What are the constituents of the Universe? Ultimately, this is a question to be decided by experiment. At the same time, it is worth noting that mathematical consistency alone severely limits what can exist in nature, even in principle. For example, Wigner famously showed that the menu of conceivable physical states is not arbitrary, but rigidly constrained by unitarity and special relativity [1]. Furthermore, modern developments in quantum field theory and scattering amplitudes have established that the perturbative dynamics of particles are almost entirely fixed by their kinematical properties. The only self-interacting theories of massless particles of spin one and spin two are gauge theory and gravity, while higher-spin massless particles are inconsistent [2–7].

These insights have demonstrated that certain states are mathematically forbidden. On the other hand, the converse possibility—that certain states might actually be mathematically required—is equally if not more intriguing. The maximalist incarnation of this idea is the notion of completeness, which is the property that all charges permitted by Dirac quantization are explicitly realized by physical states in the spectrum.

It has been conjectured that completeness is a universal feature of all consistent theories of quantum gravity [8,9], referred to collectively as the landscape. The complement of this space is the swampland, which describes the set of naively sensible gravitational effective field theories that can never actually be realized by any ultraviolet completion [10]. A well-known motivation for the completeness hypothesis is the absence of global symmetries in quantum gravity [11]. In particular, to explicitly break a global higher-form symmetry, one posits the existence of particles of all allowed charges [12,13].

In this paper, we adopt an entirely different approach to this question. Using bottom-up reasoning, we rigorously prove a version of the completeness hypothesis. Our conclusions apply to a variety of theories and are derived purely from the mathematical consistency of scattering amplitudes. The workhorse of our methodology is the analytic dispersion relation, cn(t)=12πi∮s=0dssn+1A(s,t)={s channel}+{u channel}+bn(t).(1)At fixed t, this expression extracts the Wilson coefficient cn(t) from the four-point scattering amplitude A(s,t) and relates it to a boundary contribution bn(t) at infinity plus a sum over discontinuities in the s and u channels. This equation is the key ingredient of our analysis: if we know that cn(t)≠0 and bn(t)=0, then there must be a state in either the s channel or the u channel, or both.

Physically, the conditions cn(t)≠0 and bn(t)=0 imply that there is some operator that blows up faster in the effective field theory than in the full amplitude. This is the usual situation in which the effective field theory dynamics are “unitarized” by the ultraviolet completion. Crucially, Eq. (1) is only useful for deducing the existence of states if there is a mandatory coupling cn(t)≠0 that is always unitarized, so that bn(t)=0.

Remarkably, quantum gravity furnishes the exact conditions that we desire [14]. The equivalence principle says that any pair of particles will interact gravitationally. The corresponding graviton exchange contribution appears near the forward limit of the amplitude [15], A(s,t)=-8πGNs2t+⋯,(2)so c2(t)=-8πGN/t is nonzero for any small but finite t. Notably, it has been argued from the bottom up that b2(t)=0 for t<0 under mild assumptions about gravitational scattering [16]. Furthermore, b2(t)=0 for t<0 in all known examples from string theory, where an infinite tower of higher-spin particles intervenes to unitarize gravitational scattering. The dispersion relation in Eq. (1) then implies that [20] c2(t)=-8πGNt={s channel}+{u channel},(3)so there has to be a state in the s or u channel.

In the presence of an exact symmetry, Eq. (3) can be used to prove highly nontrivial constraints on the spectrum of corresponding charges [14]. In our setup we will assume an exact symmetry group G whose maximal Abelian subgroup is H. We take G to be a finite semisimple Lie algebra, so H is the Cartan subgroup. Whether G is gauged or not will not matter for our analysis.

We then fold Eq. (3) into the simple iterative procedure shown schematically in Fig. 1. In the very first step, we specify some initial set of particles Q that are assumed to be in the spectrum and taken to be in the fundamental of G unless stated otherwise. We then scatter all pairs of particles within Q and apply Eq. (3) in order to deduce the existence of additional charged states. After adding these new states to Q, we rescatter all the elements of Q again, taking the newly deduced particles to be external states of yet another scattering process. Since these particles also interact gravitationally, Eq. (3) applies once again, and so on ad infinitum. By iterating this algorithm, we very generically discover that an infinite tower of charged particles is required purely by mathematical consistency. If the spectrum of states eventually grows to encompass the full lattice of possible charges, or weights, under H, then we say that the theory exhibits “charge completeness.”

Remarkably, we are able to prove charge completeness across a broad range of theories, though our precise conclusions depend sensitively on the degree of symmetry in G. These criteria are summarized in Table I. For example, for the Abelian symmetry group G=SO(2)≃U(1), our methods are, unfortunately, insufficient to prove anything. The same is true of G=SO(3)∼SU(2) and SO(4)∼SO(3)2. However, completeness does follow if we consider larger groups. For G=SO(N) with N≥5, the additional non-Abelian structure accommodates an explicit constructive proof of charge completeness. Meanwhile, for G=SU(N) with N≥3, we derive completeness for various infinite subsets of charges that depend on what precisely we assume for the starting representations. Our results suggest that spectral completeness is a robust property of any weakly coupled ultraviolet completion of gravity with a sufficiently large non-Abelian symmetry and a finite but appropriately chosen initial set of charged particles.

Our results have implications for the phenomenologically relevant case of grand unified theory (GUT) with G=SU(5) or Spin(10). As is well known, a proper embedding of the standard model within SU(5) GUT mandates the existence of the 5 and 10 for the matter and the 24 for the Higgs. Curiously, this set of representations is precisely sufficient to derive charge completeness, with any smaller set failing to do so. Meanwhile, Spin(10) GUT requires the 16 for the matter and the 10 for the Higgs. In this case, the phenomenologically required matter representations correspond exactly to the minimal set needed to derive charge completeness. In other words, the phenomenologically required field content of the SU(5) and Spin(10) GUTs is exactly sufficient to guarantee the completeness of electromagnetic charges.

For clarity, let us tabulate our assumptions very explicitly. Our arguments rigorously imply a version of the completeness hypothesis, provided there is (i)

an exact symmetry G,

Our final condition (iii) states that the gravitational dynamics are unitarized at tree level. Mathematically, this implies that graviton scattering is softened at high energies, so for t<0 we have b2(t)=0, justifying the application of Eq. (3). We emphasize that this last assumption is exceedingly well motivated and conservative. Any ultraviolet completion of gravity that improves the high-energy behavior of general relativity by any amount—which is the very definition of ultraviolet completion—will satisfy this condition. From the top down, this attribute is strongly motivated because it is satisfied by all perturbative string theories. This enormous class of models, which happen to be the only extant quantum gravitational theories with explicit and exhaustive predictions for scattering amplitudes, have the universal property that b2(t)=0. From the bottom up, the assumption that b2(t)=0 is also very mild and supported by general arguments [17,21]. Notably, the condition of tree-level dynamics, together with tame Regge behavior, is actually sufficient to bootstrap string amplitudes uniquely [22–24].

An important consequence of our assumptions is that the notion of completeness that we establish is both stronger and weaker than the conventional notion of completeness invoked in the swampland literature [8,9,12,13,25]. In those works, completeness makes no reference to whether the charged states are single-particle or multiparticle. That is, the presence of ultracharged states is essentially trivialized by the existence of multiparticle states, provided there already exist states of some fundamental charge. For this reason the weight of the swampland conjectures falls on whether these fundamentally charged states are present in the first place. By contrast, our analysis by fiat assumes that some minimally charged states are present, which in a sense weakens our conclusions. On the other hand, our arguments mandate the existence of ultracharged states that are single particles, which is much stronger than is required by the usual swampland conjectures. Perturbative string theory famously exhibits completeness of the spectrum by way of single-particle states, and remarkably, our methodology arrives at the very same conclusion using bootstrap methods.

The remainder of this paper is structured as follows. We begin in Sec. II by outlining a general iterative procedure for constructively deriving completeness of the charge lattice. As a warmup, we apply this algorithm to U(1) symmetry in Sec. III and explain why completeness cannot be derived in that very simplest case. Afterward, in Secs. IV D and V B we derive completeness for SO(N) and SU(N) and generalize to Spin(N), Sp(N), and E8 in Secs. VI, VII, and VIII. We then discuss the implications of these results for GUTs in Sec. IX and summarize our conclusions and future directions in Sec. X.

In this section, we outline our constructive procedure for deriving completeness. To begin, we initialize the algorithm by specifying the symmetries of the theory, together with some finite set of particles assumed to be in the spectrum. Since the full spectrum must be invariant under the action of the symmetry, we can actually exploit the symmetry to generate new charges from old ones. In particular, starting from any given charge we can apply the generators of the symmetry to construct additional families of charges from this state. We will refer to this action as “orbiting” the charge and the set of resulting charges as its “orbit.”

We then scatter pairs of particles and apply dispersion relations to deduce the existence of new charges, iterating the algorithm to consider all possible scattering processes for the additional charges that we find. At each step, we orbit the charges in hand to generate as many new ones as possible. By repeating this procedure, we incrementally populate the charge lattice. If the space of charges ultimately spans the full lattice, completeness is established and we claim victory. The algorithm is summarized in Fig. 1.

Let us consider the very simplest Abelian symmetry, corresponding to G=H=U(1). Furthermore, we assume that the spectrum is composed of a graviton, electron, and positron whose charges are Q={0,-1,+1}.(5)By construction, the spectrum is invariant under charge conjugation C. For graviton-electron scattering we have 0⊗(-1)→(-1)∨(+1),(6)so the s- and u-channel charges are -1 and +1. Since both of these are already in Q, no new charged particles are required, and this scattering process is inconclusive. Meanwhile, for electron-electron scattering we have (-1)⊗(-1)→(-2)∨0,(7)so the s- and u-channel charges are -2 and 0. Since the latter is already in Q, a new charged state is not guaranteed, so the scattering is again inconclusive. Said another way, while a doubly charged state would have been required in the s channel, that channel need not be activated. If instead only the u channel is present, then the dispersion relation is entirely accounted for by the graviton, which was already in the spectrum. Thus, in this example our algorithm fails to prove completeness.

The situation is similarly bleak for multiple Abelian factors, for example considering G=H=U(1)2. Here we assume a spectrum consisting of a graviton, electron, positron, dark electron, and dark positron with charges Q={(0,0),(-1,0),(+1,0),(0,-1),(0,+1)}.(8)Obviously, any scattering involving the graviton will be inconclusive. On the other hand, the scattering of an electron and dark electron yields (-1,0)⊗(0,-1)→(-1,-1)∨(-1,+1).(9)Both the s- and u-channel charges are new, so this pair is actually conclusive. With charge conjugation C, we know that each channel grows the spectrum by the sets {(-1,-1),(+1,+1)} or {(-1,+1),(+1,-1)}.

Concatenating either of these two sets to Q, we obtain two possible and distinct spectra. However, it is easy to verify that in the next iteration of the algorithm, all scattering pairs are inconclusive. That is, even though Eq. (9) implies the existence of new charged particles, all subsequent scattering processes can be accounted for without introducing more charged states on top of this. This difficulty persists for any Abelian symmetry.

The plot thickens when our charges are embedded within an Abelian subgroup of a non-Abelian symmetry. In this case we can actively exploit the non-Abelian generators of the symmetry to transform states of a given charge into those of a different charge.

To begin, we consider the case of a special orthogonal symmetry group G=SO(N), whose Cartan subgroup is H=U(1)⌊N/2⌋. By definition, every charged particle is a simultaneous eigenstate of the generators of H. The corresponding eigenvalues span the ⌊N/2⌋-dimensional charge lattice of integers, Λ=Z⌊N/2⌋={∑i=1⌊N/2⌋qie→i|qi∈Z},(10)where the e→i are orthonormal basis vectors. A charge vector q→∈Λ in the lattice will be denoted by q→=(q1,q2,…,q⌊N/2⌋),(11)where each entry is an orthogonal component.

Recall also that the center group is Z(SO(N))=1 or Z2 for odd or even N, respectively. In the latter case, the central charge of a given charge vector q→ is z(q→)=q1+q2+⋯+q⌊N/2⌋mod  2,(12)The central charge effectively counts the parity of the number of fundamental indices of a particular representation. We will often find it useful to classify states by their membership in the central charge sectors z=0, 1.

On top of the structures defined in Appendix A, we will on occasion make use of the outer automorphism group C, which map representations to other representations. The precise nature of C will vary case by case, but it will usually correspond to some version of charge conjugation symmetry.

Let us now consider the case of G=SU(N), whose Cartan subgroup is H=U(1)N-1. The charge lattice is Λ={q→=∑i=1N-1qiμ→i|qi∈Z},(16)where we emphasize that q→ is not expressed in terms of the orthonormal basis e→i, but rather the nonorthogonal basis of fundamental weights μ→i, which satisfy μ→i=∑j=iN-1ij(j+1)e→j.(17)In particular, the components of q→ in the basis of fundamental weights are precisely the Dynkin coordinates, q→=(q1,q2,…,qN-1).(18)For this reason, one must take special care when computing dot products. Furthermore, since e→i and μ→i are distinct, the components of q→ are not actually the eigenvalues of the Cartan generators, but are straightforwardly related to them by a linear transformation defined by Eq. (17). In spite of this mismatch, we will abuse nomenclature and glibly refer to q→ throughout this section as the charge vector. Obviously, completeness in the Dynkin coordinates defined by the components of q→ will imply completeness in the bona fide charge lattice.

As described earlier, it will be useful to be able to orbit charges using the action of the symmetry group G and the outer automorphisms C. The former is implemented by the action of the Weyl group, which acts on the simple roots of SU(N). Explicitly, these simple roots are r→i=i+1ie→i-i-1ie→i-1=-μ→i-1+2μ→i-μ→i+1,(19)where i=1,2,…,N-1 and e→0=μ→0=μ→N=0, with the normalization chosen so that r→i·μ→j=δij. In these coordinates, the Weyl group transformation reviewed in Eq. (A2) maps charge vectors via q→→q→-qir→i, which in Dynkin coordinates sends (…,qi-1,qi,qi+1,…)↓(…,qi-1+qi,-qi,qi+1+qi,…).(20)This transformation flips the sign of a given entry and adds that entry to its neighbors. On the other hand, charge conjugation C reverses the order of the Dynkin coordinates via (q1,q2,…,qN-1)→(qN-1,qN-2,…,q1).(21)For example, C swaps the fundamental and antifundamental representations of SU(N).

The center group Z(SU(N))=ZN will play an important role in the subsequent analysis. The corresponding central charges of SU(N) are known as N-ality [26], which are defined as z(q→)=q1+2q2+⋯+(N-1)qN-1mod  N.(22)Recall that the roots only transform between states within a given central charge sector, which are labeled by z=0,1,…,N-1. As noted previously, spanning all central charge sectors is a necessary but not sufficient condition for completeness. Obviously, the fundamental and antifundamental, which span the sectors z=1 and N-1, are not sufficient on their own to satisfy this criterion. This is the root of the difficulty in establishing completeness for special unitary symmetries.

The spin group Spin(N) is the well-known universal cover of the special orthogonal group SO(N). By definition, these groups share the same roots and hence the same Weyl group. They only differ in their charge lattices, with the weight lattice of Spin(N) given by Λ=Λ0∪Λ1/2Λ0=Z⌊N/2⌋={q→=∑i=1⌊N/2⌋qie→i|qi∈Z}Λ1/2=(Z+12)⌊N/2⌋={q→=∑i=1⌊N/2⌋qie→i|qi∈Z+12},(28)where Λ0 is just the integer lattice of SO(N) and Λ1/2 is the same lattice shifted by a half-integer in every direction. The latter correspond to the spinor representations of Spin(N) that are not in SO(N). The group Spin(N) admits the N-dimensional vector representation, together with the 2⌊(N-1)/2⌋-dimensional spin representations of multiplicity one and two for N odd and even respectively.

As per our earlier arguments, the existence of a vector will guarantee completeness in the Λ0 charge lattice, which corresponds to SO(N). Obviously, to ensure completeness in Λ1/2 as well, we also have to include spinor representations of Spin(N) in the spectrum.

For these reasons, we will assume that our spectrum contains the vector e→i and the spinors 12∑iσie→i for all sign choices σi∈{±1}. For even N, the cases where the σi sum to even or odd constitute the two different spinors. Equipped with these states, it is then straightforward to generate the full charge lattice.

We take the ultracharged corner states constructed in the previous section and scatter them against the spinors. In the case of Spin(4), this scattering yields (2n,0)⊗(12,12)→(2n+12,12)∨(2n-12,-12)(0,2n)⊗(12,12)→(12,2n+12)∨(-12,2n-12).(29)For Spin(N), we would use the corresponding corners of the ⌊N/2⌋-dimensional orthant. These scattering processes generate ultracharged states in the half-integer lattice, which we can then orbit inward to generate all enclosed charges within the same central charge sector. Taking the large-n limit, we then derive completeness of the full spectrum across the integers and half-integers. In conclusion, Spin(N) is complete, provided the spectrum includes the vector and spinor representations.

Consider the compact symplectic group Sp(N), which defines the set of linear transformations that leave invariant the symplectic form in even N dimensions. Here, completeness is an immediate consequence of our earlier results. To see why, note that Sp(N) has the very same charge lattice as SO(N), namely Λ=ZN/2. In particular, the vector representation of Sp(N) has the same charges as in SO(N). Furthermore, the root system of Sp(N) contains that of SO(N) as a subset, so the same is true for their respective Weyl groups. This implies that with respect to orbits, Sp(N) is availed of strictly more operations to generate new charges from old charges. Altogether, these relations imply that all of the scattering and orbit operations we performed for SO(N) are equally applicable for Sp(N). This establishes completeness of the symplectic group Sp(N).

We can make an analogous argument for E8, the largest exceptional simple Lie group. Unlike SO(N) and SU(N), the smallest nontrivial representation of E8 is the adjoint 248. We will therefore assume an initial spectrum Q comprising the graviton and the adjoint. The charge lattice of E8 is famously even and self-dual, Λ=Λ0∪Λ1/2Λ0={q→=∑i=18qie→i|qi∈Z,∑i=18qi∈2Z}Λ1/2={q→=∑i=18qie→i|qi∈Z+12,∑i=18qi∈2Z},(30)and is equivalent to its root lattice. In particular, the root system comprises R0, the set of all ±e→i±e→j with independent signs, along with R1/2, the set of all points in Λ1/2 with all |qi|=1/2. Together with the point at the origin, R0 and R1/2 are the adjoint weights of E8.

We immediately recognize Λ0 and R0 as precisely the z=0 sector weight lattice and root system of SO(16). For SO(N), the full z=0 sector is generated by our scattering algorithm if we start with the adjoint rather than the vector, as is clear beginning with the second line of Eq. (13), followed by Weyl symmetry and lowering operators as described in that section. In the exact same way, for E8 we are forced to augment Q to all of Λ0 by running the scattering algorithm starting from the adjoint weights in R0.

In particular, we note that the point 2n(e→1+e→2) is in Λ0 and hence in Q. Using the Weyl orbit of this weight, in parallel with our earlier constructions we obtain a polytope composed of the E8 adjoint weights rescaled by 2n. Applying the results reviewed in Appendix A, along with the fact that the center of E8 is trivial, we conclude that all points in the full lattice Λ contained within this polytope must also be in representations described by the polytope, so these weights must be included in Q as well. As n tends to infinity, we find that Q=Λ and hence conclude that charge completeness of E8 follows from the presence of the adjoint and the graviton.

The above analysis has direct implications for the charge completeness of GUTs. As we will see below, in these theories, the field content of the standard model actually ensures charge completeness.

We have shown that the spectrum of charges is complete across a range of theories under mild assumptions. The cornerstone of our analysis is the dispersion relation for gravitational scattering amplitudes in Eq. (1), first proposed in Ref. [14]. This remarkable formula directly relates the scattering contribution from graviton exchange to a sum over exchanges in the s and u channel. As we have emphasized, the assumption of a tree-level ultraviolet completion of gravity is an important criterion for our analysis. This condition not only zeros out the boundary term in Eq. (1), but also dictates that the unitarizing degrees of freedom are single-particle states. Under these conditions, we have arrived at a surprisingly strong conclusion: in a litany of cases, given the existence of a symmetry G and a handful of charged seed states, the sheer presence of gravity directly mandates the existence of single-particle states charged under all possible Cartan charges of G. This follows purely as a consequence of the self-consistency of scattering amplitudes. If the standard model resides in an ultraviolet completion with this symmetry G, then these assumptions require the existence of new particles with doubly, triply, etc. charged excitations of the leptons and quarks.

Our proof exploits the sequential scattering of particles and the action of the symmetry group G itself to generate the full spectrum of charges [29]. For an Abelian symmetry G, we are unable to prove completeness in any form. However, for certain choices of non-Abelian G, we show that charge completeness is mathematically required as long as the spectrum contains at least some finite handful of charged states, usually taken to be the fundamental. In our context, charge completeness is the property that the full charge lattice of the Cartan subgroup H is populated by single-particle states.

Our results suggest a number of avenues for future work. First and foremost is the question of whether it is possible to derive more general forms of completeness. In particular, given that charge completeness arises relatively straightforwardly, it is natural to ask: are all irreducible representations of a non-Abelian symmetry G required to be in the spectrum? Of course, charge completeness necessitates the presence of an infinite collection of representations of arbitrarily high weight. We have initiated a partial investigation into the question of representation completeness, yielding primarily negative results. We elaborate on our various attempts in Appendix C, based on the strategy of Ref. [14]. We present evidence that in the cases of G=SO(3),SO(4), representation completeness cannot be proven using this methodology. More generally, given that the entirety of our analysis has focused on four-point scattering, it is worth examining whether higher-point processes might afford more leverage. In particular, recent work has shown that positivity constraints on higher-point scattering are exceedingly stringent [30].

A second question relates to the precise nature of the symmetry G. As discussed earlier, the conclusions of the present work are independent of whether G is gauged or global. The latter famously runs afoul of the expectation that quantum gravity forbids exact global symmetries. This does not detract from our logic, however, since we are not claiming that a global symmetry is required. Indeed it would be interesting to study the case where the symmetry is explicitly broken. Conversely, if the symmetry is gauged then any logical implications of the dispersion relation could be relevant to the weak gravity conjecture [31].

Furthermore, while we have assumed throughout that G is internal, another interesting possibility is that G could be a spacetime symmetry, for example the Poincaré group. Since spacetime symmetries also imply conservation laws, it is natural to speculate on the completeness of spacetime charges such as physical spins. Spinning states of this kind are precisely what is needed to explicitly break the higher-form symmetries of gravity [32]. Of course, our assumptions already imply spin completeness: reproducing the 1/t pole on the left-hand side of Eq. (1) requires an infinite tower of spins on the right-hand side, since each partial wave is a polynomial in t [33]. A corollary of this fact is that the existence of particle of a given charge implies the existence of an infinite tower of higher-spin cousins with that same charge.

A third line of inquiry concerns whether our results could be strengthened or enriched by relaxing existing assumptions or adding new ones. Clearly, our strongest assumption is that the relevant dynamics are at tree level, so it would be worthwhile to understand the effects of loops. At a technical level, loops would allow for multiparticle states in the dispersion relation in Eq. (1), which naively ensures completeness trivially, starting from some initial seed of charged states. Another central assumption is the presence of the graviton, so it would be interesting to examine how important gravity truly is for our conclusions. What we fundamentally require is a particle that couples universally, so that it shows up in all of our dispersion relations, and whose amplitudes exhibit sufficiently soft high-energy behavior. Studying other setups that share these properties could be illuminating.

Alternatively, since our analysis is relatively conservative it is also reasonable to include stronger assumptions. For example, our arguments make no use of unitarity, let alone detailed kinematic information other than the dispersion relation itself. Unitarity is obviously a very weak assumption, but it would likely provide additional mileage. Furthermore, if we could somehow know for a fact that both the s and u channels of Eq. (1) are always activated, then much stronger claims could be proven. We discuss this possibility briefly in Appendix C 4.

Last but not least, our results suggest that there is untapped potential in applying the methods of the scattering bootstrap to other conjectures in quantum gravity from the bottom up. The modern amplitudes program and the bootstrap approach have recently demonstrated notable utility in addressing problems in quantum gravity, from the uniqueness of string theory to, in this work, questions in the swampland program such as the completeness hypothesis. The possibility of applying these techniques to other questions seems especially likely given the close relationships linking the completeness hypothesis to the weak gravity conjecture and the absence of global symmetries. The intersection of top-down and bottom-up approaches to quantum gravity, in applying the amplitudes bootstrap to gravitational scattering, offers a compelling program for future study, instantiating a powerful new set of tools for addressing these fundamental questions.