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Semiclassical Virasoro Symmetry of the Quantum Gravity S-Matrix

Daniel Kapec, Vyacheslav Lysov, Sabrina Pasterski, Andrew Strominger

TL;DR

The paper shows that the tree-level gravity S-matrix in four-dimensional Minkowski space carries a semiclassical Virasoro symmetry acting on the celestial sphere at null infinity, realized as a diagonal X subgroup of extended BMS^+ × extended BMS^- transformations. This symmetry is rooted in a subleading soft graviton theorem and is captured by Ward identities that relate amplitudes with and without soft gravitons. By embedding CK spaces within the X-transformation framework, the authors connect asymptotic symmetry data to classical gravitational scattering and show how these Ward identities emerge from soft theorems. The work clarifies the potential quantum fate of this Virasoro structure, discusses IR subtleties, and points to broader links with holography and celestial CFT formulations.

Abstract

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.

Semiclassical Virasoro Symmetry of the Quantum Gravity S-Matrix

TL;DR

The paper shows that the tree-level gravity S-matrix in four-dimensional Minkowski space carries a semiclassical Virasoro symmetry acting on the celestial sphere at null infinity, realized as a diagonal X subgroup of extended BMS^+ × extended BMS^- transformations. This symmetry is rooted in a subleading soft graviton theorem and is captured by Ward identities that relate amplitudes with and without soft gravitons. By embedding CK spaces within the X-transformation framework, the authors connect asymptotic symmetry data to classical gravitational scattering and show how these Ward identities emerge from soft theorems. The work clarifies the potential quantum fate of this Virasoro structure, discusses IR subtleties, and points to broader links with holography and celestial CFT formulations.

Abstract

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.

Paper Structure

This paper contains 13 sections, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Asymptotically flat geometry
  3. Metrics
  4. Constraints
  5. Extended BMS$^\pm$ transformations
  6. ${\cal X}$ transformations
  7. Christodoulou-Klainerman spaces
  8. Classical gravitational scattering
  9. ${\cal X}$ Ward identity
  10. Quantum states
  11. ${\cal X}$-invariance of ${\cal S}$
  12. Mode expansions
  13. From soft theorem to Virasoro symmetry

Figures (1)

  • Figure 1: Penrose diagram for Minkowski space. Near ${\mathcal{I}}^+$ surfaces of constant retarded time $u$ (red) are cone-like and intersect ${\mathcal{I}}^+$ in a conformal $S^2$ parametrized by $(z,{\bar{z}})$. Cone-like surfaces of constant advanced time $v$ (green) intersect ${\mathcal{I}}^-$ in a conformal $S^2$ also parametrized by $(z,{\bar{z}})$. The future (past) $S^2$ boundary of ${\mathcal{I}}^+$ is labelled ${\mathcal{I}}^+_+$ (${\mathcal{I}}^+_-$), while the future (past) boundary of ${\mathcal{I}}^-$ is labelled ${\mathcal{I}}^-_+$ (${\mathcal{I}}^-_-$).