ) Killing vectors are in involution with J m . One can check that C so (2 m ) = 2 C su ( m ) + m − m J m , (5.16)We thus find that there are only two independent singlets and K g = D C so (2 m )2 , J m E . (5.17) Here and below, the u (1) generators may be differently normalized and/or may be linear combinations of the u (1)generators in Tables B.1 and B.2, which are obtained by the rules of LieArt [47]. k ) × U(1) ⊂ SU( k + 1), which leads to the Casimirs of sequentially embedded C su ( k ) ’s, and to theKilling vectors of the U(1) factors. Using (5.16), we can recast the former in terms of C so (2 k )2 and soarrive at the following natural set of Killing vectors/tensors that are in involution: C so (2 m )2 , C so (2 m − , . . . , C so (4)2 , K , K , . . . , K m − m . (5.18)These are the ( m −
1) Casimir tensors of so (2 k )’s acting on the first 2 k Cartesian coordinates and m Cartan generators of so (2 m ). The corresponding coordinates ( θ , . . . , θ m ; φ , . . . , φ m ) are defined by X j − + i X j = x j e iφ j , j = 1 , . . . , m , (5.19)where x m = cos θ m , x m − = sin θ m cos θ m − , . . . (5.20)are given in terms of standard spherical angles on S m − . Then φ , . . . , φ m are cyclic with K j − j = p φ j , j = 1 , . . . , m . (5.21)In particular, J m = p φ + . . . + p φ m . (5.22)The Casimirs of so (2 k ), n = 1 , . . . , N + 1, are given inductively by C so (2)2 = p φ , ... C so (2 k )2 = p θ n θ n p φ n + csc θ n C so (2 k − , k = 2 , . . . m . (5.23)The nested structure of these Casimirs is precisely what is needed for separability. g = su ( m − ⊕ u (1) ⊕ u (1), 2 m = n + 1 This isometry is obtained by breaking su ( m ) in Section 5.2.4 to su ( m − ⊕ u (1). We may take the two u (1) Killing vectors as J m − and J m , (5.24)with the range of summation in (5.15) set accordingly. The five singlets correspond to K g = D C so (2 m )2 , C su ( m − , J m − , J m − J m , J m E . (5.25)The identity (5.16) allows us to use C so (2 m − instead of C su ( m − . Using the same coordinates as inSection 5.2.4, we find J m − = p φ + . . . + p φ m − , (5.26)with everything else remaining the same. This shows that the smaller isometry is still separating.23 .2.6 g = su ( m ) or su ( m − ⊕ u (1) , 2 m = n + 1 Reducing the isometry by dropping u (1) in u ( m ) in Section 5.2.4 or in u ( m −
1) in Section 5.2.5 doesnot change the invariants and the system remains separable in the same coordinates as before. g = su ( m ) ⊕ u (1), 2 m = n The invariants are K g = D C so (2 m +1)2 , C su ( m )2 , J m E . (5.27)The su ( m ) ⊕ u (1) lies inside so (2 m ) in Section 5.2.2. After using (5.11), the separation is the same asin Section 5.2.4. g = su ( m ) ⊂ su ( m ) ⊕ u (1), 2 m = n , m ≥ The isometry here is the reduction of the one in Section 5.2.7 by dropping the u (1). For m = 2, thisproduces too many invariants that fail to be in involution. However, for m ≥
3, the invariants are thesame as in (5.27), and one can separate the system using the same coordinates as in Section 5.2.7. g = g ⊂ so (7) Using an explicit representation of g as an automorphism of the algebra of unit octonions, one cancheck that C g = C so (7) . (5.28)This explains why there is only one singlet and K g = D C so (7) E . (5.29)Hence any separating coordinates on the round S can be used here. S In this section we discuss selected isometries, g ⊂ so (8), on S in Table B.2 in Appendix B. g = su (2) ⊕ su (2) ⊕ su (2) ⊕ u (1) and su (2) ⊕ su (2) ⊕ u (1) ⊕ u (1) We start with the smaller isometry which consists of su (2) (1) L × u (1) (1) R acting on Y , . . . , Y and su (2) (2) L × u (1) (2) R acting on Y , . . . , Y . The six singlets are K g = D C so (8)2 , C su (2) (1) L , C su (2) (2) L , R (1)3 R (1)3 , R (1)3 R (2)3 , R (2)3 R (2)3 E , (5.30)where R ( i )3 is a Killing vector in su (2) ( i ) L for i = 1 ,
2. The natural separating coordinates correspond tothe Killing vectors of the Cartan subalgebra of g , which is the same as the Cartan subalgebra of so (8),24nd the three Casimir invariants above. Hence we set Y + i Y = cos θ cos ζ e iφ , Y + i Y = cos θ sin ζ e iφ ,Y + i Y = sin θ cos ζ e iψ , Y + i Y = sin θ sin ζ e iψ , (5.31)which gives K = p φ , K = p φ , K = p ψ , K = p ψ , (5.32) C su (2) (1) L = 116 (cid:2) p ζ + sec ζ p φ + csc ζ p φ (cid:3) , C su (2) (2) L = 116 (cid:2) p ζ + sec ζ p ψ + csc ζ p ψ (cid:3) , (5.33)and C so (8)2 = p θ θ C su (2) (1) L + 8 csc θ C su (2) (2) L , (5.34)and the separability is manifest.We also note that there is yet another su (2) ⊕ su (2) ⊕ u (1) ⊕ u (1) subalgebra in so (8) defined by theembedding → ( , )(0 ,
0) + ( , )(1 ,
1) + ( , )(1 , −
1) + ( , )( − ,
1) + ( , )( − , − . (5.35)However, it leads to seven invariants with not all of them in involution. Hence it is not a separatingisometry. g = so (7) v,c The so (7) v,c subalgebras can be constructed using the two spinor representations of so (8), or, equiva-lently, as subalgebras preserving a selfdual tensor, C ABCD + , and an anti-selfdual tensor, C ABCD − , respec-tively (see, e.g., [4]). The C ABCD ± are tensors in R that also satisfy C ABEF ± C CDEF ± = 112 δ ABCD − C ABCD ± . (5.36)The two subalgebras are conjugate under a reflection in O(8). The singlet in each sector corresponds to C so (7) v,c = 34 C so (8)2 . (5.37)and hence one can use any separating coordinates for the round metric on S . g = g ⊂ so (7) s,v,c ⊂ so (8) One can embed g in so (8) via the three so (7)’s. Although the three embeddings are equivalent, one isled naturally to different separating coordinates depending on which one is chosen.For g ⊂ so (7) s , we can take it to be the same as in Section 5.2.9. In particular, (5.28) holds. Usingthe coordinates introduced in Section 5.2.2, we obtain separating coordinates that consist of θ in (5.11)and any separating coordinates on S . 25nother realization of g ⊂ so (7) v,c is by requiring invariance of both tensors C ABCD ± . Starting withthe two Cartan generators of g , K − K , K − K + K , (5.38)one finds that there is one more Killing vector on S in involution with those two, such that the threetogether span the Cartan subalgebra of so (6) with the generators K , K , K . (5.39)It is then straightforward to determine that the four Killing tensors, C so (4)2 , C so (6)2 , C g , C so (8)2 , (5.40)are invariant under (5.39) and in involution with each other. In fact, the seven Killing vectors/tensors in(5.39) and (5.40) satisfy both conditions C.1 and C.2, which proves that the Hamilton-Jacobi equationin the g invariant sector is separable.The corresponding separating coordinates can be defined by X + i X = Y sin θ sin θ e iφ ,X + i X = Y cos θ sin θ e iφ ,X + i X = Y cos θ e iφ ,X = Y ,X = Y , (5.41)where Y = cos ζ cos ζ ,Y = 1 √ ζ − cos ζ sin ζ ) ,Y = 1 √ ζ + cos ζ sin ζ ) . (5.42)These coordinates provide an example of polyspherical coordinates [48, 33] on S , where( θ , θ , φ , φ , φ ) parametrize an S fibered over an S with coordinates ( ζ , ζ ). The three Killingvectors (5.39) are K = p φ , K = p φ , K = p φ , (5.43)while the Killing tensors are given by C so (4)2 = p θ θ p φ + csc θ p φ ,C so (6)2 = p θ θ p φ + csc θ C so (4)2 ,C g = p ζ ζ C so (6)2 ,C so (8)2 = p ζ ζ C g . (5.44)26ence, setting the momenta and the Casimirs to constants, we obtain a separating system (2.17) forthe Hamilton-Jacobi equation. We will now discuss some examples of isometries where the separation fails. g = so ( p ) ⊕ so ( q ) ⊕ so ( r ), p + q + r = n + 1 The three rotations act block diagonally R p × R q × R r . In addition to the Casimirs, C so ( p )2 , C so ( q )2 and C so ( r )2 one also has C so ( p ) ⊕ so ( q )2 = 14 p + q X A,B =1 K AB K AB , (5.45)and the other two for so ( p ) ⊕ so ( r ) and so ( q ) ⊕ so ( r ) given by the similar formulae. One can check thatthese invariants have nonvanishing Poisson bracket among each other and hence this isometry is notseparating.This example is prototypical when g is a sum of more than two diagonal subalgebras. This leads to alarge number of invariants some of which fail to be in involution. We have omitted all those symmetriesfrom the tables in Appendix B. g = su (2) M ⊕ h The su (2) M here is the maximal subalgebra of a larger symmetry, ˜ g . The simplest example is ˜ g = so (5),where the generators of su (2) M are given by the Killing vectors K ( t ) = K + r K + r K − K ,K ( t ) = − K + r K − K − r K ,K ( t ) = 2 K + K . (5.46)The two singlets are K g = D C so (5)2 , C su (2) M E . (5.47)To determine separability one can proceed as follows: First compute all Killing vectors and Killingtensors that are in involution with the invariants (5.47) and then check whether there exists a set offour Killing vectors/tensors in involution that extends (5.47).A direct calculation shows that the only Killing vectors that are in involution with C su (2) M are thosein (5.46). The seven Killing tensors in involution with the ones in (5.47) are C so (5)2 and six bilinearsin the Killing vectors (5.46). It is then impossible to choose from this set the required four Killingvectors/tensors in involution and hence the system does not separate. See, e.g., Section 5 in [34]. su (2)’s, where it is impossible to find the required set Killing vectors/tensors satisfying conditions C.1and/or C.2 in Section 2.2. g = su (2) ⊕ su (2) ⊂ g ⊂ so (7) In this example, separability fails in an interesting way. The embedding su (2) ⊕ su (2) ⊂ so (7) is givenby the branching → ( , ) + ( , ) . (5.48)Denote the two su (2)’s as su (2) L and su (2) R , respectively. They act on Y , . . . , Y as so (4) ≃ su (2) L ⊕ su (2) R and only su (2) L acts as so (3) on ( Y , Y , Y ). The four invariants are K g = D C so (7)2 , C su (2) L , C su (2) R , C so (3)2 E . (5.49)The space of Killing tensors that are in involution with these invariants is 23-dimensional and is spannedby C so (7)2 , C so (3)2 and 21 bilinears in the Killing vectors of g . They are also in involution with any twocommuting Cartan generators of g . So, one can easily extend (5.49) to a set of six Killing vectors/tensorsin involution. This satisfies condition C.1 for separability. However, using any coordinates on S , onecan check that as endomorphisms of T S , the Killing tensors (5.49) commute only on a subspace ofdimension two. Hence to satisfy condition C.2 we would need a four-dimensional subspace in T S spanned by commuting Killing vectors, which is impossible.This truncation provides an explicit example where there is a maximal set of Killing vectors andKilling tensors that are in involution, and so one has an integrable system of geodesics, yet there areno separating coordinates for the Hamilton-Jacobi equation. g = su (3) ⊂ so (8) The two singlets span K g = D C so (8)2 , C su (3)2 E . (5.50)The Killing tensors that are in involution with those in (5.50) are C so (8)2 and bilinears in the Killingvectors of su (3). Using any convenient coordinates on S one finds that no linear combination of thosebilinears commutes, as endomorphisms, with C su (3)2 , and that the latter does not preserve the subspacesperpendicular to either one or two commuting Killing vectors. This shows that the condition C.2 isalways violated and thus (5.50) are not simultaneously separable. g = sp (4) ⊕ su (2) ⊂ so (8) There are only two singlets as one finds that C so (8)2 = 4 C sp (4)2 − C su (2)2 . (5.51)Then, similarly as in Section 5.4.4, there are no Killing tensors that are simultaneously in involutionand commute with C su (2)2 , which in turn does not preserve the orthogonal complement of the Killingvectors corresponding to the Cartan subalgebra of the isometry.28 More separable examples from holography
In this section we present some further examples of solutions belonging to consistent truncations of typeIIB supergravity and M-theory on S and S , respectively, and show that the massless Hamilton-Jacobiequation is fully separable. Just as in the simpler example in Section 4, separability is by no meansobvious given the explicit form of the metrics in local coordinates. Those metrics and additional explicitdiscussion of separability can be found in Appendix C. The two RG-flows in this section are obtained by uplifting supersymmetric domain wall solutions in the N = 8 gauged supergravity in five dimensions to solutions of type IIB supergravity in ten dimensions.The uplifted metric along S is then given by [8, 9] M ABCD = const × e V ABab e V CDcd Ω ac Ω bd , (6.1)where e V ABab are matrix elements of the scalar 27-bein that parametrizes the coset space E / USp(8),and Ω ab is the USp(8)-invariant metric. N = 1 flow This solution is a holographic dual of the RG-flow of N = 4 Yang Mills down to an N = 1 supersym-metric “Leigh-Strassler” conformal fixed point in the infra-red driven by giving a mass to one chiralmultiplet [49]. In five-dimensional supergravity the flow is defined by two scalar fields, α ( r ) and χ ( r ),and a metric function, A ( r ). The five-dimensional metric is given by the usual Poincar´e slicing: ds , = e A ( r ) (cid:0) η µν dx µ dx ν (cid:1) − dr , µ = 0 , . . . , . (6.2)The internal metric in terms of ambient coordinates is given by (3.4) in [50]: ds ( α, χ ) = a χξ ( dY A Q − AB dY B ) + a χ tanh χξ ( Y A J AB dY B ) , (6.3)and the warp factor ∆ − = ξ cosh χ . (6.4)Here Q is a diagonal matrix with Q = . . . = Q = e − α and Q = Q = e α , J is an antisymmetricmatrix with J = J = J = 1, and ξ = Y A Q AB Y A . The constant a is related to the radius, L , ofthe AdS metric (6.2), when α = χ = 0, by a = √ L . The gauge coupling constant is g = 2 /L .While the full truncation for this flow has an su (2) × u (1) symmetry, the isometry of the metric (6.3)is enhanced to su (2) × u (1) × u (1) , where su (2) × u (1) = su (2) L × u (1) R ⊂ so (4) acting on Y , . . . , Y ,while the second u (1) is the rotation in the Y , -plane. This means that this metric is of the type inSection 5.2.5 and should have a separable Hamilton-Jacobi equation. We count unbroken supersymmetries on the field theory side of the gauge/gravity duality. Note that IIB supergravity is typically written using a “mostly minus” signature. H F , corresponding to (6.3), is H F = 1 a cosh χ h ( Q AB Y A Y B )( Q CD P C P D ) − ( Q AB Y A P B )( Q CD Y C P D ) i − a sinh χ (cid:2) ( QJ ) AB Y A P B (cid:3)(cid:2) ( QJ ) CD Y C P D (cid:3) , (6.5)and indeed decomposes into the invariant Killing tensors, (5.25), as follows ( a = 1): H F = 2 ρ cosh χ C so (6)2 + 2 ρ − (1 − ρ ) cosh χ C so (4)2 − ρ − sinh χ J − ρ sinh χ J R − ρ (1 − ρ + cosh 2 χ ) R , (6.6)where, R ≡ J − J = K . (6.7)Setting C so (6)2 = c , C so (4)2 = c , J = c J , R = c R , (6.8)in (6.12), where c , c , c J , and c R are constants, we obtain the effective potential, U ( r ), for the Hamilton-Jacobi equation (3.9) on the base. For the metric (6.2), this equation is then separable.It follows from (5.16) for m = 2 that C so (4)2 = 2 C su (2) L,R . (6.9)The SU(2) L symmetry acts transitively on S . Hence the constant c represents the energy of motionon S .We refer the reader to Appendix C.1 for an elementary derivation of the effective Hamilton-Jacobiequation on B using the original local coordinates in [50]. N = 2 flow This solution represents flows of N = 4 Yang Mills down to into the infra-red N = 2 supersymmetrictheory in which mass is given to an N = 2 hypermultiplet. In five-dimensional supergravity the flow isdefined by two scalar fields, α and χ . The five-dimensional metric is given by the Poincar´e slicing (6.2).The five-dimensional flow and its IIB uplift were obtained in [9]. The uplifted metric along S has the same SU(2) × U(1) × U(1) isometry as the N = 1 flow in theprevious subsection. Hence the Hamilton-Jacobi equation is partially separable, with the Hamiltonian, H F , in the basis of invariants (6.8) given by H F = 2 ρ cosh(2 χ ) C so (6) + ρ − (cid:2) − ρ cosh(2 χ ) (cid:3) C so (4)2 + ρ − sinh (2 χ ) J + ρ (cid:2) ( ρ − cosh(2 χ ) (cid:3) R . (6.10)One may note that (6.10), unlike (6.12), involves only four out of five invariant Killing tensors. Thisis a reflection of a simpler analytic structure of this flow. We refer to Appendix C.2 for details of themetric and an explicit separation of the Hamilton-Jacobi equation in local coordinates used in [9]. For the uplifted metric, see also [51]. .2 RG-flows/Janus solutions in M-theory The two solutions in this section are obtained by an uplift from the N = 8, d = 4 gauged supergravity [52]to M-theory on S . The uplifted metric is obtained using [4] M ABCD = const × ( U ij AB + V ijAB )( U ij CD + V ijCD ) , (6.11)where U ijAB , . . . , V ijCD are matrix elements of the scalar 56-bein parametrizing the E / SU(8) cosetin the so-called SL(8 , R ) basis [53]. Using E identities, one can also invert (1.8) to obtain an explicitexpression for the internal metric as a pull-back of a deformed metric in R [54, 55]. N = 1 RG-flow in M-theory This example represents a flow of ABJM theory [56] down to a non-trivial infra-red conformal fixedpoint [57–60]. In an SU(3) × U(1)-invariant truncation of the N = 8, d = 4 gauged supergravity,the flow [61, 62] is defined by two scalar fields, α and χ . The four-dimensional metric is given bythe usual Poincar´e slicing and, just as for the analogous RG-flow in five dimensions in Section 6.1.1,the uplifted metric has an enhanced SU(3) × U(1) × U(1) isometry along S . This means that itfalls into the class of separable metrics discussed in Section 5.2.5. Indeed, the fiber Hamiltonian, H F , has exactly the same form as in (6.5), when written in the ambient space, T ∗ R , but now with Q = diag (cid:0) ρ − , ρ − , ρ − , ρ − , ρ − , ρ − , ρ , ρ (cid:1) , and the complex structure matrix J = . . . = J = 1[63]. In terms of the invariants (5.25), it is given by H F = 2 ρ cosh χ C so (8)2 + 2 ρ − ( ρ −
1) cosh χ C su (3)2 + ρ − (cid:2) ρ −
4) cosh χ (cid:3) J − ρ sinh χ J R + ρ (cid:0) ρ − cosh χ (cid:1) R , (6.12)where, R ≡ J − J = K . (6.13)Further details for this example can be found in Appendix C.3. × U(1) × U(1) invariant RG-flows/Janus solutions
This solution was obtained in [64] by an uplift of the corresponding Janus solution [45] in an SU(3) × U(1) × U(1) invariant truncation of the N = 8, d = 4 gauged supergravity. It involves a single complexscalar field, z = tanh λ e iζ , (6.14)that parametrizes the unit Poincar´e disk. The invariance of the truncation implies that the upliftedmetric has at least the same isometry and thus the fiber Hamiltonian, H F , on S should be a linearcombinations of the invariants given in Section 5.2.5.Starting with the explicit form of the uplifted metric given in (C.33), one can verify that H F = g C so (8)2 + g C su (3)2 + g J + g J R + g R , (6.15)31here R = K and g = cosh(2 λ ) + sinh(2 λ ) cos ζ ,g = − λ ) cos ζ ,g = 112 sinh(2 λ ) h ζ + 3 sin ζ (cid:0) sinh(4 λ ) − ζ sinh (2 λ ) (cid:1) i ,g = 12 sin ζ sinh(2 λ ) h sin(2 ζ ) sinh (2 λ ) + sin( ζ ) sinh(4 λ ) i ,g = 12 h(cid:0) cosh(2 λ ) + sinh(2 λ ) cos ζ (cid:1) − i(cid:0) cosh(2 λ ) + sinh(2 λ ) cos ζ (cid:1) . (6.16)Given the form of these functions it is not surprising that the corresponding metric obtained from H F should be rather involved. Yet the underlying symmetry assures that the Hamilton-Jacobi equation isseparable. We refer the reader to Appendix C.4 for an explicit treatment of that separability in localcoordinates. The simplest non-trivial examples of uplifted metrics correspond to the Coulomb flows in maximalsupergravities, for which some of the earlier examples can be found in [65, 66]. The scalar fields thatdefine those flows are given by an ( n + 1) × ( n + 1) symmetric, unimodular matrix, T AB , parameterizingthe coset space SL( n + 1 , R )SO( n + 1) . (6.17)The tensor, M ABCD , is then of a Ricci type and the internal Hamiltonian is simply H F = 14 T AC T BD K AB K CD . (6.18)The corresponding metric along the fiber is then [17] (see, also [4, 6, 15]) ds F = ∆ β T − AB dY A dY B , ∆ − βn = T AB Y A Y B , (6.19)where β = 2 n − d − d − , (6.20)and the metric is that of a conformally rescaled ellipsoid.In specific examples [65–68], one takes T AB , to be diagonal,( T AB ) = diag( a , a , . . . , a n +1 ) , a a . . . a n +1 = 1 , (6.21)so that H F has the same form as in (6.5) with J = 0. The isometry of the metric (6.19) depends on howmany identical eigenvalues arises in (6.21) and the separability of the corresponding Hamilton-Jacobiequation follows from the discussion in Sections 5.2.2, 5.2.3 and 5.4.2. In particular, the Hamilton-Jacobi equation is partially separable if there are only two distinct eigenvalues in (6.21), but fails to beseparable if there are more than two as for instance for the flows discussed in [69,68]. The reason is that k different eigenvalues break the isometry to a product SO( n ) × . . . × SO( n k ), n + . . . + n k = n + 1 as in32ection 5.4.2 and the resulting invariant Killing tensors are not in involution. This lack of separabilitymight seem counterintuitive as the Hamilton-Jacobi equation for the metric (6.19) on an ellipsoid with constant matrix, T AB , does separate in the elliptic Jacobi coordinates for any values of a , . . . , a n +1 .The point is that in our problem those eigenvalues are functions on the base, a i = a i ( x ), and lead to anon-trivial mixing between the base and the fiber when k > In this section we present the (1 , m, n ) multi-mode superstrata solutions. These are solutions to D = 6, N = (1 ,
0) supergravity, coupled to two anti-self dual tensor multiplets and can be consistently truncatedon a three-sphere, while retaining non-trivial gauge vectors, see [21] for details. The appearance of gaugevectors modifies the analysis of separability relative to Section 5.
Recall that the massless Hamilton-Jacobi equation, for a truncation with non-trivial gauge vectors A µ i ( x ), can be cast in the form of (3.12): H = H B ( x ; π ) − A µ i ( x ) π µ K i + X ω g ω ( x ) K ω ( y ; p ) , (7.1)where X ω g ω ( x ) K ω ( y ; p ) = 12 (cid:16) U IJ + A I σ A J σ (cid:17) K I m K J n . (7.2)In order to discuss when (7.1) will separate for a deformed spherical fiber with non-trivial gauge-vectors, we can apply much of the discussion of Section 5, but with a couple of twists. To do so it willbe convenient to: • Define e H F and f M ABCD implicitly by: e H F = 18 f M ABCD K AB K CD = 12 (cid:16) U IJ + A I σ A J σ (cid:17) K I K J . (7.3)Note that if the gauge-vectors vanish the identification ( e H F , f M ABCD ) → ( H F , M ABCD ) is madewith our previous discussion. • Suppose again that the uplifted geometry is invariant under g = h ⊕ . . . ⊕ h s ⊕ u (1) ⊕ . . . ⊕ u (1) ν , (7.4)where h i are simple, and g is embedded in so ( n + 1) through a chain of maximal subalgebras, so ( n + 1) ≡ g ⊃ g ⊃ . . . ⊃ g k ≡ g . (7.5) See, e.g., [31] Define ˜ g ⊂ g as the Lie algebra for the group under which the reduction gauge vectors A µ i ( x )transform and in which they are valued.Assuming that the the set of functions ( A µ i ( x ) π µ , g ω ( x )) on B are linearly independent, then sepa-ration for a given g will generically be impeded unless:(i) The Lie algebra, g , corresponds to one of the “separating isometries” discussed in section 5 (orsummarized in Appendix B) which separates the corresponding H F with vanishing gauge vectors.Since f M ABCD has the same index structure as M ABCD they transform in the same representationof g , hence e H F will separate when the corresponding H F separates for the same g .(ii) The Lie algebra ˜ g , is of the form:˜ g = ˜ h ⊕ . . . ⊕ ˜ h s ⊕ u (1) ⊕ . . . ⊕ u (1) ν , (7.6)where the ˜ h i are abelian subalgebras of the corresponding h i . This is required so that condition(i) discussed below (3.14) is satisfied, ensuring the K i are in involution and can be consistentlyset to constants.(iii) The Killing vectors, K i , generating the subalgebra ˜ g ⊂ g , are in involution with the set of Killingtensors/vectors required to separate the the e H F part of the Hamiltonian, i.e. those required toseparate the corresponding H F discussed in Section 5.This discussion makes it clear that for separation with non-trivial gauge fields, one should start byimposing one of the “separating isometries,” g , from Section 5 on H e F and restricting the gauge vectorsto some abelian subalgebra ˜ g ⊂ g . We defer the full analysis of the necessary conditions on ˜ g to futurework, but note that there may be conspiracies between the functions ( A µ i ( x ) π µ , g ω ( x )) on B whichwould alter the above discussion. , m, n ) superstrata The (1 , m, n ) superstrata, [21], belong to a three sphere reduction of D = 6, N = (1 ,
0) supergravity,coupled to two anti-self dual tensor multiplets. The solutions are microstate geometries for the D1/D5/Pblack hole.We will use the standard parametrization ( θ, φ , φ ) of the three-sphere which is embedded in theambient R as Y = sin θ sin φ , Y = sin θ cos φ , Y = cos θ sin φ , Y = cos θ cos φ . (7.7)In addition, it will be convenient to identify the generators of an su (2) ⊕ u (1) subalgebra of so (4): L = K + K , L = K − K , L = K + K ,R = K − K , (7.8)where K AB are the standard Killing vectors which generate so (4) in the coordinates (7.7) and identify34he Casimir: C so (4)2 = 14 X A,B =1 K AB K AB . (7.9)Introducing the light cone coordinates: u = 1 √ t − y ) and v = u = 1 √ t + y ) , (7.10)where y ∼ y + 2 πR y is the periodic direction along which of the branes of underlying D1/D5/P systemintersect, then the coordinates on B are specified by ( u, v, r ) where r is a radial direction. The (1 , m, n )superstrata solutions can then be parametrized in terms of the positive real constant a , the inverseradius of the three-sphere at infinity, g − , and by the specification of pair of holomorphic functions F = ∞ X n =1 b n ξ n and F = ∞ X n =1 d n ξ n , (7.11)where the b n and c n are real constants and ξ is the complex variable, ξ ≡ r √ a + r e i √ Ry v . (7.12)It will be convenient to introduce the scalars: χ A = − ag R y p a + r ) (cid:18) iF , F , − ie i √ Ry v F , e i √ Ry v F (cid:19) + c.c. , (7.13)and the combination Ω ≡ R y g (4 − χ A χ A ) = 2 g R y − a a + r (cid:16) | F | + | F | (cid:17) . (7.14)The full (1 , m, n ) superstrata can then be fully specified by: H B ( x ; π ) = − π a g R y + 12 a R y r ( a + r )Ω r π g + 4 a R y r π ( π − π ) g + a R y ( π − π ) + 2 a R y r ( a + r ) π g ! , (7.15) − A µi ( x ) π µ K i = 1 √ g R y Ω " a r ( π − π )( R + L ) + a a + r − a g R y ! π − π ! ( R − L ) + π √ a g R y Ω [( χ χ − χ χ ) L − ( χ χ + χ χ ) L ]+ π / a g R y Ω (cid:2) ( χ + χ )( R + L ) + (cid:0) χ + χ (cid:1) ( R − L ) (cid:3) , (7.16) There is no need to consider C su (2)2 separately since it is proportional to C so (4)2 , see (6.9) and (5.16). This radius is proportional to the D1/D5 charges via g = ( Q Q ) − / . X ω g ω ( x ) K ω ( y ; p ) = 14 g R y Ω (cid:20) C so (4)2 + a r ( L + R ) − a a + r ( L − R ) (cid:21) , (7.17)There are two possible impediments to separability to note at this juncture: • The gauge vectors identified in (7.16) belong to the non-abelian su (2) ⊕ u (1) ⊂ so (4), so the K i will not be in involution. • The base part of the massless Hamilton-Jacobi equation, (7.15), may not separate on B , since Ω may mix all three base coordinates ( u, v, r ) in a highly no-trivial manner.We will show in the next subsection that each of these points can be overcome by appropriatelyrestricting F and F , leading to partially and fully separable massless Hamilton-Jacobi equations forsubfamilies of the (1 , m, n ) superstrata. × U(1) subsectors
In order to produce a partially separable solution between the base and fiber, points one and two inthe list of the previous subsection must be addressed. This is simply done by restricting the isometrygroup and gauge vectors to a U(1) × U(1) subgroup. There are two distinct ways of accomplishing this,by either setting F = 1 or F = 0, giving the (1 , , n ) and (1 , , n ) subfamilies respectively. Denotingthe generators of U(1) × U(1) by (
L, R ), they can be identified in our three-sphere conventions as: L = K and R = K . (7.18)The corresponding Casimirs are then given by: C u (1) L = L and C u (1) R = R . (7.19)The most general U(1) × U(1) invariant P ω g ω ( x ) K ω ( y ; p ) in concert with the most general expansionof − A µi ( x ) π µ K i for U (1) × U (1) gauge vectors then takes the form: − A µi ( x ) π µ K i = h L ( x ) L + h R ( x ) R , (7.20) X ω g ω ( x ) K ω ( y ; p ) = g ( x ) C so (4)2 + g ( x ) C u (1) L + g ( x ) C u (1) R , (7.21)where h L,R ( x ) and g ω ( x ) are arbitrary functions of the base.Since all of the ( K i , K ω ) appearing in (7.20)-(7.21) are in involution and are sufficient in num-ber, the fiber is guaranteed to be fully separable and partially separate from the base in the masslessHamilton-Jacobi equation. This is purely a consequence of the U(1) × U(1) isometry imposed on P ω g ω ( x ) K ω ( y ; p ) = (cid:16) U IJ + A I σ A J σ (cid:17) K I m K J n and restricting the gauge group to the abelianU(1) × U(1) group. 36 .3.1 The (1 , , n ) superstrata The (1 , , n ) subfamily of superstrata is given by setting F = 0, or equivalent χ = χ = 0, in the(1 , m, n ) family of Section 7.2, giving: H B ( x ; π ) = − π a g R y + 12 a R y r ( a + r )Ω , ,n ) r π g + 4 a R y r π ( π − π ) g + a R y ( π − π ) + 2 a R y r ( a + r ) π g ! , (7.22) − A µi ( x ) π µ K i = √ g R y Ω , ,n ) " a r ( π − π ) L + a a + r π a g R y + π − π ! R + √ a g R y − g R y Ω , ,n ) ! π R , (7.23)and X ω g ω ( x ) K ω ( y ; p ) = 1 g R y Ω , ,n ) (cid:20) C so (4)2 + a r C u (1) L − a a + r C u (1) R (cid:21) , (7.24)where Ω , ,n ) = 12 R y g (cid:0) − χ − χ (cid:1) = 2 g R y − a a + r | F | . (7.25)From these expressions we can see that the fiber will separate internally and partially from the baseby setting: L = c L , R = c R , C so (4)2 = c . (7.26) , , n ) superstrata The (1 , , n ) subfamily of superstrata is given by setting F = 0, or equivalent χ = χ = 0, in the(1 , m, n ) family of Section 7.2, giving: H B ( x ; π ) = − π a g R y + 12 a R y r ( a + r )Ω , ,n ) r π g + 4 a R y r π ( π − π ) g + a R y ( π − π ) + 2 a R y r ( a + r ) π g ! , (7.27) − A µi ( x ) π µ K i = √ g R y Ω , ,n ) " a r ( π − π ) L + a a + r π a g R y + π − π ! R − √ a g R y − g R y Ω , ,n ) ! π L , (7.28)37nd X ω g ω ( x ) K ω ( y ; p ) = 1 g R y Ω , ,n ) (cid:20) C so (4)2 + a r C u (1) L − a a + r C u (1) R (cid:21) , (7.29)where Ω , ,n ) = 12 R y g (cid:0) − χ − χ (cid:1) = 2 g R y − a a + r | F | . (7.30)From these expressions we can see that the fiber will separate internally and partially from the baseby setting: L = c L , R = c R , C so (4)2 = c . (7.31) Consider again the the base part of the Hamiltonian for the (1 , m, n ) superstrata (7.15): H B ( x ; π ) = − π a g R y + 12 a R y r ( a + r )Ω r π g + 4 a R y r π ( π − π ) g + a R y ( π − π ) + 2 a R y r ( a + r ) π g ! . (7.32)This part of the Hamiltonian is independent of and so separates trivially in u . The only dependence on v comes though (7.14):Ω = 2 g R y − a a + r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 b n (cid:18) r √ a + r e i √ Ry v (cid:19) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 d n (cid:18) r √ a + r e i √ Ry v (cid:19) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (7.33)which clearly obstructs separation between ( v, r ) for generic F = P ∞ n =1 b n ξ n and F = P ∞ n =1 d n ξ n .However if only a single term appears in each of the sums, say b n and d n , then the dependence on v will be removed: Ω = 2 g R y − a a + r (cid:20) b n (cid:18) r a + r (cid:19) n + d n (cid:18) r a + r (cid:19) n (cid:21) , (7.34)so that H B ( x ; π ) is trivially completely separable on B .Thus, if one restricts to single (1 , , n ) or (1 , , n ) the six-dimensional massless Hamilton-Jacobiequations will be fully separable. This separability has been extensively exploited in [70–73]. In this paper we have explored the interplay between consistent truncation and separability of theHamilton-Jacobi equation in supergravity solutions. This work was stimulated by the discovery ofseparability in single-mode superstrata in [70–73], however, we have now shown that this was far from38n accident: the structure of the uplifted metric of any consistent truncation is well-adapted to possibleseparability of both the wave equation and the Hamilton-Jacobi equation. This does not mean a genericconsistent truncation will result in separability, and we have exhibited several, relatively simple exampleswhere such separability fails.To analyze this situation we introduced the idea of a separating isometry, which implies that ifa consistent truncation without reduction gauge vectors preserves this isometry, then the metric willnecessarily have a separable Hamilton-Jacobi equation. We then classified the separating isometrieson sphere truncations, S n , for n = 2 , . . . ,
7. The surprise is that there are, in fact, many non-trivialseparable isometries, and this is what we mean when we say that consistent truncations are “well-adapted” to separability. When reduction gauge vectors are present, in addition to the requirementof a separating isometry, the gauge group must be restricted to an abelian subgroup of the specificseparating isometry in order to not impede separability.Obviously, specifying isometries will imply the existence of Killing vectors, and the associated con-served momenta, but the separable isometries on sphere are “too small” for the Killing vectors to fullyseparate the Hamilton-Jacobi equation. This means that there must be rank-two Killing tensors tocomplete the separation, and in this paper we have classified and constructed these rank-two Killingtensors using bilinears of the Killing vectors on the sphere.While we have classified separability using generic consistent truncations that preserve a specifiedisometry, this does not mean we have a complete classification of separability, even for consistenttruncations. We used “genericity” of the background to show that the Killing tensors appearing in theexpansion of the inverse metric must all be in involution with one another. As we noted earlier, for non-separating isometries there can be specialized loci of the consistent truncation on which the Hamilton-Jacobi equation becomes separable. In particular, this can happen at points in the moduli space atwhich the actual symmetry is enhanced to a separating isometry. However, there are, almost certainlyless trivial examples that do not involve symmetry enhancement, or could be based on symmetryenhancement that involves discrete isometries. We have not classified these possibilities.The important practical conclusion here is that, if a consistent truncation has a separating isometrythen the Hamilton-Jacobi equation is always separable in the absence of reduction gauge vectors.
Further,if reduction gauge vectors are present, they must be restricted to an abelian subgroup of the separatingisometry in order to not impede separability.
If a consistent truncation only has a non-separating isom-etry then the Hamilton-Jacobi equation is generically non-separable, but there can still be specializedconfigurations in which the Hamilton-Jacobi equation becomes separable.Combining the mathematics of separability with the metrics arising through consistent truncationsis not only a very interesting formal problem, but it also has important impact on supergravity, stringcompactifications and holography.In any circumstance in which one creates a supergravity background to address a physics problem,one of the first priorities is to probe that background, and geodesics and scalar waves are the simplest ofsuch probes. Analysis then leads to issues of separability and this determines how one tries to solve forthe dynamics of probes, and the extent to which such analysis can be achieved. Consistent truncationsarise in situations in which the higher dimensional physics of string theory, as well as holography arisingout of string theory, can be reduced to some simpler, lower-dimensional system. Thus our analysis39asts a broad light on the extent to which one can analyze probes of a rich families of supergravitybackgrounds. Indeed, in Section 6 we catalogued families of important holographic backgrounds inwhich the Hamilton-Jacobi equation is separable.Beyond this utility, this work has two immediate applications:(i) Given a supergravity solution, one can check whether the massless Hamilton-Jacobi equationpartially separates. If so it may be a sign the the solution is part of a consistent truncation whichmay subsequently be identified. A first step in this identification would be to check that thegeometry conforms to the uplifted form of (3.6).(ii) Given a pair of supergravity theories related by consistent truncation, the separating isometries(see the tables in Appendix B) may be used to constrain ans¨atze for solutions which will necessarilyhave a partially separable massless Hamilton-Jacobi equations between the fiber and base in theuplift. Such solutions will be more amenable to analytic investigation due to the additionalsymmetry.The first application was precisely the historical development that occurred with the (1 , m, n ) su-perstrata, [19–21], which motivated this work. Examples which give weight to the second application,include the microstate geometry probe calculations of [70–73] and the calculation of the spin 2 spectrumof the solution of Section 6.1.1 in [74]. This later calculation was used as a check against the resultscoming directly from exceptional field theory using the methods of [75, 76]. It is interesting to note thatthe examples where purely analytic results for the spin 2 spectrum can be found in [77, 74, 78], possessseparating isometries. This is suggestive that studying solutions with separating isometries within theframework of exceptional field theory may prove particularly useful.There are also other interesting avenues of investigation that follow on from our work. We havenot examined the separability of the scalar wave equation in any great detail. We noted in Section 2that separability of the wave equation is a stronger constraint than separability of the Hamilton-Jacobiequation, and so the latter is necessary for the former. It would be interesting to generalize our discussionto the wave equation and see if the concept of a separating isometry has similar leverage in separatingthe wave equation. It is also natural to broaden the question to consider a notion of separabilitythat encompasses charged scalars that transform in finite-dimensional representations of some non-abelian gauge symmetry, and how that gauge symmetry might interact with a separating isometry. InAppendix A we make some first steps in looking at this broader notion of separability and, once again,show that consistent truncations are well-adapted to separability.It is also interesting to consider separability in a broader context than just consistent truncations.
Given a manifold which can be written as a fibration, is it possible to develop a set of conditions, thatif satisfied by the fiber, lead to partial separability of the Hamilton-Jacobi equation between the base andfiber?
In this work the fiber was a deformed sphere restricted as necessary for a consistent truncationwithin supergravity, while the conditions amounted to a preservation of separating isometries and arestriction of the gauge group to some abelian subgroup of the separating isometry group. However, instring theory, compactifications on Calabi-Yau manifolds are ubiquitous, so it is potentially extremelyuseful to determine whether there is a comparable analysis of Calabi-Yau fibers which are adapted for40eparability. To begin addressing this problem one could consider simple K¨ahler or hyper-K¨ahlerfibers, so that conditions can be readily imposed via K¨ahler potentials.Thus we have shown that the observations of [70–73] made in the context of microstate geometriesin six dimensions was the tip of a much larger ice-berg that extends across all consistent truncations andhas a broad impact for holographic field theory. Indeed, it means that solving the dynamics of probesin supergravity backgrounds could be a rather more tractable problem that one might have naivelyexpected. Acknowledgments
We are grateful to Nikolay Bobev and Jesse van Muiden for discussions. The work of KP and NW wassupported in part by the DOE grant DE-SC0011687. NW and RW were supported by the ERC Grant787320 - QBH Structure. RW was also supported by the Research Foundation - Flanders (FWO). KPand RW are grateful to the IPhT Saclay for hospitality during the initial stage of this project.
A Separating the scalar wave equation
As we remarked in Sections 1 and 2, the geometric optics approximation to the wave equation, andits realization through WKB methods, mean that the separability of the scalar wave equation impliesthe separability of the Hamilton-Jacobi equation, but the converse is not true. This link is furtherweakened if the scalars are coupled to electromagnetic fields. Moreover, if, instead of considering asingle scalar, we broaden the notion of separability to include finite families of scalars that transformin a non-trivial representations of a non-abelian gauge symmetry, then one really has lost the link withthe Hamilton-Jacobi equation.Charged scalar waves are also very interesting probes of supergravity solutions and this is an issuewell worth exploring in the future: what can we say about consistent truncations, and the broader notionof separability for the wave equation? In this appendix we make a short excursion into these ideas andonce again find that there is a very powerful synergy between the metrics of consistent truncation andthe broader notion of separability.The massive scalar wave equation on M is given by: ∇ M ∇ M Ψ = 1 p | G | ∂ M (cid:16)p | G | G MN ∂ N Ψ (cid:17) = − m Ψ . (A.1)The Kaluza-Klein ansatz, (3.1), for the metric can be written in terms of frames as E M A = E µα B µm e ma e ma , (A.2) For an example of such a manifold where separability proved useful see the conifold solutions of [79]. α and a are flat along B and F , respectively. The rescaling of the frames corresponding to (3.2) is E µα = ∆ − d − e µα . (A.3)so that ∆ ≡ det( e ma )det( ◦ e ma ) = s det( g mn )det( ◦ g mn ) , (A.4)where ◦ e ma is a frame for the round metric ◦ g mn . From (A.3) and (A.4) we see that, p | G | = ∆ − d − det( e µα ) det( ◦ e ma ) = ∆ − d − q − det( g µν ) det( ◦ g mn ) , (A.5)and hence∆ − d − ∇ M ∇ M Ψ = 1 p | g | ∂ µ h p | g | g µν (cid:0) ∂ ν − A I ν K I m ∂ m (cid:1) Ψ i + 1 p ◦ g ∂ m (cid:20) q ◦ g (cid:0) − g µν A I ν K I m ∂ µ Ψ (cid:1) + (cid:0) U IJ + g ρσ A I ρ A J σ (cid:1) K I m K J n ∂ n Ψ (cid:21) , (A.6)where ◦ g is the determinant of the round metric on F . This reduces to: ∇ M ∇ M Ψ = ∆ d − h g µν (cid:0) ∇ µ − A I µ L I (cid:1) (cid:0) ∇ ν − A J ν L J (cid:1) Ψ + U IJ L I L J Ψ i , (A.7)where, g and ∇ µ are the determinant of the metric and the covariant derivative on B respectively. Theexpression: L I F ≡ K I m ∂ m F , (A.8)denotes the Lie derivative. To establish (A.7) one must use the Killing equation for K I m on the “round”compact manifold to remove the divergences of K I m .Observe that the operator U IJ ( x ) L I L J , (A.9)is a deformation of the Laplacian of the “round” metric on F and can be simplified in terms of theeigenmodes of this operator. Indeed, suppose the scalar, Ψ( x, y ), can be written in terms of a set ofharmonics that form a representation of the isometries on the round sphere:Ψ( x, y ) = X j ψ j ( x ) Y j ( y ) . (A.10)Then there is a constant representation matrix, T I ij , such that L I Y i ( y ) = T I ij Y j ( y ) , (A.11)and (A.7) reduces to∆ − d − ∇ M ∇ M Ψ= h g µν (cid:0) δ jk ∇ µ − A I µ T I j k (cid:1) (cid:0) δ ij ∇ ν − A J ν T J ij (cid:1) ψ i ( x ) + U IJ T I j k T J ij ψ i ( x ) i Y k ( y ) . (A.12)42here are several things to note at this juncture. First, modulo the warp factor, the wave operatorhas separated and reduced to an operator entirely on B . The dependence on F has been reduced to grouprepresentations of harmonics on the round metric. This is therefore a “consistent truncation” for anychoice of harmonics, Y j ( y ). This is what we mean when we refer to a “broader notion” of separabilityfor the wave equation. The standard (stronger) form of separability involves a single function on B anda single function on F . Here the separation of variables involves a finite space of modes.If the background vector fields, A I µ , are restricted to abelian U(1) subgroups of the isometries, A ˆ I µ , with charges q ˆ I . Then one can construct a Y ( y ), giving a one-dimensional eigenspace under theseU(1)’s, with: L ˆ I Y ( y ) = iq ˆ I Y ( y ) . (A.13)When this occurs, the first part of the operator in (A.7), reduces to: h g µν (cid:0) ∇ µ − iq ˆ I A ˆ I µ (cid:1) (cid:0) ∇ ν − iq ˆ J A ˆ J ν (cid:1) ψ ( x ) i Y ( y ) . (A.14)This part of the Laplacian has now separated in the more traditional (strong) sense: Ψ( x, y ) = ψ ( x ) Y ( y ).As with the Hamilton-Jacobi equation, the massive wave equation, (A.1), is only separable if ∆ − d − separates into the sum of a function of x and a function of y . Such a separation is possible if U IJ ∝ δ IJ ,but generically it will only be the massless wave equation that will be separable.Finally, we note that the broader notion of separability described here overlaps strongly with the ideaof consistent truncation: in both instances some higher-dimensional dynamics can be reduced exactlyto dynamics in lower dimensions. It is thus an obvious question as to whether the broader concept ofseparability explored here goes beyond the idea of consistent truncation. The answer is obviously yes: aconsistent truncation typically involves a precisely specified spectrum of higher-dimensional fields witha very limited set of representations of the isometry group involved in the dimensional reduction. Herewe are demonstrating broader separability for a scalar field in any representation and in any consistenttruncation. For example, eleven-dimensional supergravity doesn’t even have a scalar field, but theresults presented here show that an eleven-dimensional scalar, in any representation of the underlyingreduction isometry, is “broadly separable” in either of the standard sphere reductions. There is thus aninteresting set of open questions about the “broader separability” of the dynamics of fields that maynot be part of the consistent truncation, and yet may be very useful probes of those backgrounds. B Summary tables
In Tables B.1 and B.2 in this appendix we list chains of isometries on n -spheres, S n , n = 2 , . . . , S n , the lie algebra, g , of an isometry imposedon the uplifted metric is given through a chain of maximal subalgebras of so ( n + 1), with each subse-quent step indicated by an indentation. For example, on S , both so (3) and su (2) ⊕ u (1) are maximal43ubalgebras of so (4), while u (1) ⊕ u (1) is a maximal subalgebra of su (2) ⊕ u (1). A particular embeddingof g in so ( n + 1) can be also determined directly from the branching of the vector representation, ,of so ( n + 1) given in the third column. In the next three columns we give the number of singlets of g in the branching of the three irreps of so ( n + 1) that comprise the metric, see (3.21). The last but onecolumn indicates whether the Hamilton-Jacobi equation is separable in the sense of Section 5, while thelast column gives a reference to a subsection where the separability or the lack thereof is discussed inmore detail.The lists of separating isometries in Tables B.1 and B.2 are complete in the sense that by imposinga smaller isometry onto the uplifted metric will generically lead to a Hamilton-Jacobi equation thatis not partially separable. This does not mean that there are no Hamilton-Jacobi equations with thatisometry that are separable, but only that the isometry itself does not guarantee separability for allmetric functions, M ABCD ( x ), in (1.8) with that invariance.Smaller subalgebras may arise in multiple chains of maximal subalgebras. To simplify the tables weusually list them only once. Similarly, we usually omit duplicates of subalgebras containing u (1) factorsthat differ either by the normalization or more generally linear combinations of the u (1) charges, seefor example the su (2) ⊕ u (1) ⊕ u (1) on S in Table B.1 or su (3) ⊕ u (1) ⊕ u (1) in Table B.2. They areequivalent as far as our problem is concerned.Additional subtlety is present on S , where an isometry subalgebra, g , may be embedded into so (8)acting on R through one of its eight-dimensional irrepses: s , v or c . Since the branchings of thetensor products of v and of the products of c are related by a reflection in R , they are equivalentas far as separability is concerned. In the first column in Table B.2, we have indicated the triality ofthe embedding of g into so (8). For the three non-separable examples indicated by the ∗ , the analysis inSection 5.4.5 is not completely exhaustive. We use the same conventions as in [46,47], with the branchings s → + , v,c → under the standard so (7) ⊂ so (8). • SEP Sec. S so (3) u (1) (2) + (0) + ( − − S so (4) so (3) + su (2) ⊕ u (1) (1) + ( −
1) 1 0 1 Y 5.2.4 u (1) ⊕ u (1) (1 ,
1) + (1 , −
1) + ( − ,
1) + ( − , −
1) 2 1 1 Y 5.2.5 S so (5) so (4) + su (2) ⊕ u (1) (0) + (1) + ( −
1) 1 1 1 Y 5.2.7 so (3) ⊕ so (2) (2) + ( −
2) + (0) 1 1 1 Y 5.2.3 so (3) S so (6) su (3) ⊕ u (1) ( −
2) + (2) 1 0 1 Y 5.2.4 su (3) + su (2) ⊕ u (1) ⊕ u (1) (2 ,
2) + ( − , −
2) + (1 , −
2) + ( − ,
2) 3 1 1 Y 5.2.5 su (2) ⊕ u (1) (2) + ( −
2) + ( −
2) + (2) 3 1 1 Y 5.2.6 so (3) ⊕ u (1) ( −
2) + (2) 2 0 1 N 5.4.2 su (2) ⊕ su (2) ⊕ u (1) ( , )(2) + ( , )( −
2) + ( , )(0) 1 1 1 Y 5.2.3 su (2) ⊕ u (1) ⊕ u (1) (0 ,
2) + (0 , −
2) + (1 ,
0) + ( − ,
0) 3 1 1 Y 5.2.5 so (5) + su (2) + so (3) ⊕ so (3) ( , ) + ( , ) 1 1 1 Y 5.2.3 S so (7) so (6) + su (3) ⊕ u (1) (0) + ( −
2) + (2) 1 1 1 Y 5.2.7 su (3) + + so (3) ⊕ u (1) (0) + ( −
2) + (2) 2 1 1 N 5.4.2 su (2) ⊕ su (2) ⊕ su (2) ( , , ) + ( , , ) 1 1 1 Y 5.2.3 su (2) ⊕ su (2) ⊕ u (1) ( , )(1) + ( , )( −
1) + ( , )(0) 2 1 1 Y 5.2.8 so (5) ⊕ so (2) (2) + ( −
2) + (0) 1 1 1 Y 5.2.3 g su (3) + + su (2) su (2) ⊕ su (2) ( , ) + ( , ) 2 1 1 N 5.4.3 Table B.1: Separating isometries on n -spheres, S n , n = 2 , . . . , • SEP Section S so (8) s, v, c su (2) ⊕ su (2) ⊕ su (2) ⊕ su (2) ( , , , ) + ( , , , ) 1 1 1 Y 5.2.3 su (2) ⊕ su (2) ⊕ su (2) ⊕ u (1) ( , , )(1) + ( , , )( −
1) + ( , , )(0) 2 1 1 Y 5.3.1 su (2) ⊕ su (2) ⊕ u (1) ⊕ u (1) ( , )(1 ,
0) + ( , )( − ,
0) + ( , )(0 ,
1) + ( , )(0 , −
1) 4 1 1 Y 5.3.1 s so (6) ⊕ so (2) (2) + ( −
2) + (0) 1 1 1 Y 5.2.3 su (3) ⊕ u (1) ⊕ u (1) (0 ,
2) + (0 , −
2) + (2 ,
0) + ( − ,
0) 3 1 1 Y 5.2.5 v, c su (4) ⊕ u (1) (1) + ( −
1) 1 0 1 Y 5.2.4 su (3) ⊕ u (1) ⊕ u (1) (3 , −
1) + ( − ,
1) + (1 ,
1) + ( − , −
1) 3 1 1 Y 5.2.5 s, v, c su (3) s so (7) + v, c so (7) su (4) + su (3) ⊕ u (1) ( )(3) + ( )( −
3) + ( )(1) + ( )( −
1) 3 1 1 Y 5.2.6 su (2) ⊕ su (2) ⊕ su (2) ( , , ) + ( , , ) 2 1 1 N ∗ sp (4) ⊕ u (1) (1) + ( −
1) 2 0 1 N ∗ s, v, c g + s so (5) ⊕ so (3) ( , ) + ( , ) 1 1 1 Y 5.2.3 v, c sp (4) ⊕ su (2) ( , ) 1 0 1 N ∗ Table B.2: Examples of (non-)separating isometries on S . C Explicit examples
In this appendix we show explicitly how the massless Hamilton-Jacobi equation (1.2): G MN ∂S∂x M ∂S∂x N = 0 , (C.1)separates for the metrics introduced in Sections 6 and 7. C.1 The N = 1 flow in Section 6.1.1 The metric for the uplift of the N = 1 holographic RG-flow in Section 6.1.1 in terms of local coordinatesin [50] reads: ds = X / ρ cosh χ (cid:18) ds , − a e ds (cid:19) , (C.2)where ds , = e A ( r ) (cid:0) η ij dx i dx j (cid:1) − dr , i, j = 0 , . . . , , (C.3)and e ds = sech χρ h dθ + ρ X ( σ + σ ) + sin (2 θ )4 X (cid:16) dφ − ρ σ (cid:17) i + ρ X (cid:16) sin θ dφ + 12 cos θ σ (cid:17) . (C.4)46he scalar fields, α ( r ) and χ ( r ), depend only on the radial coordinate, r , and X ( r, θ ) = cos θ + ρ ( r ) sin θ , ρ = e α ( r ) . (C.5)The σ j are the standard SU(2)-invariant forms, satisfying dσ i = σ j ∧ σ k . Explicitly, one can take Eulerangles, ( ϕ , ϕ , ϕ ), on SU(2) and write: σ = cos( ϕ ) dϕ + sin( ϕ ) sin( ϕ ) dϕ ,σ = sin( ϕ ) dϕ − sin( ϕ ) cos( ϕ ) dϕ ,σ = dϕ + cos( ϕ ) dϕ . (C.6)The metric has an SU(2) × U(1) × U(1) isometry, with the two U(1)’s given by a φ -translation and a ϕ -translation, which is a rotation between σ and σ . It was shown in [63] that the metric is that of adeformed Hopf fibration over stretched CP , with the Hopf fiber given by ω = sin θ dφ + cos θ σ .Assuming a separable ansatz of the form: S = S x ( x ) + S θ ( θ ) + S ϕ ( ϕ ) + S ϕ ( ϕ ) + S ϕ ( ϕ ) + S φ ( φ ) , (C.7)then the massless Hamilton Jacobi equation (C.1), reduces to the following system of differential equa-tions.Three trivial ODEs along the fiber: S ′ ϕ = c ϕ , S ′ ϕ = c ϕ , S ′ φ = c φ , (C.8)corresponding to the three commuting Killing vectors of the isometries along the ϕ , ϕ and φ coordi-nates of the metric (C.2).Two non-trivial ODEs along the fiber: S ′ ϕ ( ϕ ) + c ϕ + (cid:18) c ϕ − c ϕ cos ϕ sin ϕ (cid:19) − c ϕ = 0 , (C.9) S ′ θ ( θ ) + c φ sin θ + 4 c ϕ cos θ − c θ = 0 , (C.10)corresponding to (conformal) Killing tensors.The remaining differential equation purely on the base: g µν ∂S x ∂x µ ∂S x ∂x ν = − g ρ h − c ϕ + c ϕ ) + ( c φ − c θ + 4 c ϕ − c φ c ϕ ) ρ − c φ ρ + (cid:2) c ϕ + (cid:0) c φ − c θ + 4 c φ c ϕ (cid:1) ρ + 4 c ϕ ( ρ − (cid:3) cosh 2 χ i , (C.11)which agrees with (6.12). C.2 The N = 2 flow in Section 6.1.2 The type IIB supergravity uplift of the metric [9] for the N = 2 flow is given by: ds = Ω ds , − ds , (C.12)47here ds , is given in (6.2) and the internal metric is given by : ds = L (cosh(2 χ ) X X ) / ρ (cid:18) dθ cosh(2 χ ) + 14 ρ cos θ (cid:16) σ cosh(2 χ ) X + σ + σ X (cid:17) + sin θ dφ X (cid:19) (C.13)The radius, L , of AdS metric, (6.2), for vanishing scalars is related to gauge coupling constant, L = √ /g . The functions X , are defined by X ( r, θ ) ≡ cos θ + ρ ( r ) cosh(2 χ ( r )) sin θ ,X ( r, θ ) ≡ cosh(2 χ ( r )) cos θ + ρ ( r ) sin θ , (C.14)and the warp factor is Ω ≡ (cosh(2 χ ) X X ) / ρ . (C.15)The metric ds has an SU(2) × U(1) × U(1) isometry. The first U(1) is defined by φ -translations andthe second by rotating σ and σ . The symmetry action has co-dimension one on the compactificationmanifold, and this is manifest in the explicit dependence on the coordinate θ .Assuming a separable ansatz of the form: S ( x, y ) = S x ( x ) + S θ ( θ ) + S ϕ ( ϕ ) + S ϕ ( ϕ ) + S ϕ ( ϕ ) + S φ ( φ ) , (C.16)then the massless Hamilton Jacobi equation (C.1), reduces to the following system of differential equa-tions.Three trivial ODEs along the fiber: S ′ ϕ ( ϕ ) = c ϕ , S ′ ϕ ( ϕ ) = c ϕ , S ′ φ ( φ ) = c φ (C.17)corresponding to the three commuting Killing vectors of the isometries along the ϕ , ϕ and φ coordi-nates of the metric (C.2).Two non-trivial ODEs along the fiber: S ′ ϕ ( ϕ ) + c ϕ − c ϕ + 2 c ϕ c ϕ cos ϕ sin ϕ − c ϕ = 0 , (C.18) S ′ θ ( θ ) + c φ sin θ + 4 c ϕ cos θ − c θ = 0 . (C.19)corresponding to (conformal) Killing tensors.The remaining differential equation purely on the base: g µν ∂S x ∂x µ ∂S x ∂x ν = 1 L ρ (cid:2) c ϕ − c ϕ + c φ ρ + (cid:0) c θ − c φ − c ϕ (cid:1) ρ cosh 2 χ + 2 c ϕ cosh 4 χ (cid:3) . (C.20)which agrees with (6.10). We have rescaled the σ j in [9] by a factor of two and interchanged σ ↔ σ , so as to bring them into line with (C.6) .3 The N = 1 flow in Section 6.2.1 This example represents a flow of ABJM theory down to a non-trivial infra-red conformal fixed point[57, 58]. In four-dimensional supergravity the flow [61, 62] is defined by two scalar fields, α and χ . Thefour-dimensional metric is given by the usual Poincar´e slicing; ds , = dr + e A ( r ) (cid:0) η ij dx i dx j (cid:1) , i, j = 0 , , . (C.21)The M-theory uplift of this flow is given in [63]. The metric is given by : ds = X / ρ / cosh / χ (cid:0) ds , + ds (cid:1) , (C.22)with ds = L sech χρ ( dµ + ρ cos µX " ds CP + ρ X sin µ (cid:18) dψ − ρ − dφ −
12 sin θ σ (cid:19) + cosh χX cos µ (cid:18) dψ + tan µ dφ −
12 sin θ σ (cid:19) , (C.23)where the radii of the “round” AdS and S are L/ L respectively. The functions, ρ and X , aredefined by: ρ ≡ e α , X ( r, µ ) ≡ cos µ + ρ ( r ) sin µ . (C.24)The σ j are the standard SU(2)-invariant one-forms (C.6) and ds CP = dθ + 14 sin θ (cid:0) σ + σ + cos θ σ (cid:1) , (C.25)is the standard metric for CP .The metric, (C.22) has an SU(3) × U(1) × U(1) isometry. The SU(3) action is transitive on thestretched S defined by the Hopf fiber of ( dφ − sin θ σ ) over CP . The first U(1) is defined by ψ -translations and the second is a rotation between σ and σ . The symmetry action has co-dimensionone on the compactification manifold, and this is manifest in the explicit dependence on the coordinate µ . Applying the discussion of Section 3.2, one can now investigate the separability of the masslessHamilton-Jacobi equation.Assuming a separable ansatz of the form: S ( x, y ) = S x ( x ) + S µ ( µ ) + S θ ( θ ) + S ϕ ( ϕ ) + S ϕ ( ϕ ) + S ϕ ( ϕ ) + S φ ( φ ) + S ψ ( ψ ) , (C.26)then the massless Hamilton Jacobi equation (C.1), reduces to the following system of differential equa-tions.Four trivial ODEs along the fiber: S ′ ϕ ( ϕ ) = c ϕ , S ′ ϕ ( ϕ ) = c ϕ , S ′ φ ( φ ) = c φ , S ′ ψ ( ψ ) = c ψ , (C.27) The coordinates employed here differ to those appearing in (4.23) of [63] by φ → − ψ and ψ → − ( φ + ψ ). ϕ , ϕ , φ and ψ coordinates of the metric (C.22).Three non-trivial ODEs along the fiber: S ′ ϕ ( ϕ ) + c ϕ + c ϕ − c ϕ c ϕ cos ϕ sin ϕ − c ϕ = 0 , (C.28) S ′ θ ( θ ) + 4 c ϕ sin θ + (cid:18) c ϕ + c ψ cos θ (cid:19) − c θ = 0 , (C.29) S ′ µ ( µ ) + (cid:18) c φ sin µ (cid:19) + c θ cos µ − c µ = 0 . (C.30)corresponding to (conformal) Killing tensors.The remaining differential equation purely on the base: g µν ∂S x ∂x µ ∂S x ∂x ν = 12 L n (cid:2) c θ ( ρ − − c ψ (cid:3) ρ − − (cid:2) c µ + c φ (2 c ψ + 2 c φ ρ − c φ ) (cid:3) ρ + " c ψ − c θ ρ + ( c θ − c µ + c φ ( c φ + 2 c ψ )) ρ cosh 2 χ o , (C.31)which agrees with (6.12). C.4 The solution in Section 6.2.2
The full metric for this example is in [64] and may be written as: ds = X / Σ / ds , + ds e S , (C.32)where ds e S is a deformed seven-sphere, parametrized by the coordinates, ( χ, θ, ϕ , ϕ , ϕ , φ ), with metric: ds e S = m − (cid:18) Σ X (cid:19) / " dχ + X Σ cos χ ds CP + X Σ sin χ (cid:18) dψ + 12 sin θ σ + Ξ X dφ (cid:19) ! + 1Σ (cid:18) dφ + cos χ (cid:18) dψ + 12 sin θ σ (cid:19)(cid:19) , (C.33) ds , = e A ( r ) ds AdS + dr ., (C.34)and the CP metric is given by (C.25). In these expressions ( X, Ξ , Σ) control the deformation of theseven-sphere through the four dimensional scalar fields, λ ( r ) and ζ ( r ): X ( r ) = cosh 2 λ + cos ζ sinh 2 λ , Ξ( r ) = 2 cos ζ sinh 2 λ , Σ( r, χ ) = cosh 2 λ − cos ζ sinh 2 λ cos 2 χ . (C.35)Assuming a separable ansatz of the form: S ( x, y ) = S x ( x ) + S χ ( χ ) + S θ ( θ ) + S ϕ ( ϕ ) + S ϕ ( ϕ ) + S ϕ ( ϕ ) + S φ ( φ ) + S ψ ( ψ ) , (C.36)50hen the massless Hamilton Jacobi equation (C.1), reduces to the following system of differential equa-tions.Four trivial ODEs along the fiber: S ′ ϕ ( ϕ ) = c ϕ , S ′ ϕ ( ϕ ) = c ϕ , S ′ φ ( φ ) = c φ , S ′ ψ ( ψ ) = c ψ , (C.37)corresponding to the four commuting Killing vectors of the isometries along the ϕ , ϕ , φ and ψ coordinates of the metric (C.33).Three non-trivial ODEs along the fiber: S ′ ϕ ( ϕ ) + (cid:18) c ϕ − c ϕ cos ϕ sin ϕ (cid:19) − c ϕ = 0 , (C.38) S ′ θ ( θ ) − c ψ + 4 c ϕ + c ϕ sin θ ! + (cid:18) c ψ − c ϕ cos θ (cid:19) − c θ = 0 , (C.39) S ′ χ ( χ ) − c φ + (cid:18) c φ − c ψ sin χ (cid:19) + c θ + c ψ cos χ ! − c χ = 0 . (C.40)corresponding to (conformal) Killing tensors.The remaining differential equation purely on the base: g µν ∂S x ∂x µ ∂S x ∂x ν = m n c θ Ξ( r ) − X ( r ) h c χ + ( c φ X ( r ) − c ψ Ξ( r )) io . (C.41)which agrees with (6.15). C.5 The (1 , , n ) superstata solution in Section 7.3.1 The full metric for the (1 , , n ) superstrata may be written as: ds = ∆ g Ω ( − R y ( du + dv ) dv + 2Σ a R y (cid:18) R y ( a + r )Ω − r g (cid:19) dv + g R y Ω a + r )∆ dr − √ a R y ( du + dv ) (cid:0) sin θ dφ − cos θ dφ (cid:1) − √ a R y ( a + r )Ω − r g R y ! cos θ dv dφ + g R y Ω dθ + 2 g R y sin θ dφ + Ω cos θ dφ ) , (C.42)where Ω = 12 g R y (cid:0) − χ − χ (cid:1) , ∆ = 4 − χ − χ √ p − χ − χ + ( χ + χ ) cos 2 θ . (C.43)and Σ = r + a cos θ . (C.44)In these expressions the details of the microstate geometry are specified by the scalars ( χ , χ ), whichalways appear in the combination: χ + χ = 2 g R y (cid:18) a a + r (cid:19) | F | , (C.45)51here F is a tunable holomorphic function, depending only on the base mixing the ( u, v ) coordinates(7.11).Assuming a separable ansatz of the form: S ( x, y ) = S u ( u ) + S vr ( v, r ) + S θ ( θ ) + S φ ( φ ) + S φ ( φ ) , (C.46)then the massless Hamilton Jacobi equation (C.1), reduces to the following system of differential equa-tions.One trivial ODE along the base and two along the fiber: S ′ u ( u ) = c u , S ′ φ ( φ ) = c φ , S ′ φ ( φ ) = c φ , (C.47)corresponding to the three commuting Killing vectors of the isometries along the u , φ and φ coordinatesof the metric (C.52).A non-trivial ODE along the fiber: S ′ θ ( θ ) + c φ sin θ + c φ cos θ − c θ = 0 , (C.48)corresponding to a conformal Killing tensor.The remaining differential equation purely on the base: g µν ∂S∂x µ ∂S∂x ν − A I µ K I m ∂S∂x µ ∂S∂y m = − R y g Ω (cid:18) c θ − a a + r c φ + a r c φ (cid:19) , (C.49)where − A I µ K I m = 2 √ R y g Ω a r r a + r − − r a + r
10 0 0 + 2 √ a g R y g R y Ω (cid:18) r a + r (cid:19) − ! (C.50)and g µν is the inverse of the reduced metric on the base given by: ds = − a R y g du + dv + 2 r a R y g dv ! + g " R y a + r dr + 2 r a (cid:18) r a (cid:19) dv , (C.51)note that this metric is in the form of a time-like K¨ahler fibration, see [21, 80].52 .6 The (1 , , n ) superstata solution in Section 7.3.2 The full metric for the (1 , , n ) superstrata may be written as: ds = ∆ g Ω ( − R y ( du + dv ) dv + 2Σ a R y (cid:18) R y r Ω + 2 g ( a − r ) (cid:19) dv + g R y Ω a + r )∆ dr − √ a R y ( du + dv ) (cid:0) sin θ dφ − cos θ dφ (cid:1) − √ a R y (cid:20) g (cid:0) r sin θ dφ + a cos θ dφ (cid:1) − R y r Ω sin θ dφ (cid:21) dv + g R y Ω dθ + Ω sin θ dφ + 2 g R y cos θ dφ ) (C.52)where Ω = 12 g R y (cid:0) − χ − χ (cid:1) , ∆ = 4 − χ − χ √ p − χ − χ + ( χ + χ ) cos 2 θ . (C.53)and Σ = r + a cos θ . (C.54)In these expressions the details of the microstate geometry are specified by the scalars ( χ , χ ), whichalways appear in the combination: χ + χ = 2 g R y (cid:18) a a + r (cid:19) | F | , (C.55)where F is a tunable holomorphic function, depending only on the base mixing the ( u, v ) coordinates(7.11).Assuming a separable ansatz of the form: S ( x, y ) = S u ( u ) + S vr ( v, r ) + S θ ( θ ) + S φ ( φ ) + S φ ( φ ) , (C.56)then the massless Hamilton Jacobi equation (C.1), reduces to the following system of differential equa-tions.One trivial ODE along the base and two along the fiber: S ′ u ( u ) = c u , S ′ φ ( φ ) = c φ , S ′ φ ( φ ) = c φ , (C.57)corresponding to the three commuting Killing vectors of the isometries along the u , φ and φ coordinatesof the metric (C.52).A non-trivial ODE along the fiber: S ′ θ ( θ ) + c φ sin θ + c φ cos θ − c θ = 0 , (C.58)corresponding to a conformal Killing tensor.The remaining differential equation purely on the base: g µν ∂S∂x µ ∂S∂x ν − A I µ K I m ∂S∂x µ ∂S∂y m = − R y g Ω (cid:18) c θ − a a + r c φ + a r c φ (cid:19) , (C.59)53here − A I µ K I m = 2 √ R y g Ω a r r a + r − − r a + r
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