4D Scattering Amplitudes and Asymptotic Symmetries from 2D CFT

TL;DR

This work proposes a holographic-like duality for 4D Minkowski space by foliating spacetime into AdS3 and dS3 slices, recasting 4D scattering amplitudes as 2D celestial CFT correlators. The core insight is that soft theorems and asymptotic symmetries in 4D map to Ward identities and infinite-dimensional algebras (Kac-Moody and Virasoro) in a 2D CFT, with massless modes described by 3D Chern-Simons gauge theories that encode soft radiation and memory effects. The paper develops a detailed correspondence: AdS3/CFT2 for the current sector, AdS3 gravity via CS theory for gravitons, and a unifying interpretation of memory, Aharonov-Bohm phases, and black-hole horizon toy models in this framework. It also discusses the implications for flat-space holography, unitarity in the 2D dual, and potential loop-level generalizations. Overall, the work provides a concrete, technically rich bridge between 4D scattering, soft theorems, and 2D conformal structure, with deep connections to asymptotic symmetries and topological field theory.

Abstract

We reformulate the scattering amplitudes of 4D flat space gauge theory and gravity in the language of a 2D CFT on the celestial sphere. The resulting CFT structure exhibits an OPE constructed from 4D collinear singularities, as well as infinite-dimensional Kac-Moody and Virasoro algebras encoding the asymptotic symmetries of 4D flat space. We derive these results by recasting 4D dynamics in terms of a convenient foliation of flat space into 3D Euclidean AdS and Lorentzian dS geometries. Tree-level scattering amplitudes take the form of Witten diagrams for a continuum of (A)dS modes, which are in turn equivalent to CFT correlators via the (A)dS/CFT dictionary. The Ward identities for the 2D conserved currents are dual to 4D soft theorems, while the bulk-boundary propagators of massless (A)dS modes are superpositions of the leading and subleading Weinberg soft factors of gauge theory and gravity. In general, the massless (A)dS modes are 3D Chern-Simons gauge fields describing the soft, single helicity sectors of 4D gauge theory and gravity. Consistent with the topological nature of Chern-Simons theory, Aharonov-Bohm effects record the "tracks" of hard particles in the soft radiation, leading to a simple characterization of gauge and gravitational memories. Soft particle exchanges between hard processes define the Kac-Moody level and Virasoro central charge, which are thereby related to the 4D gauge coupling and gravitational strength in units of an infrared cutoff. Finally, we discuss a toy model for black hole horizons via a restriction to the Rindler region.

Figures (8)

• Figure 1: Minkowski space is divided into Milne and Rindler regions which are time-like and space-like separated from the origin, respectively. Each region is then foliated into a family of warped slices, each at a fixed proper distance from the origin.
• Figure 2: Equivalence of 4D scattering amplitudes and 2D correlators for the special case of multiple soft boson gauge emission and multiple conserved current insertion.
• Figure 3: The celestial sphere houses a region $R$ whose boundary $\partial R$ encircles the trajectory of a hard particle. The single helicity Aharonov-Bohm phase around $\partial R$ is simultaneously i) the cumulative charge of hard tracks threading $R$, ii) the integrated velocity kick experienced by test charges along $\partial R$, i.e. the electromagnetic memory effect, and iii) the Ward identity for the holomorphic conserved current of the 2D CFT. Here $dy^i$ is the infinitesimal vector tangent to $\partial R$ while $dy_\perp^i$ is the infinitesimal vector orthogonal to $\partial R$ but still on the celestial sphere.
• Figure 4: Single emission of an abelian gauge boson and multiple emission of non-abelian gauge bosons. In both cases, external legs connect to bulk-boundary propagators $K_{\mu}$. In the non-abelian case, these soft emissions accumulate into a soft branch described by the field $A_\mu$.
• Figure 5: Schematic depicting soft, single helicity non-abelian gauge bosons coupling to hard sources. Each soft branch is initiated by a set of $(+)$ helicity soft gauge bosons, so the corresponding field configuration is self-dual.