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$(2,2)$ Scattering and the Celestial Torus

Alexander Atanasov, Adam Ball, Walker Melton, Ana-Maria Raclariu, Andrew Strominger

TL;DR

By analytic continuation to Klein space, the paper shows the celestial boundary becomes a torus over a null interval and the full infinity forms an S^3 composed from I,i^0,i'. It develops the SL(2,R)_L x SL(2,R)_R symmetry and constructs L- and H- primaries for massless scalars, revealing a discrete spectrum labeled by conformal weights and descendant levels. It also establishes a celestial state-operator correspondence and expresses L-primary scattering as weighted integrals of Mellin-transformed amplitudes, linking to poles in conformal weights discussed in Minkowski amplitudes. Collectively, this framework advances celestial holography in split signature and suggests a discrete, torus-based structure for boundary observables.

Abstract

Analytic continuation from Minkowski space to $(2,2)$ split signature spacetime has proven to be a powerful tool for the study of scattering amplitudes. Here we show that, under this continuation, null infinity becomes the product of a null interval with a celestial torus (replacing the celestial sphere) and has only one connected component. Spacelike and timelike infinity are time-periodic quotients of AdS$_3$. These three components of infinity combine to an $S^3$ represented as a toric fibration over the interval. Privileged scattering states of scalars organize into $SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R$ conformal primary wave functions and their descendants with real integral or half-integral conformal weights, giving the normally continuous scattering problem a discrete character.

$(2,2)$ Scattering and the Celestial Torus

TL;DR

By analytic continuation to Klein space, the paper shows the celestial boundary becomes a torus over a null interval and the full infinity forms an S^3 composed from I,i^0,i'. It develops the SL(2,R)_L x SL(2,R)_R symmetry and constructs L- and H- primaries for massless scalars, revealing a discrete spectrum labeled by conformal weights and descendant levels. It also establishes a celestial state-operator correspondence and expresses L-primary scattering as weighted integrals of Mellin-transformed amplitudes, linking to poles in conformal weights discussed in Minkowski amplitudes. Collectively, this framework advances celestial holography in split signature and suggests a discrete, torus-based structure for boundary observables.

Abstract

Analytic continuation from Minkowski space to split signature spacetime has proven to be a powerful tool for the study of scattering amplitudes. Here we show that, under this continuation, null infinity becomes the product of a null interval with a celestial torus (replacing the celestial sphere) and has only one connected component. Spacelike and timelike infinity are time-periodic quotients of AdS. These three components of infinity combine to an represented as a toric fibration over the interval. Privileged scattering states of scalars organize into conformal primary wave functions and their descendants with real integral or half-integral conformal weights, giving the normally continuous scattering problem a discrete character.

Paper Structure

This paper contains 7 sections, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Null, spacelike and timelike infinity
  3. Symmetries
  4. Primary scalar states
  5. Primary scalar operators
  6. Comments on scattering
  7. Mapping $H$-primaries to $L$-primaries

Figures (1)

  • Figure 1: Toric Penrose diagram for signature $(2,2)$ Klein space. 45$^{\rm o}$ lines are null as usual. A Lorentzian torus is fibered over every point in the diagram. The spacelike cycle of the torus degenerates along the timelike line $U=V$, while the timelike cycle degenerates along the spacelike line $U=-V$. Neither cycle degenerates at null infinity $\cal I$ which is the interval $- {\pi \@@over 2}<U< {\pi \@@over 2},~V={\pi \@@over 2}$. Spacelike infinity $i^0$ is at $(U,V)=(-{\pi \@@over 2},{\pi \@@over 2})$ and has the conformal geometry of signature $(1,2)$ AdS$_3/\mathbb{Z}$. Timelike infinity $i^\prime$ is at $(U,V)=({\pi \@@over 2},{\pi \@@over 2})$ and has the conformal geometry of signature $(2,1)$ AdS$_3/\mathbb{Z}$. The blue lines are lines of constant $w{\bar{w}} -z{\bar{z}}$ with $\tau=0$ at $U=0$.