Celestial Klein spaces

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 $(\ensuremath{-}+++\ensuremath{\cdots})$ signature, now the null infinity $\mathcal{I}$ has only one connected component. The spatial and timelike infinities (${i}^{0}$ and ${i}^{\ensuremath{'}}$) are quotients of generalizations of AdS spaces to nonstandard signature. Together, $\mathcal{I}$, ${i}^{0}$, and ${i}^{\ensuremath{'}}$ combine to produce the topological boundary ${S}^{p+q\ensuremath{-}1}$ as an ${S}^{p\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{S}^{q\ensuremath{-}1}$ 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 ${S}^{p\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{S}^{q\ensuremath{-}1}$ fixed. These correspond to massless particles that emerge at that point and are Mellin transforms of plane wave states.