Celestial Klein Spaces

TL;DR

This paper explores the structure of Klein spaces with various signatures, analyzing their conformal boundaries, infinities, and the representation of scattering states through highest weight states and Mellin transforms.

Contribution

It introduces a detailed geometric and group-theoretic analysis of Klein spaces with arbitrary signatures, including their boundaries and scattering state representations.

Findings

The conformal boundary of Klein spaces has a unique connected component.

Infinities are described as quotients of generalized AdS spaces.

Scattering states are represented by highest weight states and Mellin-transformed plane waves.

Abstract

We consider the analytic continuation of $(p+q)$-dimensional Minkowski space (with $p$ and $q$ even) to $(p,q)$-signature, and study the conformal boundary of the resulting "Klein space". Unlike the familiar $(-+++..)$ signature, now the null infinity ${\mathcal I}$ has only one connected component. The spatial and timelike infinities ($i^0$ and $i'$) are quotients of generalizations of AdS spaces to non-standard signature. Together, ${\mathcal I}, i^0$ and $i'$ combine to produce the topological boundary $S^{p+q-1}$ as an $S^{p-1} \times S^{q-1}$ fibration over a null segment. The highest weight states (the $L$-primaries) and descendants of $SO(p,q)$ with integral weights give rise to natural scattering states. One can also define $H$-primaries which are highest weight with respect to a signature-mixing version of the Cartan-Weyl generators that leave a point on the celestial $S^{p-1} \times S^{q-1}$ fixed. These correspond to massless particles that emerge at that point and are Mellin transforms of plane wave states.