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

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.

We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries and Verlinde lines, TDLs together with their attached defect operators provide models of fusion categories without braiding. We study the crossing relations of TDLs, discuss their relation to the 't Hooft anomaly, and use them to constrain renormalization group flows to either conformal critical points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.

Abstract:We conjecture a precise relationship between the Schur limit of the superconformal index of four-dimensional N = 2 field theories, which counts local operators, and the spectrum of BPS particles on the Coulomb branch. We verify this conjecture for the special case of free field theories, N = 2 QED, and SU(2) gauge theory coupled to fundamental matter. Assuming the validity of our proposal, we compute the Schur index of all Argyres-Douglas theories. Our answers match expectations from the connection of Schur operators with two-dimensional chiral algebras. Based on our results we propose that the chiral algebra of the generalized Argyres-Douglas theory (A k−1 , A N −1 ) with k and N coprime, is the vacuum sector of the (k, k + N ) W k minimal model, and that the Schur index is the associated vacuum character.

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.