Laplace operators on Sasaki-Einstein manifolds

We decompose the de Rham Laplacian on Sasaki-Einstein manifolds as a sum over mostly positive definite terms. An immediate consequence are lower bounds on its spectrum. These bounds constitute a supergravity equivalent of the unitarity bounds in dual superconformal field theories. The proof uses a generalization of Kahler identities to the Sasaki-Einstein case.

(1) The right hand side of equation (1) features the Lefschetz operator, the action of the Reeb vector, the tangential Cauchy-Riemann operator as well as their adjoints. Full definitions will be given in section I. The equation ∆ = 2∆∂ can be derived from the Kähler identities, commutators between the Dolbeault and Lefschetz operators and their adjoints [1,2]. Our proof of (1) will follow a similar route; we will obtain Kähler-like identities that are valid on Sasaki-Einstein manifolds. These are summarized in appendix A.
This note is written with two audiences in mind: physicists working in gauge/string duality or supergravity and mathematicians interested in Sasaki-Einstein geometry. Therefore we split the discussion into two separate parts, giving a proof of both (1) and the identities in section I, while discussing their motivation by and relevance to physics in section II. Readers who want to focus on the mathematical aspects can ignore section II. Those not interested in the full proof should read section I up to equations (5) before skipping ahead to section II.

I. THE PROOF
The tangential Cauchy-Riemann operator∂ B and the associated Kohn-Rossi cohomology groups H p,q ∂B (S) were first introduced in [3][4][5]. Given a complex manifold with boundary, Lewy, Kohn, and Rossi considered under what circumstances functions on the boundary can be extended to holomorphic functions in the bulk. They have to satisfy the projection of the Cauchy-Riemann equations onto the boundary, hence the name for∂ B . The Kohn-Rossi cohomology groups feature also in the work by Yau and collaborators on the complex plateau * johannes.schmude@riken.jp problem [6][7][8]. This concerns the question when a real manifold is also the boundary of a complex manifold.
The tangential Cauchy-Riemann operator has properties akin to those of a Dolbeault operator on a Kähler manifold. Yet it is not the only differential operator on a Sasaki-Einstein manifold with this characteristic. The simplest example is the basic differential∂ bsc , which arises in the context of basic forms. 1 These are transverse to the Reeb vector and carry no charge under the Lie derivative along the Reeb. If one drops the zerocharge condition, it is possible to introduce a transverse differential∂ T . This has been studied by Tievsky [9], who obtained transverse Kähler identities, which are a special case of the identities derived in section I and summarized in section A. Note that we will refer to transverse forms as horizontal.
In what follows, we will give a full proof of (1) after setting the stage by giving all necessary definitions. Since the proof is based on the equivalent considerations in the Kähler case, our discussion will follow [1,2] very closely.

A. Exterior calculus on Sasaki-Einstein manifolds
Consider a d = 2n + 1 dimensional Sasaki-Einstein manifold S. Given the Reeb vector ξ and the contact form η, the tangent bundle splits as T S = D ⊕ L ξ . 2 Furthermore, there is a two-form J = 1 2 dη with i ξ J = 0. J defines an endomorphism on T S which satisfies J 2 = −1 + ξ ⊗ η. Since η(D) = 0, one can decompose the complexified tangent bundle as . This in turn induces a corresponding decomposition on the complexified cotangent bundle which also extends to the exterior algebra 1 Our notation is non-standard. The basic differential is often denoted as∂ B , with the tangential Cauchy-Riemann operator being∂ b . 2 L ξ is the line tangent to ξ. In what follows we will set L ξ = ξ and L * ξ = η. See section 1 of [10] for a review of Sasaki-Einstein geometry.
Elements of Ω * C that vanish under the action of i ξ are called horizontal, while those annihilated by η∧ are vertical.
The decomposition (2) induces a decomposition of the exterior derivative, ∂ B is the tangential Cauchy-Riemann operator. ∂ B and gives rise to the Kohn-Rossi cohomology groups H p,q ∂B (S). Continuing with the theme of generalizing concepts from Kähler geometry to Sasaki-Einstein manifolds, we define the Lefschetz operator L : allows us to define adjoints for the above operators when acting on Ω k C : Recall that on odd-dimensional manifolds ⋆ satisfies ⋆⋆ = 1.
When restricted to D, the action of J becomes that of an almost complex structure I which acts as I(α) = J n m α n dx m and I(X) = X m J n m ∂ n . Of course Ω 1,0 = {α ∈ Ω 1 |I(ω) = ıω}. We also define which makes use of the projection Π p,q : Ω * C → Ω p,q . It will turn out useful to distinguish between the rank of a form on Ω * C and on * D * . Hence we define the 3 In components By definition, d 0 commutes with L η . A first example of the uses of d 0 is given by the notion of primitive forms. α ∈ Ω * C with d 0 ≤ n is primitive if and only if Λα = 0. Essentially, the idea of primitivity on * D * is the same as on Kähler manifolds, the contact one-form just comes along for the ride and there is in no difference between horizontal and vertical forms. We define P k as the set of primitive elements of k D * .
Next we introduce an orthonormal frame e i on D * . Defining z i = e 2i−1 + ıe 2i and imposing I(z i ) = ız i , consistency requires that I(e 2i−1 ) = −e 2i and I(e 2i ) = e 2i−1 . Then In what follows, we will make use of two results concerning the Hodge star. To begin, assume that (V, , ) is a Euclidean vector space admiting a decomposition V = W 1 ⊕ W 2 that is compatible with the metric , . For simplicity we assume that dim R W i ∈ 2Z. The metrics , i induce Hodge star operators One can use identical considerations to decompose the action of ⋆ on Ω * C into seperate operations on D * and η. Introducing a hodge dual • on D * , one finds The starting point for our discussion of Lefschetz decomposition is the commutator The proof is via induction in n. Consider d = 3, n = 1.
To proceed we follow [2]. Restricting to * D * one can copy all results from proposition 6.20 to lemma 6.24. The most important of these results is Lefschetz decomposition. Given α ∈ k D * , there is a unique decomposition The decomposition is compatible with the bidigree decomposition and with the decomposition into horizontal and vertical components. Moreover, is an isomorphism and the primitivity condition is equivalent to L n−k+1 α = 0. The Lefschetz decomposition becomes incredibly useful when used together with the Bidigree decomposition, equation (7) and the identity ∀α ∈ P k , •L j α = F (n, j, k)L n−k−j I(α), Since no differential operators are involved and α ∈ k D * , one can copy the proof in [1] after adjusting for conventions. Once the dust settles, the only difference is in the k-dependent prefactor. One finds {L η , Λ η } = 1 by direct calculation using the The most involved calculation is that of the commutator Before we turn to the proof, let us try to interpret this re-  η). Similarly, the right hand side annihilates η while the left hand side does not. One can guess the correct result by considering the action of both sides on J and η, adding suitable terms on the right hand side to achieve equality.
The proof of (12) is once again an elaboration on the proof for Kähler manifolds in [1]. Let us first consider horizontal forms. Here, it is sufficient to explicitly evaluate the action of (12) L i α for α ∈ P k ; the result will generalize for generic elements of * D * due to Lefschetz decomposition. Furthermore one applies Lefschetz decomposition to∂ B α = α 0 + Lα 1 + L 2 α 2 + . . . . We have α ∈ P k and thus 0 = j L n−k+1+j α j and finally L n−k+1+j α j = 0. Using equation (10) it follows that most of the α j vanish and∂ B α = α 0 + Lα 1 .

D. Beyond Kähler identities?
Since we found Sasaki-Einstein equivalents of both ∆ = 2∆∂ and the Kähler identities, it is tempting to ask how much more of Kähler geometry can be generalized. For example, since ∆∂ is self-adjoint and elliptic, one can show that Ω k C = H k ⊕∆∂ (Ω k C ) which implies Hodge's theorem. Similarly, the relation between the de Rham and Hodge Laplacians allows for an isomorphism between the respective spaces of harmonic forms. However, it turns out that ∆∂ B is not elliptic. We will sketch the calculation leading to this result. Recall that ∆∂ B is elliptic if the symbol σ ∆∂ B ∈ Hom(Ω k C , Ω k C ) ⊗ S 2 (T * S) maps any non-zero ω ∈ T * S to an automorphism on Ω k C . When calculating the symbol one essentially keeps only those terms of ∆∂ B that are of highest order in derivatives. In the context of the tangential Cauchy-Riemann operator, this means that ∂ B and∂ B can be taken to be anticommuting and that the overall result is essentially the same as for the symbol of the Dolbeault Laplacian on a Kähler manifold, provided one substitutes Therefore, σ ∆∂ B (ξ) = 0 and ∆∂ B is not elliptic. Tievsky's discussion of a transverse Laplacian ∆ T arrives at a similar result. In that case, it turns out that ∆ T − (Λ η d) 2 is elliptic. A similar result should hold here, possibly after replacing Λ η d with £ ξ [9]. El Kacimi-Alaoui has studied elliptic operators acting on basic forms [11].

II. MOTIVATION AND APPLICATIONS
Both equation (1) as well as the identities in appendix A find application in the AdS/CFT correspondence. Freund-Rubin compactification on Sasaki-Einstein manifolds yields supergravity duals of superconformal field theories [12][13][14][15][16]. The AdS/CFT dictionary links the conformal energy of SCFT operators to the spectrum of ∆, their R-charge to that of the Lie-derivative along the Reeb vector, £ ξ . The conformal energy, R-charge, and spin of any SCFT operator have to satisfy the unitarity bounds [17], which should be reflected on the supergravity side in the spectrum of ∆. We will argue shortly that equation (1) allows us to re-derive the unitarity bounds from supergravity when considered in conjunction with the calculations in [18,19].
First, note that the Kähler-like identities allow for a study of the eigenmodes of ∆. In the case where the Sasaki-Einstein manifold has a coset structure, this has been done using harmonic analysis [20]. [18,19] obtained the structure of the Kaluza-Klein spectrum of generic Sasaki-Einstein manifolds using a construction similar to that in [21], which can be nicely summarized in terms of the identities in appendix A: Given any eigen k-form ω of ∆, one diagonalizes the action of ∆ on the k + 1-forms {∂ B ω,∂ B ω, L η ω, Lω, ∂ B∂B ω, . . . }. The resulting eigenstates fill out representations of the superconformal algebra, equivalence classes in Kohn-Rossi cohomology groups H p,q ∂B (SE) correspond to short multiplets. Whereas the original calculations were based on a rather tiresome direct approach, the methods developed in this note are expected to simplify that kind of anlysis considerably.
With this in mind, we turn to the spectral problem for ∆. Consider a k-form ω with £ ξ ω = ıq, q ≥ 0, and d 0 ≤ n. All terms on the right hand side of (1) are positive definite except for the mixed term M = ı(L η∂ * B − ∂ B Λ η ) = N + N * . M is self-adjoint and its spectrum is real. Moreover, N 2 = 0 and N ( * D * ) ⊂ * D * ∧ η and N ( * D * ∧η) = 0. That is, N maps horizontal to vertical forms and annihilates the latter. N * behaves accordingly and it follows that ω, M ω vanishes if ω is horizontal or vertical. This is also the case if ω is neither horizontal nor vertical yet holomorphic. 4 As long as we restrict to one of these cases, (1) takes the form of a bound on the spectrum of ∆. This was conjectured and partially shown in the context of the calculations of the superconformal index [22,23] in [18,19]. Here, the spectrum was constructed from primitive elements of Ω p,q . For such forms, (1) implies with equality if and only if∂ B ω =∂ * B ω = 0. In the Kähler case, the latter of these is implied by transversalityd * ω = 0. Here however, d * ω = 0 leads only to the vanishing of the horizontal component of∂ * B ω. Indeed, which vanishes since ω was assumed to be primitive. Assuming that every element of H p,q ∂B (S) has a representative closed under∂ * B , the bound (13) is saturated on the elements of H p,q ∂B (S). These are the forms that correspond to the short multiplets in the SCFT, and (13) together with the expressions for the derived eigenmodes of ∆ given in [18,19] allows to recover the unitarity bounds from supergravity. Note that (13) and a precursor to (1) were already conjectured in those references. Furthermore, the appendix of [19] contains an argument that every element of H p,q ∂B (S) is either primitive, carrying zero charge, or both. For the cases of interest in the context of that paper it turned out that all elements are primitive.
A further application of (1) is the stability analysis of Pilch-Warner solutions in [24]. In the absence of general theorems concerning Laplace operators on Sasaki-Einstein manifolds, the authors constructed explicitly examples of primitive, basic (1, 1)-forms whose existence renders these solutions perturbatively unstable. Assuming that the calculations in [24] generalize to generic transverse forms, our results might allow for a continuation of their analysis to manifolds where explicit constructions are not feasible.
Finally, it would be interesting to extend the results presented here beyond the Sasaki-Einstein case. As long as there is a dual SCFT, there is a unitarity bound meaning that there should be some equivalent of (1) or at least (13).