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

TL;DR

This paper explores the structure of null infinity in (2,2) spacetime, revealing a celestial torus and a discrete set of scattering states organized by conformal symmetry, offering new insights into scattering amplitudes.

Contribution

It introduces a novel geometric framework for (2,2) spacetime, replacing the celestial sphere with a celestial torus and demonstrating a discrete structure in scattering states.

Findings

Null infinity becomes a product of a null interval and a celestial torus.

Spacelike and timelike infinity are time-periodic quotients of AdS3.

Scattering states form a discrete set with conformal weights.

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.