Bootstrap Bounds on Closed Einstein Manifolds

A compact Riemannian manifold is associated with geometric data given by the eigenvalues of various Laplacian operators on the manifold and the triple overlap integrals of the corresponding eigenmodes. This geometric data must satisfy certain consistency conditions that follow from associativity and the completeness of eigenmodes. We show that it is possible to obtain nontrivial bounds on the geometric data of closed Einstein manifolds by using semidefinite programming to study these consistency conditions, in analogy to the conformal bootstrap bounds on conformal field theories. These bootstrap bounds translate to constraints on the tree-level masses and cubic couplings of Kaluza-Klein modes in theories with compact extra dimensions. We show that in some cases the bounds are saturated by known manifolds.


Introduction
There are several cases of historically intractable problems in physics which have seen significant recent progress arise from a numerical bootstrap approach. To employ such an approach, one starts with some general principles, derives from them consistency conditions that the observable quantities of interest must satisfy, and then solves these consistency conditions numerically on a computer. This gives general constraints on the possible values that the observables can take. Examples of interesting physical systems that have been recently studied using such methods include strongly coupled conformal field theories (CFTs) [1], scattering amplitudes [2,3], and random matrix models [4]. These studies have given information about these systems which is generally out of reach of traditional perturbative methods.
In this paper we show that it is possible to apply a similar numerical bootstrap approach to obtain nontrivial mathematical results about closed Einstein manifolds. A closed Einstein manifold M is a compact smooth manifold without boundary that is equipped with a metric g mn whose Ricci curvature satisfies R mn = λg mn for some constant λ. We limit our consideration to closed Einstein manifolds that are connected and orientable, as well as the quotients of such manifolds by isometry subgroups. We find new bounds on the geometric data associated with such manifolds. Specifically, the bounds involve the spectrum of eigenvalues of various Laplacian operators and the triple overlap integrals of their eigenmodes.
Our bounds come from studying certain consistency conditions involving the geometric data. For a simple example of these consistency conditions, consider the scalar Laplacian, ∆. A scalar eigenfunction ψ a with eigenvalue λ a satisfies the eigenvalue equation ∆ψ a = λ a ψ a . For any three eigenfunctions we can construct the following triple overlap integral: (1.1) The eigenfunctions can be chosen so that they form an orthonormal basis, so we can write the product ψ a 1 ψ a 2 as a sum over eigenmodes ψ a 1 ψ a 2 = a g a 1 a 2 a ψ a , (1.2) where orthonormality gives the coefficients as the triple overlap integrals (1.1). By repeatedly applying (1.2) to pairs of eigenmodes, integrals involving products of four or more eigenfunctions can be reduced to sums of products of the triple overlap integrals. For example, we can write M ψ a 1 ψ a 2 ψ a 3 ψ a 4 = a g a 1 a 2 a g a 3 a 4 a , where we used the completeness relation (1.2) on ψ a 1 ψ a 2 and ψ a 3 ψ a 4 . Now the key point is that this expansion can be done in other ways by pairing different eigenfunctions. Equating the resulting expressions then gives consistency conditions for the triple overlap integrals. For example, exchanging a 2 ↔ a 3 in Eq. (1.3) and equating with the original result gives a g a 1 a 2 a g a 3 a 4 a = a g a 1 a 3 a g a 2 a 4 a . (1.4) Consistency conditions of the form (1.4) will be the basic relations that we exploit to get bootstrap bounds on the geometric data. By considering quartic overlap integrals involving derivatives of scalar eigenfunctions, we will find consistency conditions that are much more complicated and involve various triple overlap integrals and eigenvalues. The consistency conditions usually cannot be solved directly, but we can extract nontrivial bounds by reformulating them as a semidefinite program. An example of such a bound is shown in Figure 1, which shows numerical upper bounds on the ratio of the first two distinct non-vanishing eigenvalues of the scalar Laplacian on closed Einstein manifolds with non-negative scalar curvature as a function of the smallest Lichnerowicz eigenvalue on transverse traceless tensors. We obtain many different bounds of this sort on eigenvalues and triple overlap integrals, both numerical and exact.
There is a sharp analogy between this geometry bootstrap and the modern conformal bootstrap [1] (see Ref. [5] for a recent review), which heavily inspires our approach. The geometric data we consider is analogous to the conformal data that defines a CFT: the eigenmodes are analogous to CFT operators, their eigenvalues are analogous to the conformal dimensions, and their triple overlap integrals are analogous to the operator product expansion (OPE) coefficients. The eigenfunction expansions such as (1.2) are like the OPE in a CFT and the consistency conditions such as (1.4) are like the crossing relations that form the basis of the modern CFT bootstrap. See Table 1  There are many existing results concerning the spectrum of eigenvalues of Laplacian operators on compact Riemannian manifolds [6], but bounds on the triple overlap integrals are less well studied. As far as we know, the bounds we find for Einstein manifolds are not known in the mathematical literature. Often it is difficult to determine explicitly the eigenvalues and eigenfunctions of a given manifold (see Appendix A for some exceptions). For example, for compact Calabi-Yau manifolds there are no nontrivial examples for which the spectrum of the scalar Laplacian is known analytically and numerical methods must be used [7]. One numerical method for computing the low-lying eigenvalues is to use Donaldson's algorithm to approximate the Ricci-flat Kähler metric [8,9], as in Ref. [10]. This is what one might call a Monte Carlo approach, where accurate results are obtained for a specific manifold. In contrast, the bootstrap approach is able to find bounds that apply to all Einstein manifolds satisfying a few assumptions.
While the results we derive can be considered as purely mathematical results about closed Einstein manifolds, there is also a physical interpretation in terms of Kaluza-Klein theories. Starting with a higher-dimensional theory on a product spacetime, one can integrate over any compact extra dimensions to obtain an equivalent lower-dimensional theory. This lower-dimensional the-ory will contain towers of particles of various spins which correspond to the excitations of the higher-dimensional fields in the extra dimensions. The masses of these Kaluza-Klein modes are determined by the eigenvalues of Laplacians on the internal manifold and the strengths of their three-point interactions are determined by the triple overlap integrals of the eigenmodes. In the case where the internal manifold is Einstein, the mathematical bounds on eigenvalues and overlap integrals we obtain thus translate into bounds on the tree-level masses and cubic interactions of these Kaluza-Klein modes. This is the case, for example, in reductions of pure gravity and for Calabi-Yau reductions of supergravity.
Conventions: We use the Einstein summation convention for spatial indices n, m, . . . but not for the indices a, i, I labelling eigenfunctions. We (anti)symmetrize with weight one. The curvature conventions are those of Ref. [11].

Geometry review
We start by setting our notation and reviewing some basic Riemannian geometry that we need for the rest of the paper. We also review the geometric consistency conditions discussed in Ref. [12].

Einstein manifolds
Let M be an N -dimensional Einstein manifold, with metric g mn , that is connected, orientable, and closed (compact without boundary). We denote the volume of M by V and its Ricci curvature by R mn . By definition, an Einstein manifold satisfies where λ is a real constant. Taking the trace of Eq. (2.1) shows that the Ricci scalar R is also constant, given by R = N λ. Einstein manifolds with R = 0 are called Ricci-flat manifolds.
Einstein manifolds are solutions to the vacuum Einstein equations with a possibly nonzero cosmological constant. For this work we are interested in the case of Riemannian Einstein manifolds and their application as the compact spatial dimensions of a Lorentzian Kaluza-Klein product spacetime. The case N = 1 would require a separate treatment for many of our formulae and is already completely understood (it is just the circle), so henceforth we restrict to N ≥ 2. See Ref. [13] for a classic reference on Einstein manifolds.

Laplacians
We need to introduce three different Laplacian operators on the closed Einstein manifold M: the scalar Laplacian, the vector Hodge Laplacian, and the Lichnerowicz Laplacian on tensors. These are self-adjoint elliptic operators (with respect to the appropriate inner products), so we can use standard results of spectral theory on compact manifolds. For example, on any given manifold each Laplacian has a discrete spectrum that is bounded from below and unbounded from above and there exists a basis of real orthonormal eigenmodes. In Kaluza-Klein theories these eigenmodes correspond to different particles and their eigenvalues determine the masses of the particles (see, e.g., Ref. [14] for details).

Scalar Laplacian
We denote by ψ a an orthonormal basis of non-constant real eigenfunctions of the scalar Laplacian on M, where a is a discrete index labelling the different eigenfunctions. These eigenfunctions satisfy where λ a > 0 is the corresponding eigenvalue. Orthonormality implies that where M denotes the integral over M with the canonical volume form. The normalized constant eigenfunction is V −1/2 . This is the unique zero mode for the scalar Laplacian and we now treat it separately from the non-constant eigenfunctions (unlike in the introduction). Completeness tells us that any L 2 -normalizable scalar function φ on M can be expanded as where c 0 = V −1/2 M φ and c a = M φψ a . There are special eigenfunctions of the scalar Laplacian called conformal scalars, which are defined as those scalars whose gradients are conformal Killing vectors that are not Killing vectors. We index these by the set I conf. . Conformal scalars satisfy the equation and they exist only on the round spheres [15]. The conformal scalars are precisely the L = 1 spherical harmonics on S N if N > 1 (see Appendix A). 1 On Einstein manifolds with R > 0 the Lichnerowicz bound gives [17] and this is saturated only by conformal scalars. This is analogous to a CFT unitarity bound.

Vector Laplacian
Denote by Y m,i an orthonormal basis of real, transverse eigenvectors of the one-form Hodge Laplacian on M, where i is a discrete index labelling the different eigenvectors. These eigenvectors satisfy where λ i ≥ 0 is the corresponding eigenvalue. Orthonormality implies that Using completeness and the Hodge decomposition, any one-form V m on M can be expanded as Killing vectors are special transverse eigenvectors that generate the isometries of the metric. We index them by the set I Killing . They satisfy the Killing equation, (2.10) There is another lower bound on the eigenvalues λ i for closed Einstein manifolds, which is saturated only by Killing vectors. Since λ i ≥ 0 there are no nontrivial Killing vectors on closed Einstein manifolds with R < 0.

Lichnerowicz Laplacian
Lastly, we denote by h T T mn,I an orthonormal basis of real transverse traceless eigentensors of the Lichnerowicz Laplacian on M, where I is a discrete index labelling the different eigentensors. On an Einstein manifold, these eigentensors satisfy where ∆ L is the Lichnerowicz Laplacian and λ I is a Lichnerowicz eigenvalue. 2 Orthonormality implies that M h T T mn,I 1 h mn,T T (2.13) Using completeness and the tensor decomposition, any symmetric tensor field T mn on M can be expanded as (see, e.g., Ref. [14] for details) where the coefficients are There is no general lower bound on Lichnerowicz eigenvalues, so a finite number of them can be negative on any given manifold. For example, the Böhm metrics on S 3 × S 2 can have large negative Lichnerowicz eigenvalues [18]. However, some lower bounds do exist if we make additional assumptions. For example, on closed Einstein manifolds with R > 0 that admit a Killing spinor, there is the following lower bound [18,19]: .
Another distinguished Lichnerowicz eigenvalue is λ I = 2R/N . Eigentensors h T T mn,I with this eigenvalue are called infinitesimal Einstein deformations and correspond to directions in the moduli space of Einstein structures of M, i.e., the space of Einstein metrics on M modulo diffeomorphisms and volume rescalings. They give rise to massless scalars, called shape moduli, in Kaluza-Klein reductions of gravity. Einstein manifolds with λ I > 2R/N are therefore rigid, 3 i.e., isolated points in moduli space. Examples of rigid manifolds are the compact symmetric spaces discussed in Appendix A [24].

Eigenmode expansions
Since the eigenfunctions of the scalar Laplacian are complete, we can expand any product of two eigenfunctions as a sum over eigenfunctions using Eq. (2.4), where the coefficients are given by the triple overlap integrals of eigenfunctions, Triple overlap integrals such as these are natural objects associated to any closed manifold. 4 We can also expand products involving derivatives of eigenfunctions, Similarly, using the expansions for vectors and symmetric tensors in Eqs. (2.9) and (2.14) we obtain where we have defined two additional triple overlap integrals, (2.23)

Consistency conditions
We now introduce the consistency conditions that are the main tool in the bootstrap. These conditions arise from using completeness to evaluate quartic overlap integrals of eigenfunctions in multiple ways. For example, we have the identity where the Wick contraction notation denotes that the indicated pair of fields is replaced by the appropriate eigenmode expansion from Section 2.3. Evaluating the contractions gives This is a consistency condition that must be satisfied by the geometric data on any connected and orientable closed manifold. It appears in a slightly different context in Ref. [25].
For closed Einstein manifolds we can derive additional consistency conditions by evaluating more complicated integrals. We consider the following two additional identities: Including Eq. (2.25), the resulting consistency conditions are equivalent to 5 where we have defined (2.32) When R = 0, these consistency conditions are the sum rules responsible for the good high-energy behavior of the four-point tree amplitudes of massive Kaluza-Klein excitations of the graviton in general relativity (GR) with Ricci-flat compact extra dimensions [12]. In these cases the sums correspond to sums over the exchanged particles in the four-point amplitude. (See Refs. [26][27][28] for related recent work on the scattering of Kaluza-Klein states in a Randall-Sundrum model with one extra dimension.) These consistency conditions must be satisfied by the geometric data on any closed Einstein manifold M that is connected and orientable. In fact, they also hold for the quotients of such manifolds, M/Γ, where Γ is any subgroup of the isometry group of M. This is because the Γinvariant eigenmodes form a closed subsector of the consistency conditions. These quotient spaces include certain non-orientable manifolds, such as RP N with even N , as well as certain orbifolds. The quotients also include spaces that are not orbifolds. In the following sections we will usually just write "closed Einstein manifolds" instead of "quotients of connected and orientable closed Einstein manifolds." Note that we do not get additional independent consistency conditions from integrals involving more than four eigenfunctions if we consider all consistency conditions with four eigenmodes.

Bootstrap bounds
In analogy to the conformal bootstrap bounds on conformal data, we can use the consistency conditions from Section 2.4 to find bounds on the geometric data of closed Einstein manifolds. To do this we postulate some candidate geometric data, i.e., a collection of eigenvalues and triple overlap integrals, and then search for a constant vector α ∈ R 3 such that the condition can never be satisfied by this data. If such an α exists, then the candidate geometric data cannot correspond to any closed Einstein manifold. Although we have far fewer constraints than in typical conformal bootstrap problems, we will see that it is still possible to get some nontrivial bounds.

Eigenvalue bounds
We begin by looking for bootstrap bounds on the eigenvalues of the scalar Laplacian on closed Einstein manifolds. These translate to bounds on the tree-level masses of Kaluza-Klein modes.

Low-lying eigenvalues
Suppose we are given a closed Einstein manifold M with R ≥ 0. Let ψ a 1 be an eigenfunction of the scalar Laplacian on M whose eigenvalue λ a 1 is the smallest nonzero eigenvalue and let λ a 2 be the next distinct eigenvalue in order of increasing size. The nonzero scalar eigenvalues therefore satisfy We can write the eigenfunction expansion of ψ 2 a 1 as Similarly, we let λ I 1 be the smallest eigenvalue of the Lichnerowicz Laplacian on M acting on transverse traceless tensors.
The three eigenvalues λ a 1 , λ a 2 , and λ I 1 define a candidate low-lying spectrum on M. Given such a candidate spectrum, our goal is to determine if it is inconsistent. To do this, we search for a constant vector α ∈ R 3 such that the following conditions are satisfied: If such an α exists, then the candidate spectrum could not have come from a closed Einstein manifold since it is inconsistent with the consistency conditions (3.1). To reach this conclusion we have used the positivity of g 2 a 1 a 1 a and g 2 a 1 a 1 I , which follows from the reality of the eigenmode basis. Alternatively, if no such α can be found, then we conclude nothing.
The problem of finding an α that satisfies Eqs. (3.4) can be formulated as a semidefinite program (SDP). The general procedure for doing this is reviewed in Refs. [29,30] on the conformal bootstrap. Once we have the problem formulated as an SDP, we can solve it using numerical solvers. The resulting bounds are then rigorous up to the precision of the numerics. For the numerical computations in this work we mostly used the specialized program SDPB [31,32]. Some of the simpler SDPs can also be solved in Mathematica using the function SemidefiniteOptimization.
A simplifying feature of this problem is that the consistency conditions depend only on the two ratios λ a 2 /λ a 1 and λ I 1 /λ a 1 , which ensures that the resulting constraints are independent of constant rescalings of the metric. For each value of λ I 1 /λ a 1 we solve a series of SDPs to obtain an upper bound on λ a 2 /λ a 1 . These upper bounds are shown in Figure 1 for N = 2, . . . , 8. When λ I 1 vanishes we find that λ a 2 ≤ 4λ a 1 ; this a special case of the bound found in Ref. [12] and is saturated by flat tori whose first eigenvalues are identical to those of a circle, which we call "long flat tori." For λ I 1 > 0, the upper bounds decrease linearly until λ I 1 = λ a 1 , at which point they exhibit kinks and thereafter become constant. The upper bounds grow linearly for negative λ I 1 , so we cannot rule out large values for the ratio of the first two nonzero eigenvalues of the scalar Laplacian on manifolds with large negative Lichnerowicz eigenvalues. In the plot we have also marked the points corresponding to long flat tori and the projective spaces RP N for N = 3, . . . , 8 and CP N/2 for N = 4, 6, 8 with their standard metrics (there are no transverse traceless tensors on RP 2 and CP 1 ≈ S 2 ). These projective spaces all fall comfortably below the bounds. It would be interesting to find out if there are manifolds that live at the kinks of this plot, just as nontrivial CFTs often appear at such kinks in the conformal bootstrap.
An additional assumption we can make is that the triple overlap integrals g a 1 a 1 a vanish when λ a = λ a 1 , so that the first sum in Eq. (3.3) vanishes. This follows, for example, if there is a Z 2 symmetry under which ψ a 1 is odd, as is the case for parity on S N . This means that we can drop the constraints in Eq. (3.4d), so it becomes easier to rule out a point and hence the resulting bounds are stronger. We plot these bounds for N = 2, . . . , 8 in Figure 2. As with the previous upper bounds, the trend is an approximately linear decrease until some critical λ I 1 /λ a 1 is reached, now given by 2(N +1)/N , above which the bound is constant. We also marked the points corresponding to S N for N = 3, . . . , 8, which lie on the boundaries of the allowed regions and away from the kinks. Round spheres thus have the largest possible ratio of the first two distinct nonzero eigenvalues of the scalar Laplacian amongst closed Einstein manifolds with R ≥ 0, a Z 2 symmetry, and a sufficiently large value of λ I 1 /λ a 1 . Now let us consider manifolds with either R > 0 or R < 0. In these cases we can use the rescaling freedom to set R = ±1 when computing bounds. For R > 0, the bounds are stronger compared to the R ≥ 0 bounds we considered previously, but the price we pay is the introduction of an additional scale in the problem. In either case, a candidate spectrum is determined by λ a 1 , λ a 2 , and λ I 1 , where for R > 0 we also ensure consistency with the Lichnerowicz bound (2.6). Given such a spectrum, we try to rule it out by searching for a constant vector α ∈ R 3 such that Since we now have an extra scale, we must fix λ I 1 to make a two-dimensional plot of the allowed eigenvalues λ a 1 and λ a 2 . In Figure 3 we show the upper bounds on λ a 2 /λ a 1 for each value of λ a 1 /|R| on closed Einstein manifolds with N = 2, . . . , 8 and R > 0 or R < 0, assuming that λ I ≥ 0. When λ a 1 is of order the curvature scale, the R > 0 bounds are stronger than what we found above, but they approach the R ≥ 0 bound (the dashed line) as λ a 1 /R → ∞. Intuitively, this is because the effects of curvature become negligible for high-frequency modes. The bounds for R < 0 approach λ a 2 /λ a 1 = 4 from above.
We now show how the bounds change if we make a stronger assumption about the size of the first Lichnerowicz eigenvalue. In Figure 4 we show the bounds on λ a 2 /λ a 1 for closed Einstein manifolds with R > 0 and N = 2, . . . , 8 when λ I satisfies the inequality in Eq. (2.16), the lower bound implied by the existence of a Killing spinor. We also plot the points corresponding to S N , which saturate the bounds for N = 2, . . . , 5 but not for N = 6, 7, 8 (they would if we made the stronger assumption that λ I ≥ 4R/(N − 1)).
Next we explore a slightly different type of question about eigenvalues. Suppose we assume a particular value for the third nonzero scalar eigenvalue and ask what values the first and second eigenvalues can take. In particular, let us set N = 4 and assume the following: which is consistent with both S 4 and CP 2 with their standard metrics. With these assumptions,  the values of λ a 1 /R and λ a 2 /R allowed by the consistency conditions (2.28) are given by the blue region in Figure 5. We see that the 4-sphere lives at the tip of a pointed part of the allowed region.
To compute these bounds we used two copies of the consistency conditions (2.28), one for λ a 1 and one for λ a 2 . The plot is invariant under a reflection along the diagonal since λ a 1 and λ a 2 are treated symmetrically. The results so far have been general bounds for all manifolds satisfying a handful of assumptions. We can also try to obtain constraints for specific manifolds by inputting additional information. Suppose that we know the scalar eigenfunctions on a given manifold but not necessarily the Lichnerowicz spectrum. We can then write the consistency conditions as where F 1 combines the first and third terms of Eq. (3.1) and is assumed to be some known constant vector for a given choice of ψ a 1 . We can then find an upper bound on λ I 1 , the smallest Lichnerowicz eigenvalue on transverse traceless tensors, by finding the smallest λ I 1 for which there exists a vector α ∈ R 3 satisfying We have carried out this procedure for the compact symmetric spaces of rank one restricted to the subsector of zonal spherical functions (see Appendix A), taking ψ a 1 to be the lightest nontrivial zonal spherical function. For these cases, the equations can be solved analytically and therefore the bounds are exact. This gives the following finite bounds for n > 1: • For CP n we find λ I 1 ≤ 8(n + 2), which is saturated for n = 2 [33,34].

General eigenvalues
The above bounds constrain just the smallest eigenvalues, but we can also look for bounds on general eigenvalues. Let ψ a 1 now be a generic eigenfunction of the scalar Laplacian, with eigenvalue λ a 1 , and define λ a 2 as the smallest eigenvalue bigger than λ a 1 , i.e., the scalar eigenvalues satisfy (3.10) The eigenfunction expansion of ψ 2 a 1 can then be written as We again define λ I 1 as the smallest eigenvalue of the Lichnerowicz Laplacian on transverse traceless tensors.
We want to determine how large λ a 2 /λ a 1 can be for a given choice of λ I 1 /λ a 1 . For a closed Einstein manifold with R ≥ 0, we can deduce that a given choice of eigenvalues is not feasible if we can find a constant vector α ∈ R 3 such that To be able to formulate this last condition as an SDP we can write it as The resulting upper bounds on λ a 2 /λ a 1 are shown in Figure 6 for N = 2, . . . , 8. The bounds grow linearly for λ I 1 < 0 and are constant for λ I 1 ≥ 0, given by λ a 2 /λ a 1 ≤ 4 in all dimensions. This result in the case λ I 1 = 0 was already given in Ref. [12].

Bounds on cubic couplings
We have seen that the consistency conditions can be used to derive bounds on Laplacian eigenvalues. Now we show that they can also give upper bounds on the triple overlap integrals of eigenfunctions of Laplacian operators, which correspond to cubic couplings in Kaluza-Klein theories. 6 These couplings must be real and can take either sign, but we will see that sometimes their absolute values are bounded from above by fixed multiples of geometric invariants.

Spin-2 self-interactions
We start by looking for bounds on the integral of the cube of an eigenfunction of the scalar Laplacian, (3.14) In Kaluza-Klein reductions of GR down to a d-dimensional flat spacetime, the massive spin-2 excitations of the graviton in the lower-dimensional theory interact with one another through treelevel cubic amplitudes of the form [12] where M d is the lower-dimensional Planck mass, p i are the momenta, and i ⊗ i are polarizations. The quantities g a 1 a 2 a 3 √ V thus control the strengths of these interactions relative to the strength of gravity. In particular, g a 1 a 1 a 1 √ V gives the relative strength of the self-interaction.
We assume that the scalar eigenvalues satisfy so that λ a 2 is second smallest nonzero scalar eigenvalue if it is greater than λ a 1 , otherwise it is the smallest nonzero eigenvalue. We again define λ I 1 as the smallest Lichnerowicz eigenvalue on transverse traceless tensors. We want to use the consistency conditions to find an upper bound on |g a 1 a 1 a 1 | √ V for a given choice of the low-lying eigenvalues λ a 1 , λ a 2 , and λ I 1 . This can be achieved with the same strategy used to bound OPE coefficients in the conformal bootstrap [35]. If R ≥ 0, we look for a vector α ∈ R 3 that satisfies the following constraints: Rearranging Eq. (3.1) and using these constraints, we get i.e., an upper bound on the rescaling invariant combination |g a 1 a 1 a 1 | √ V . For simplicity we use the second inequality in Eq. (3.18), although the first inequality is stronger when λ a 1 is degenerate. To find the best upper bound, we search for an α that maximizes the objective function α · F 1 while satisfying the constraints in Eqs. (3.17). This is again a problem that can be formulated as an SDP [29]. This SDP, and all others in the remainder of this section, are simple enough that they can be solved analytically using the Minimize function in Mathematica.
We start by searching for a bound on |g a 1 a 1 a 1 | √ V with λ a 2 = λ a 1 , i.e., assuming only that the nonzero scalar eigenvalues satisfy λ a ≥ λ a 1 . We also assume for now that λ I ≥ 0. With these assumptions, we find finite upper bounds on |g a 1 a 1 a 1 | √ V for N ≤ 13. These bounds are independent of λ a 1 , as required by rescaling invariance. We plot the bounds together with their rounded numerical values for N ≤ 12 in Figure 7. For N = 13 the upper bound is ≈ 28.5. The exact bounds are algebraic numbers, such as 16/ √ 111 for N = 3, but mostly their exact forms are not very enlightening. We also mark the points corresponding to the lightest nontrivial zonal spherical functions on the projective spaces RP N , CP N/2 , and HP N/4 with their standard metrics. The overlap integrals of these eigenfunctions can be calculated by integrating products of Jacobi polynomials, as described in Appendix A. The upper bound increases with N , so for manifolds satisfying our assumptions we can get a lower bound on the number of extra dimensions if we know the strength of the cubic self-interaction of the lightest Kaluza-Klein state relative to the strength of gravity.
We now generalize the upper bounds on |g a 1 a 1 a 1 | √ V to allow for different values of λ I 1 /λ a 1 . These are shown in Figure 8 for N = 2, 4, . . . , 16, where the restriction to even N is just to keep the plots readable. The intercepts of the curves with the vertical axis give the bounds of Figure 7. There Next we explore the effect of varying λ a 2 /λ a 1 , assuming again that λ I ≥ 0. In Figure 9 we plot the resulting bounds for λ a 2 /λ a 1 ∈ (0, 4] for manifolds with R ≥ 0 and N = 2, 4, . . . , 10. When λ a 2 /λ a 1 = 1, we recover the bounds of Figure 7. When λ a 2 /λ a 1 = 4, which is the maximum possible value and is achieved by long flat tori, we find that the self-coupling must vanish. This is consistent with the vanishing of all self-couplings on flat tori, which follows from their U (1) N symmetry. The eigenfunction ψ a 1 becomes completely general in the limit λ a 2 /λ a 1 → 0, so it is not surprising that in this limit there is no finite bound for N > 2. More precisely, a finite bound exists for λ a 2 /λ a 1 > x N , where x N is a constant that is positive for N > 2. For N ≤ 13 the bounds approach a finite value as λ a 2 /λ a 1 → x + N , so to high numerical precision they appear to be discontinuous at these points. It would be interesting if there is an explanation for this in terms of explicit manifolds. We list numerical approximations to x N for N = 2, . . . , 14 and the corresponding bounds in Table 2  V as a function of λ I 1 /λ a 1 for closed Einstein manifolds with R ≥ 0 and N = 2, 4, . . . , 16, assuming that λ a ≥ λ a 1 and λ I ≥ λ I 1 .

Interactions with multiple spin-2 fields
We now look for bounds on triple overlap integrals of the form In Kaluza-Klein reductions of gravity, these integrals give the strengths of the tree-level cubic interactions between two spin-2 particles of different masses.
We assume again that R ≥ 0 and that the scalar eigenvalues satisfy Eq. (3.16). The procedure is similar to before. We choose the vector α ∈ R 3 to satisfy the following constraints: With these constraints satisfied, we search for an α ∈ R 3 that maximizes the objective function α · F 1 , giving the bound  Table 2: Upper bounds on the size of the cubic self-coupling |g a 1 a 1 a 1 | √ V for closed Einstein manifolds with R ≥ 0 and λ I ≥ 0 as λ a 2 /λ a 1 → x + N , where λ a ∈ {λ a 1 } ∪ [λ a 2 , ∞) and x N is the smallest value of λ a 2 /λ a 1 below which there is no finite bound.
Taking λ I ≥ 0 gives the bounds shown in Figure 10 for λ a 2 /λ a 1 ∈ (0, 4] and N = 2, 4, . . . , 20. The upper bounds do not decrease monotonically as we increase λ a 2 /λ a 1 , unlike the previous bound. This is possible due to the fact that the normalization condition in Eq. (3.20a) depends on λ a 2 , so a solution α for some λ a 2 is not necessarily a solution for larger λ a 2 . The bounds all approach 1/ √ 2 as λ a 2 /λ a 1 → 4, which is the value attained by long flat tori and is marked with a cross in the plot. The bounds diverge for λ a 2 /λ a 1 < x N , where x N are the constants encountered above. We also mark the points corresponding to the overlap of the two lightest nontrivial zonal spherical functions on RP N , CP N/2 , and HP N/4 with their standard metrics.
We can get stronger bounds by further imposing that g a 1 a 1 a vanishes if λ a = λ a 1 . This is the case when the eigenfunctions with eigenvalue λ a 1 are odd under a Z 2 symmetry. Suppose also that the eigenfunctions with eigenvalue λ a 2 are even, so the eigenvalues of the scalar Laplacian satisfy where λ ± a are the nonzero eigenvalues of the even/odd eigenfunctions. We implement this by dropping condition (3.20c). The resulting bounds are shown in Figure 11 for N = 2, 4, . . . , 20. We also mark the points corresponding to the overlap of the lightest nontrivial zonal spherical harmonics on S N , which lie on the boundaries of the allowed regions.   √ V as we vary λ a 2 /λ a 1 for closed Einstein manifolds with R ≥ 0 and a Z 2 symmetry, assuming that λ I ≥ 0, λ − a ≥ λ a 1 , and λ + a ≥ λ a 2 . The cross again corresponds to modes on long flat tori. The filled circles correspond to the overlaps of the lightest nontrivial zonal spherical harmonics on S N . The dashed line is 2(3λ a 2 /λ a 1 − 4) −1/2 , which the bounds approach as N → ∞. This plot is identical to Figure 10 in the region λ a 2 /λ a 1 ≤ 1.

Spin-2 and scalar interactions
We now look for bounds on triple overlap integrals involving an eigenfunction of the scalar Laplacian and a transverse traceless eigentensor of the Lichnerowicz Laplacian, In Kaluza-Klein reductions of gravity, these integrals control the strengths of the cubic interactions between a massive spin-2 mode and a spin-0 Lichnerowicz mode.
We look for bounds involving the lightest eigentensor, h T T mn,I 1 . We assume that λ a ∈ {λ a 1 } ∪ [λ a 2 , ∞). For the case R ≥ 0, we require the vector α ∈ R 3 to satisfy the following constraints: This then gives the inequality Maximizing the objective function α · F 1 subject to the above constraints then gives the best upper bound on the combination |g a 1 a 1 I 1 |λ −1 The simplest case to consider is λ I 1 = 0 and λ a 2 = λ a 1 , so we have λ I ≥ 0 and λ a ≥ λ a 1 . This is relevant, for example, to Ricci-flat manifolds with shape moduli and a parallel spinor. The resulting upper bounds are finite for N ≤ 13 and are shown in Figure 12. We also plot the couplings for certain modes on flat N -tori, 7 |g a 1 a 1 I 1 |λ −1 Interestingly, these tori modes saturate the upper bounds for N = 2, . . . , 6, but not for N ≥ 7. The bounds do not change if we assume instead that R = 0. We can also find bounds when we vary λ a 2 /λ a 1 , keeping λ I 1 fixed at zero. We show these in Figure 13 for N = 2, 4, . . . , 20. We also mark the points corresponding to the maximum couplings on flat N -tori given by Eq. (3.27) at λ a 2 /λ a 1 = 4, which saturate the bounds for every N . A finite bound exists for λ a 2 /λ a 1 > x N , where x N are the constants from Section 3.2.1. For N ≤ 4 the bound as λ a 2 /λ a 1 → x N is finite, given by 1/ √ 2, ≈ 1.0491, or ≈ 1.9773 for N = 2, 3, or 4, respectively.
We can similarly find bounds when we vary λ I 1 /λ a 1 , assuming that λ a ≥ λ a 1 . We show these 7 For example, on the square flat N -torus we can take where θ k ∈ [0, 2π) parameterizes the k th circle in (S 1 ) N . By elongating the N th circle we get λa 2 = 4λa 1 .
bounds for closed Einstein manifolds with R ≥ 0 and N = 2, 4, . . . , 16 in Figure 14. A finite bound exists when N ≤ 17 and λ I 1 /λ a 1 > w N , where w N are the constants from Section 3.2.1.

Consistency conditions with two fixed eigenfunctions
The consistency conditions we have been studying so far involve one fixed scalar eigenfunction. These correspond to sum rules for the scattering of identical massive spin-2 Kaluza-Klein particles in dimensional reductions of pure GR when R = 0. We now study bounds obtained from more general consistency conditions involving two different fixed scalar eigenfunctions, which are related to four-point amplitudes with two different Kaluza-Klein external states. These are the analogue of mixed correlator bootstrap equations for CFTs [30].
The consistency conditions involving two fixed scalar eigenfunctions, ψ a 1 and ψ a 2 , are given explicitly in Appendix B, where we also explain how to derive them. When combined with the V as we vary λ I 1 /λ a 1 for closed Einstein manifolds with R ≥ 0 and N = 2, 4, . . . , 16, assuming that λ a ≥ λ a 1 and λ I ≥ λ I 1 .
earlier consistency conditions, they can be put in the following form: g a 1 a 1 I g a 2 a 2 I F 2 g a 1 a 1 I g a 2 a 2 I + 1 λ 2 We include these F k 's explicitly in the ancillary notebook. We have assumed that a 1 = a 2 by dropping terms proportional to δ a 1 a 2 .
We can determine whether a given spectrum is consistent with these consistency conditions by again formulating the problem as an SDP, following the approach of the mixed correlator conformal bootstrap [30].

Bounds on the third eigenvalue
First we investigate how large we can make the third nonzero scalar eigenvalue relative to the first two eigenvalues. Suppose that the scalar eigenvalues satisfy and let ψ a k be an eigenfunction with eigenvalue λ a k for k = 1, 2. We also take λ i ≥ λ i 1 and λ I ≥ λ I 1 .
Now consider closed Einstein manifolds with R ≥ 0 and a candidate low-lying spectrum defined by the five eigenvalues λ a 1 , λ a 2 , λ a 3 , λ i 1 , and λ I 1 . We can rule out such a spectrum by finding an α ∈ R 14 satisfying the following constraints: where M 0 means that the matrix M is positive semidefinite. If λ a 1 = λ a 2 , we additionally require that α · F 6 ≥ 0, ∀ λ i ≥ 0, (4.5) which ensures that the contributions from any Killing vectors are non-negative. Similarly, if we want to include round spheres then we must also ensure that the contributions from conformal scalars are non-negative, α · F 11 ≥ 0, ∀ λ a > 0. (4.6) In practice, it is simpler to just manually append the sphere data to any exclusion plots if it is excluded by the other constraints. Following Refs. [30,36], we can also implement the symmetry of g a 1 a 2 a 3 by replacing the eight constraints in Eq. (4.4e) with the four constraints which can result in stronger bounds.
We take λ I ≥ 0 and λ i ≥ 0. For each λ a 2 /λ a 1 ∈ [1, 4] we find an upper bound on λ a 3 /λ a 1 as shown in Figure 15 for N = 2, 3, . . . , 8, where the solid lines are the bounds obtained when the symmetry of g a 1 a 2 a 3 is implemented through Eqs. (4.7) and (4.8) and the dashed lines instead use Eq. (4.4e). The purple region, including the various purple lines, corresponds to the moduli space of flat 2-tori. 8 The cross at (4,9) corresponds to flat N -tori whose first three distinct nonzero eigenvalues coincide with those of a circle, which we call "very long flat tori"-for N = 2, these are the tori lying in the region of the canonical fundamental domain with |τ | ≥ 3-and the cross at (2,4) corresponds to the square flat 2-torus. We also mark the Fermat quintic threefold using the eigenvalues found numerically in Ref. [10]. When using SDPB to produce these bounds, we found cases where to obtain the optimal bound it was necessary to disallow the program from terminating due to detection of a primal feasible solution (cf. the discussion in Ref. [37]). Figure 15: Upper bounds on λ a 3 /λ a 1 on closed Einstein manifolds with R ≥ 0 and N = 2, . . . , 8, where λ a ∈ {λ a 1 , λ a 2 } ∪ [λ a 3 , ∞) and we assume that λ I ≥ 0. The bounds represented by the solid lines take into account the symmetry of g a 1 a 2 a 3 , while the bounds represented by the dashed lines do not. The purple region, including the various purple lines, covers the image of the moduli space of flat 2-tori. The crosses mark the square 2-torus and very long flat N -tori, while the pentagon marks the eigenvalues of the Fermat quintic threefold found numerically in Ref. [10].
Suppose now that the manifold has a Z 2 symmetry such that ψ a 1 is odd and ψ a 2 is even and let the nonzero scalar eigenvalues satisfy where λ ± a are the eigenvalues of the even/odd eigenfunctions. The consistency conditions (4.1) allow us to probe the ratio of the first two odd eigenvalues, λ a 3 /λ a 1 . We again consider closed Einstein manifolds with R ≥ 0. To find bounds we solve the SDP obtained from the subset of the conditions in Eqs. (4.4) that are consistent with the Z 2 symmetry (imposing the symmetry of g a 1 a 2 a 3 does not help in this case). We obtain an upper bound on λ a 3 /λ a 1 for each λ a 2 /λ a 1 ∈ (0, 4] as shown in Figure 16, where we take λ i ≥ 0 and either λ I ≥ 0 (solid lines) or λ I /λ a 1 ≥ 4 (dashed lines). The very long flat N -tori at position (4,9) saturate the λ I ≥ 0 bounds, as marked by the cross, and the round spheres lie at the edge of the λ I /λ a 1 ≥ 4 bounds, as shown by the filled circles.

Bounds on vector couplings
Now we use the consistency conditions (4.1) to find bounds on the triple overlap integrals g a 1 a 2 i . These integrals were not accessible with the earlier consistency conditions since they are antisymmetric in a 1 and a 2 . Here we will focus on the case where the manifold has R ≥ 0 and the transverse eigenvector corresponds to a Killing vector, so that λ i = 2R/N . This is possible when the manifold has a continuous symmetry. The quantity g a 1 a 2 i for i ∈ I Killing can be non-vanishing only if λ a 1 = λ a 2 . In the case of GR with a Ricci-flat internal manifold with a continuous symmetry, this quantity measures the Kaluza-Klein charge of the massive spin-2 states, q = √ 2V g a 1 a 2 i M Following the same approach as in Section 3.2, we can use the consistency conditions to obtain upper bounds on the following scale invariant combination: (4.10) In Figure 17 we plot the upper bounds on the square root of this quantity for closed Einstein manifolds with R > 0 and N = 2, . . . , 8, where we take λ a 1 = λ a 2 as the smallest nonzero eigenvalue of the scalar Laplacian and we assume that λ I ≥ 0. We also mark points corresponding to certain modes on the round N -spheres, 9 which lie just below the upper bounds. For N > 13, the bounds diverge above some finite value of λ a 1 /R. The dashed line gives the upper bound for Ricci-flat manifolds with N ≤ 13; this bound is saturated by flat tori, which is related to the saturation of the weak gravity conjecture by Kaluza-Klein modes in toroidal compactifications of gravity [38].

Discussion
We have developed a bootstrap approach to find bounds on the geometric data of closed Einstein manifolds. Specifically, our bounds constrain the eigenvalues and triple overlap integrals of eigenfunctions of various Laplacians. These bounds are found by starting with associativity conditions that must be satisfied by various higher-point integrals of eigenfunctions, then formulating them as 9 These have |ga 1 a 2 i|λ semidefinite programming problems which can be solved numerically or analytically on a computer. This approach mirrors the conformal bootstrap. As far as we know, the bounds we find are new results about Einstein manifolds that have not appeared in the mathematical literature. Physically, these bounds can be thought of as constraints on the possible masses and three-point couplings of massive Kaluza-Klein modes in dimensional reductions of gravity.
There are several interesting directions that could be explored in the future. We have presented examples that illustrate some of the different types of bounds that can be deduced from the consistency conditions, but there should be many other examples that are worth exploring. It would also be worthwhile to try to find more powerful versions of the constraints studied here by looking for additional consistency conditions coming from higher-derivative integrands. Having a systematic or recursive method for finding these would be useful. This may not be possible without further assumptions, such as restricting to quotients of maximally symmetric manifolds (which still includes nontrivial spaces such as compact hyperbolic manifolds). On a technical level, the obstruction is due to the inevitable appearance of Riemann tensors in higher-derivative integrands. More conceptually, this is connected to the restrictions imposed by the consistent propagation of higher-spin particles in nontrivial gravitational backgrounds [39]. Perhaps string theory backgrounds satisfy additional consistency conditions that give more powerful bootstrap constraints.
Our results take the form of general bounds that apply to many Einstein manifolds. It would be interesting to see if consistency conditions are powerful enough to isolate rigid manifolds in "bootstrap islands" with only a small number of assumptions about the eigenvalues above the lightest modes, like what happens in the CFT bootstrap for strongly coupled CFTs such as the 3D Ising model at criticality [30,36], and whether bootstrapping can be an efficient method to calculate the geometric data of manifolds of interest. This seems likely to require additional consistency conditions beyond those studied here. Other interesting directions would be to look for consistency conditions involving higher p-forms or fermions, as would occur in dimensional reductions of supergravity; to search for consistency conditions for spaces other than closed manifolds, such as bounded domains in Euclidean space; and to incorporate symmetry constraints from more complicated isometry subgroups.
numbers C, the quaternions H, and the octonians O, each with their standard metrics. These are the compact symmetric spaces of rank one.

Round spheres
Consider the N -dimensional round sphere with unit radius, S N . This is the set of points that are unit distance from the origin in R N +1 with the metric induced from the standard Euclidean metric. It can also be described as the symmetric space SO(N + 1)/SO(N ). For (x 1 , . . . , x N +1 ) ∈ R N +1 , we can parameterize the sphere with spherical coordinates where 0 ≤ θ 1 < 2π and 0 ≤ θ k ≤ π for 2 ≤ k ≤ N . The metric in these coordinates is The Ricci scalar and volume are The eigenfunctions of the scalar Laplacian on S N , i.e, the (higher-dimensional) spherical harmonics, are obtained from the harmonic homogeneous polynomials in R N +1 restricted to the sphere [40]. They can be written in spherical coordinates in terms of associated Legendre polynomials [41] where M k for k = 1, . . . , N are integers satisfying The spherical harmonics with a given value of L transform into each other under rotations, forming the rank-L symmetric traceless tensor representation of SO(N +1). Under parity they transform by an overall factor of (−1) L . In general they are complex, but we can define real spherical harmonics as (restricting to N > 1) A special subset of the spherical harmonics are the zonal spherical harmonics. These are eigenfunctions that are invariant under the SO(N ) rotations leaving a given point fixed. They are an example of zonal spherical functions, which are defined for general homogeneous spaces and always form a closed subsector of the consistency conditions. Taking the fixed point as (0, . . . , 0, 1) ∈ R n+1 , the zonal harmonics can be written in terms of x 2 ≡ x 2 1 + · · · + x 2 N and x N +1 . For example, the first few nontrivial zonal spherical harmonics on S 3 with unit L 2 -norm are L = 1 : (A. 10) In general, the normalized zonal spherical harmonics are the spherical harmonics with M k = 0 for k = 1, . . . , N − 1 and can be written in terms of Gegenbauer polynomials, Vector and tensor eigenmodes on S N can be obtained from contractions of certain mixedsymmetry constant tensors with the coordinates of R N +1 [42].

Real projective space
The N -dimensional real projective space RP N is S N /Z 2 , where Z 2 acts as the antipodal map on S N . The eigenmodes on RP N are thus proportional to the parity-even eigenmodes on S N , i.e., the scalar spherical harmonics and transverse traceless eigentensors with even L and the transverse eigenvectors with odd L.

Complex projective space
The complex projective space CP n with the Fubini-Study metric is the symmetric space U (n + 1)/(U (n)×U (1)). Equivalently, it is the quotient space S 2n+1 /S 1 obtained from the Hopf fibration.
It has real dimension N = 2n. The Ricci scalar and volume are R = 4n(n + 1), V = π n n! . (A.14) The complex projective line CP 1 is homothetic to S 2 , i.e., isometric to a constant rescaling of S 2 .
The eigenfunctions of the scalar Laplacian on CP n are obtained by projecting the spherical harmonics on S 2n+1 ⊂ C n+1 that are invariant under U (1) = S 1 , where the action of U (1) on (z 1 , . . . , z n+1 ) ∈ C n+1 is multiplication by a phase [40]. The U (1) invariants are thus z izj . From Eq. (A.7), we get that the eigenvalues of the scalar Laplacian on CP n are λ k = 4k(k + n), k ∈ Z ≥0 . (A.15) The zonal spherical functions fixing the point (0, . . . , 0, 1) ∈ C n+1 are the eigenfunctions that are further invariant under the action of U (n) on (z 1 , . . . , z n ), so they can be written in terms of | z | 2 ≡ z 1z1 + · · · + z nzn and |z n+1 | 2 . For example, the first few nontrivial zonal spherical functions on the complex projective plane CP 2 with unit L 2 -norm are The smallest Lichnerowicz eigenvalue on transverse traceless symmetric 2-tensors on CP n with the Fubini-Study metric is λ I 1 = 32 for n = 2 and λ I 1 = 8n for n ≥ 3 [33,34]. 10
(A. 19) The quaternionic projective line HP 1 is homothetic to S 4 .
The eigenfunctions of the scalar Laplacian on HP n are obtained by projecting the spherical harmonics on S 4n+3 ⊂ H n+1 that are invariant under Sp(1) = S 3 , where the action of Sp(1) on (q 1 , . . . , q n+1 ) ∈ H n+1 is right multiplication by unit quaternions. Writing a quaternion as a pair of complex numbers, q i = z i + w i j, with quaternion conjugateq i =z i − w i j, the Sp(1) invariants of interest are z izj + w iwj [44]. From Eq. (A.7), we get that the eigenvalues of the scalar Laplacian on HP n are λ k = 4k(k + 2n + 1), k ∈ Z ≥0 . (A.20) The zonal spherical functions on HP n fixing the point (0, . . . , 0, 1) ∈ H n+1 are the eigenfunctions that are further invariant under the action of Sp(n) on (q 1 , . . . , q n ), so they can be written in terms of | q | 2 ≡ q 1q1 + · · · + q nqn and |q n+1 | 2 ≡ q n+1qn+1 . For example, the first few nontrivial zonal spherical functions on the quaternionic projective plane HP 2 with unit L 2 -norm are Relatively little has been published about the Lichnerowicz eigenvalues of transverse traceless tensors on HP n for n > 1. It is possible to get lower bounds on the first such eigenvalue using representation theory. From the results of Refs. [24,45] we get λ I ≥ 16n.