We prove that symmetry in the presence of gravity implies a version of the completeness hypothesis. For a broad class of theories, we demonstrate that the existence of finitely many charged particles logically necessitates the existence of infinitely many charged particles populating the entire charge lattice. Our conclusions follow from the consistency of perturbative gravitational scattering and require the following ingredients: (1) a weakly coupled ultraviolet completion of gravity, (2) a non-Abelian symmetry <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi></math>, gauged or global, whose Cartan subgroup generates the Abelian charge lattice, and (3) a spectrum containing some finite set of charged representations, in the simplest cases taken to be a single particle in the fundamental. Under these conditions, the Abelian charge lattice is completely filled by single-particle states for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi><mo>=</mo><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>≥</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi><mo>=</mo><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>≥</mo><mn>3</mn></math>, which in turn implies completeness for other symmetry groups such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>p</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>E</mi><mn>8</mn></msub></math>. Curiously, a corollary of our results is that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></math> grand unified theories have precisely the minimal field content needed to derive completeness using our methodology.

United States

Francesco Calisto

Clifford Cheung

Grant N. Remmen

Spain

Francesco Sciotti

Michele Tarquini

Completeness from gravitational scattering

Is field space infinite? If not, it either loops back on itself or ends altogether. Periodic boundary conditions are of course familiar, but field space endpoints—which appear in real-world systems—are far less explored. In this paper we argue that boundaries in field space are generic, radiatively stable structures that allow for new physics at very low scales not ruled out by experiment. Such boundaries are delocalized in field space from the vacuum, so they can only be accessed by coherent fields or high multiplicity processes, both of which are weakly constrained observationally. Low multiplicity interactions do not detect the boundary and instead perceive a “mirage cutoff” that is parametrically higher than the true cutoff of the theory. Hence, field space boundaries are deformations of the standard model that are Lorentz invariant, local, unitary at low energies, and experimentally unconstrained. We comment on the possibility of field space boundaries on the long-range force carriers and the Higgs, as well as possible implications for the hierarchy problem.

Ira Z. Rothstein

Hiding new physics at the end of field space

Lorentz invariance, unitarity, and causality enforce powerful constraints on the theory space of physical scattering amplitudes. However, virtually all efforts in this direction have centered on the very simplest case of four-point scattering. In this work, we derive an infinite web of “multipositivity bounds” that nonlinearly constrain all tree-level higher-point scattering amplitudes under similarly minimal assumptions. Our construction rules out several deformations of the string and implies mixed-multiplicity bounds on the Wilson coefficients of planar effective field theories. Curiously, an infinite class of multipositivity bounds is exactly saturated by the amplitudes of the open string.

Multipositivity bounds for scattering amplitudes

The scattering amplitudes of string theory exhibit many extraordinary properties. But are they the unique mathematical objects to do so? Recently, it has been shown how the spectrum and amplitudes of open string theory follow directly from the assumptions of faster than power-law falloff at high energies and a property dubbed level truncation. At present there is no analogous principle for closed string scattering, which is famously rigid and naively impervious to modification. In this paper we analytically bootstrap the spectrum and four-point amplitudes of the closed string—together with a parametrized space of deformations—from conditions on high-energy falloff and level truncation. While these deformations exhibit the same Regge scaling as pure gravity, in the tensionless limit they reproduce remarkable extremal amplitudes that have appeared in bottom-up studies of positivity.

Aaron Hillman

Uniqueness criteria for the Virasoro-Shapiro amplitude

The super BKP (SBKP) hierarchy in fermionic picture is constructed by embedding tensor product of infinite-dimensional Lie superalgebra of type B and Grassmann algebra into Clifford superalgebra. On the basis of bosonic counterpart, we construct the super Baker functions of the SBKP hierarchy and give a bilinear identity represented by it. The super differential operator is introduced and the bilinear identity in terms of the super pseudo-differential operator is shown. Finally, the solutions of non group orbits in the super fermionic Fock space are discussed.

China

Huizhan Chen

The super BKP hierarchy from Clifford super module

We show that the Veneziano amplitude of string theory is the unique solution to an analytically solvable bootstrap problem. Uniqueness follows from two assumptions: faster than power-law falloff in high-energy scattering and the existence of some infinite sequence in momentum transfer at which higher-spin exchanges cancel. The string amplitude—including the mass spectrum—is an output of this bootstrap. If the amplitude merely vanishes at high energies, the solution is a three-parameter family containing the Veneziano, Coon, and hypergeometric amplitudes, and more.

Bootstrap Principle for the Spectrum and Scattering of Strings

We explore generalized symmetry in the context of nonlinear dynamical gravity. Our basic strategy is to transcribe known results from Yang-Mills theory directly to gravity via the tetrad formalism, which recasts general relativity as a gauge theory of the local Lorentz group. By analogy, we deduce that gravity exhibits a one-form symmetry implemented by an operator U  α  labeled by a center element α of the Lorentz group and associated with a certain area measured in Planck units. The corresponding charged line operator W  ρ  is the holonomy in a spin representation ρ, which is the gravitational analog of a Wilson loop. The topological linking of U  α  and W  ρ  has an elegant physical interpretation from classical gravitation: the former materializes an exotic chiral cosmic string defect whose quantized conical deficit angle is measured by the latter. We verify this claim explicitly in an AdS-Schwarzschild black hole background. Notably, our conclusions imply that the standard model exhibits a new symmetry of nature at scales below the lightest neutrino mass. More generally, the absence of global symmetries in quantum gravity suggests that the gravitational one-form symmetry is either gauged or explicitly broken. The latter mandates the existence of fermions. Finally, we comment on generalizations to magnetic higher-form or higher-group gravitational symmetries.

Maria Derda

Joon-Hwi Kim

Vinicius Nevoa

Ira Rothstein

Nabha Shah

Generalized symmetry in dynamical gravity

We explore a recently proposed effective field theory describing electromagnetically or gravitationally interacting massive particles in an expansion about their mass ratio, also known as the self-force (SF) expansion. By integrating out the deviation of the heavy particle about its inertial trajectory, we obtain an effective action whose only degrees of freedom are the lighter particle together with the photon or graviton, all propagating in a Coulomb or Schwarzschild background. The 0SF dynamics are described by the usual background field method, which at 1SF is supplemented by a “recoil operator” that encodes the wobble of the heavy particle, and similarly computable corrections appearing at 2SF and higher. Our formalism exploits the fact that the analytic expressions for classical backgrounds and particle trajectories encode dynamical information to all orders in the couplings, and from them we extract multiloop integrands for perturbative scattering. As a check, we study the two-loop classical scattering of scalar particles in electromagnetism and gravity, verifying known results. We then present new calculations for the two-loop classical scattering of dyons, and of particles interacting with an additional scalar or vector field coupling directly to the lighter particle but only gravitationally to the heavier particle.

France

Julio Parra-Martinez

Jordan Wilson-Gerow

Gravitational scattering and beyond from extreme mass ratio effective field theory

Bethe Ansatz was discovered in 1932. Half a century later its algebraic structure was unearthed: Yang-Baxter equation was discovered, as well as its multidimensional generalizations [tetrahedron equation and d-simplex equations]. Here we describe a universal method to solve these equations using Clifford algebras. The Yang-Baxter equation (<math><mi>d</mi><mo>=</mo><mn>2</mn></math>), Zamolodchikov's tetrahedron equation (<math><mi>d</mi><mo>=</mo><mn>3</mn></math>) and the Bazhanov-Stroganov equation (<math><mi>d</mi><mo>=</mo><mn>4</mn></math>) are special cases. Our solutions form a linear space. This helps us to include spectral parameters. Potential applications are discussed.

India

Pramod Padmanabhan

Vladimir Korepin

Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras

Is string theory uniquely determined by self-consistency? Causality and unitarity seemingly permit a multitude of putative deformations, at least at the level of two-to-two scattering. Motivated by this question, we initiate a systematic exploration of the constraints on scattering from higher-point factorization, which imposes extraordinarily restrictive sum rules on the residues and spectra defined by a given amplitude. These bounds handily exclude several proposed deformations of the string: the simplest “bespoke” amplitudes with tunable masses and a family of modified string integrands from “binary geometry.” While the string itself passes all tests, our formalism directly extracts the three-point amplitudes for the low-lying string modes without the aid of worldsheet vertex operators.