We study solutions of the Klein-Gordon, Maxwell, and linearized Einstein equations in R 1,d+1 that transform as d-dimensional conformal primaries under the Lorentz group SO(1, d + 1). Such solutions, called conformal primary wavefunctions, are labeled by a conformal dimension ∆ and a point in R d , rather than an on-shell (d + 2)-dimensional momentum. We show that the continuum of scalar conformal primary wavefunctions on the principal continuous series ∆ ∈ d 2 + iR of SO(1, d + 1) spans a complete set of normalizable solutions to the wave equation. In the massless case, with or without

It is shown that the tree-level S-matrix for quantum gravity in four-dimensional Minkowski space has a Virasoro symmetry which acts on the conformal sphere at null infinity.

The four-dimensional (4D) Lorentz group SL(2, C) acts as the two-dimensional (2D) global conformal group on the celestial sphere at infinity where asymptotic 4D scattering states are specified. Consequent similarities of 4D flat space amplitudes and 2D correlators on the conformal sphere are obscured by the fact that the former are usually expressed in terms of asymptotic wavefunctions which transform simply under spacetime translations rather than the Lorentz SL(2, C). In this paper we construct on-shell massive scalar wavefunctions in 4D Minkowski space that transform as SL(2, C) conformal primaries. Scattering amplitudes of these wavefunctions are SL(2, C) covariant by construction. For certain mass relations, we show explicitly that their three-point amplitude reduces to the known unique form of a 2D CFT primary three-point function and compute the coefficient. The computation proceeds naturally via Witten-like diagrams on a hyperbolic slicing of Minkowski space and has a holographic flavor.1 The subleading soft theorem has a one-loop exact anomaly [9-12] whose effects remain to be understood but are recently discussed in [13,14].2 One may hope that ultimately 4D quantum gravity scattering amplitudes are found to have a dual holographic representation as some exotic 2D CFT on CS 2 , but at present there are no proposals for such a construction.

The conventional gravitational memory effect is a relative displacement in the position of two detectors induced by radiative energy flux. We find a new type of gravitational 'spin memory' in which beams on clockwise and counterclockwise orbits acquire a relative delay induced by radiative angular momentum flux. It has recently been shown that the displacement memory formula is a Fourier transform in time of Weinberg's soft graviton theorem. Here we see that the spin memory formula is a Fourier transform in time of the recently-discovered subleading soft graviton theorem. Contents

It was shown by Low in the 1950s that the subleading terms of soft-photon S-matrix elements obey a universal linear relation. In this Letter, we give a new interpretation to this old relation, for the case of massless QED, as an infinitesimal symmetry of the S matrix. The symmetry is shown to be locally generated by a vector field on the conformal sphere at null infinity. Explicit expressions are constructed for the associated charges as integrals over null infinity and shown to generate the symmetry. These charges are local generalizations of electric and magnetic dipole charges.

Recently, spin-one wavefunctions in four dimensions that are conformal primaries of the Lorentz group SL(2, C) were constructed. We compute low-point, tree-level gluon scattering amplitudes in the space of these conformal primary wavefunctions. The answers have the same conformal covariance as correlators of spin-one primaries in a 2d CFT. The BCFW recursion relation between three-and four-point gluon amplitudes is recast into this conformal basis.

We provide a unified treatment of conformally soft Goldstone modes which arise when spin-one or spin-two conformal primary wavefunctions become pure gauge for certain integer values of the conformal dimension ∆. This effort lands us at the crossroads of two ongoing debates about what the appropriate conformal basis for celestial CFT is and what the asymptotic symmetry group of Einstein gravity at null infinity should be. Finite energy wavefunctions are captured by the principal continuous series ∆ ∈ 1 + iℝ and form a complete basis. We show that conformal primaries with analytically continued conformal dimension can be understood as certain contour integrals on the principal series. This clarifies how conformally soft Goldstone modes fit in but do not augment this basis. Conformally soft gravitons of dimension two and zero which are related by a shadow transform are shown to generate superrotations and non-meromorphic diffeomorphisms of the celestial sphere which we refer to as shadow superrotations. This dovetails the Virasoro and Diff(S2) asymptotic symmetry proposals and puts on equal footing the discussion of their associated soft charges, which correspond to the stress tensor and its shadow in the two-dimensional celestial CFT.

Asymptotic symmetries of theories with gravity in d = 2m+2 spacetime dimensions are reconsidered for m > 1 in light of recent results concerning d = 4 BMS symmetries.Weinberg's soft graviton theorem in 2m + 2 dimensions is re-expressed as a Ward identity for the gravitational S-matrix. The corresponding asymptotic symmetries are identified with 2m + 2-dimensional supertranslations. An alternate derivation of these asymptotic symmetries as diffeomorphisms which preserve finite-energy boundary conditions at null infinity and act non-trivially on physical data is given. Our results differ from those of previous analyses whose stronger boundary conditions precluded supertranslations for d > 4. We find for all even d that supertranslation symmetry is spontaneously broken in the conventional vacuum and identify soft gravitons as the corresponding Goldstone bosons.