A Song of Spice and Ice: A diagonal dual to dS$_3$ & its Galilean flat space limit

Abstract

We propose a diagonal CFT dual to asymptotically de Sitter spacetimes in three dimensions (3d), constructed from the holomorphic part of the CFT dual at the future boundary and the antiholomorphic part of the CFT dual at the past boundary, in part inspired by an antipodal map between the two CFT copies. This gives equal and opposite imaginary central charges for the holomorphic and antiholomorphic sectors. The corresponding diagonal holographic energy-momentum tensor expectation values are constructed from the Wavefunction at the future boundary and its conjugate. This enables a recovery of the entropy of $dS_3$, as well as a family of complex Kerr $dS_3$ spacetimes with two real horizons, by a Cardy formula. We then explore the flatspace limit of our dS construction. With the central charges as above, the flatspace limit naturally induces a Galilean contraction on the dual diagonal 2d CFT, in contrast to the more familiar Carrollian limit when 3d Asymptotically Flat Spacetime (AFS) is approached from $AdS_3$. The corresponding limit for the complex Kerr family above leaves the inner horizon intact, resulting in flat space complex black holes ($\mathbb{C}$BH) that are closely related to flat space cosmologies which arise from BTZ black holes in AdS$_3$. We recover the entropy of these $\mathbb{C}$BHs by the BMS-Cardy formula.