James Bonifacio, PhD
Case Western Reserve University
In this post we’ll explore some of the motivation and background for a recent preprint about a new method for finding bounds on Einstein manifolds which is inspired by scattering amplitude calculations and the conformal bootstrap.
To begin, let’s review some facts about Riemannian geometry. Suppose that someone hands you a compact Riemannian manifold, such as the double torus depicted in the figure below. Given such a manifold, there are all sorts of interesting geometric quantities that you could
compute: its dimension, its volume, the lengths of its closed geodesics, the eigenvalues of the scalar Laplacian*, and so on. As a simple example, take the two-dimensional sphere, S^2, with unit radius and the usual round metric. This space has volume 4π, its closed geodesics all have length 2π, and the eigenvalues of its scalar Laplacian are given by L(L + 1), where L = 0, 1, 2, . . . and the Lth value is repeated 2L + 1 times. Some other quantities that you could compute for a compact manifold are integrals over the manifold of products of three eigenfunctions,
which we’ll call triple overlap integrals. (It turns out that integrals of products of more than three eigenfunctions can all be written in terms of these.) For example, on S^2 the eigenfunctions are the spherical harmonics and their triple overlap integrals can be written in terms of Clebsch–Gordan coefficients. We collectively call these various quantities “geometric data” of the manifold.
Now suppose that instead of giving you an actual manifold, someone gives you the next best thing, namely a sequence of numbers that they claim corresponds to the geometric data of a compact manifold. For example, they may give you a sequence of putative eigenvalues of the scalar Laplacian given by L^3 for L = 0, 1, . . . , where the Lth value is repeated L times. How can you tell whether or not these really could be the eigenvalues of the scalar Laplacian on some manifold? Well, there are various checks you can do. One simple condition that these eigenvalues should satisfy is that the smallest eigenvalue is zero, corresponding to the constant function. As you can easily check, our example satisfies this condition. A more subtle requirement is that the eigenvalues must obey Weyl’s law, which dictates how the number of eigenvalues grows as they tend to infinity. More precisely, if n(λ) is the total number of eigenvalues less than λ, Weyl’s law says that the following limit must be a constant:
where N is the dimension of the manifold. Applying this formula to our example eigenvalues, we find that it holds only if N = 4/3. Since the dimension of a manifold has to be an integer, we can conclude that our original list of candidate eigenvalues could not have come from a manifold. This illustrates that not every list of numbers can correspond to the geometric data of a manifold. If we modify our example so that the Lth value is now repeated L^8 times, it becomes consistent with Weyl’s law with N = 6. Could this sequence correspond to the scalar eigenvalues of some six-dimensional manifold? We’ll come back to this later.
Extra dimensions and graviton scattering
It turns out that there are some additional consistency conditions that the Laplacian eigenvalues and triple overlap integrals of a manifold have to satisfy. To give a sense of why these consistency conditions exist, it’s useful to discuss a physical interpretation of the geometric data of a manifold. Consider a hypothetical scenario in which there exist additional spatial dimensions beyond the three large spatial dimensions that we observe. This is required, for example, in string theories, since they only really make sense if the total number of spacetime dimensions is ten or eleven. One way for extra dimensions to have escaped our attention so far is if they are compact and small enough that we could not have resolved them with any terrestrial experiment, a scenario that goes by the name of Kaluza–Klein theory. If gravity is dynamical then the space describing the extra dimensions should be a solution of the gravitational equations of motion; in the simplest cases, this means that the extra dimensions are described by an Einstein manifold, a solution to the vacuum Einstein equations, and from now on we’ll only consider such manifolds. In the case of ten-dimensional string theories, a common choice of manifold to describe the six extra spatial dimensions is a particular type of Einstein manifold called a Calabi–Yau manifold.
In a Kaluza–Klein theory, every light particle in the lower-dimensional world is accompanied by an infinite tower of massive copies of itself. The particles in this tower correspond to versions of the original particle that have excited vibrational modes in the extra dimensions. These vibrational modes are described by the eigenmodes of various Laplacian operators in the extra dimensions and the corresponding eigenvalues determine the masses of the particles. For example, the graviton—the massless spin-2 particle that mediates the gravitational force—is associated with an infinite tower of massive spin-2 particles whose masses squared are given by the eigenvalues of the scalar Laplacian on the manifold describing the extra dimensions. Additionally, the strength of the interaction between any three of these graviton excitations is determined by the triple overlap integral of the corresponding eigenfunctions of the scalar Laplacian. The geometric data of a manifold thus corresponds physically to properties of particles in a Kaluza–Klein theory.
Now suppose you want to perform an experiment where you collide together two massive excitations of the graviton and two other excitations come out. The probabilities for the different possible outcomes of this experiment are encoded in a scattering amplitude. This scattering amplitude depends, among other things, on the energy of the collision, E. When you ordinarily look at scattering amplitudes involving massive spin-2 particles, for large values of the energy they grow quite fast as a function of energy, like E^10. However, since these massive spin-2 particles are secretly just components of a higher-dimensional graviton, their amplitudes should inherit the relatively soft high-energy behavior of graviton amplitudes, which grow instead like E^2. From the perspective of the lower-dimensional amplitude, achieving this softening requires miraculous-looking cancellations between the different Feynman diagrams that contribute to the amplitude. For example, the E^8 part of the amplitude only vanishes because the contribution from the exchange of the infinite number of graviton excitations precisely cancels the contribution from exchanging the massless graviton. These cancellations occur because of subtle relations between the masses and interaction strengths of the different particles that contribute to the scattering amplitude. Since the masses and interaction strengths are in turn determined by the geometric data of the manifold describing the extra dimensions, this geometric data must satisfy certain nontrivial relations to ensure the amplitudes have the expected high-energy behavior. These relations are the additional consistency conditions alluded to above.
Consistency conditions and bootstrap bounds
We have just argued that the geometric data of a manifold must satisfy consistency conditions to ensure that the scattering amplitudes of the massive excitations of the graviton in Kaluza–Klein theories have the high-energy behavior that we would expect based on their extra-dimensional origin. An example of one of these consistency conditions is the following:
where i labels the non-constant eigenfunctions ψi of the scalar Laplacian with eigenvalues λi, ψ1 is a fixed eigenfunction, g11i is the integral of ψ1^2 ψi, and V is the volume of the manifold. This is the consistency condition responsible for the vanishing of the E^8 part of the amplitude mentioned above. The problem is that the consistency conditions involve potentially infinite sums over all of the particles contributing to the amplitude, so it’s not obvious a priori how to extract useful information from them. Fortunately, there is an analogous and well-studied problem that arises in a different context in physics, that of conformal field theories. It has been known since the 70’s that conformal correlators must satisfy certain nontrivial consistency conditions, but only over the last decade or so have effective computational tools been developed to extract useful information from these consistency conditions in more than two dimensions. This approach is called the conformal bootstrap and most modern numerical implementations use a form of optimization called semidefinite programming. We can take some of the techniques that have been developed for the conformal bootstrap and directly apply them to the consistency conditions satisfied by the geometric data of a manifold to extract some useful geometric information. This useful information usually takes the form of bounds on the geometric data, so we call them geometric bootstrap bounds.
Now that we have our consistency conditions and we know how to extract useful information from them, we can go ahead and search for geometric bootstrap bounds. An example of the type of bound that we get from this approach is the following: the ratio of any two consecutive eigenvalues of the scalar Laplacian can be at most four on Einstein manifolds with a non-negative Ricci scalar**. Physically, this means that the mass of an excitation of the graviton can be at most twice the mass of the next-lightest such excitation. Going back to the second example of candidate scalar Laplacian eigenvalues that we considered earlier, we see that its first and second distinct nonzero eigenvalues differ by more than a factor of four. We can therefore conclude that this sequence could not correspond to the spectrum of graviton excitations in a Kaluza–Klein theory with six extra dimensions satisfying our assumptions, such as what you would get from string theory with an internal Calabi–Yau manifold. Another result for Einstein manifolds with a non-negative Ricci scalar is that there is an upper bound on the size of the triple overlap integral of the lightest eigenfunction of the scalar Laplacian times the square root of the volume of the manifold, |g111| sqrt(V) . For example, on manifolds with six dimensions or fewer this quantity cannot exceed 10/3. This means that the strength of the self-interaction of the lightest massive excitation of the graviton can not be more than 10/3 times the strength of gravity when there are six or fewer extra dimensions. The fact that we can get such strong bounds from consistency conditions is quite surprising. As a final example, we show in the figure below an upper bound on the triple overlap integral for the lightest two scalar eigenfunctions in terms of the ratio of their eigenvalues and for different values of N, the dimension of the manifold, assuming the manifold satisfies the same conditions as in the previous two examples.
It is an old dream to constrain and solve physical theories using consistency conditions. This idea has experienced a recent resurgence thanks to advances in our understanding and computing power, giving new insights in several areas of theoretical physics. Here we have seen how related ideas coming from the conformal bootstrap and amplitudes can also teach us some new things about geometry and the physics of extra dimensions. For more details and for references to the literature, see the preprint available here.