Topology of Future Infinity in dS/CFT

TL;DR

This work evaluates the dS/CFT correspondence for Vasiliev gravity in de Sitter space by computing the Wheeler–deWitt wave function for universes with nontrivial future boundary topology via the dual 3D vector models. It finds a pronounced genus-growth in the WdW amplitude, driven by the Chern–Simons singlet constraint that yields $Z_{S^1\times\Sigma_g}\sim k^{(g-1)N^2}$ and hence $|\\Psi|^2\sim\exp\left[{1\over G_N^2}(g-1)\ln k\right]$, signaling an instability toward increasingly complex future topology. The paper assesses several proposed renormalization strategies and finds them unsatisfactory within a consistent dS/CFT framework, while arguing for stringent consistency conditions on the dual CFT, including a restricted form of nonunitarity and a universal finite-temperature structure. It also compares Vasiliev results with Einstein gravity saddles, showing that different contour choices yield qualitatively distinct genus dependences, and concludes that a robust resolution likely requires a deeper understanding of the closed-string sector and observer-dependent interpretations. Overall, the work highlights a potentially universal instability toward topological complexity in de Sitter quantum gravity and outlines key open questions for a coherent holographic description.

Abstract

The dS/CFT proposal of Anninos, Hartman, and Strominger relates quantum Vasiliev gravity in dS_4 to a large N vector theory in three dimensions. We use this proposal to compute the Wheeler-de Witt wave function of a universe having a particular topology at future infinity. This amplitude is found to grow rapidly with the topological complexity of the spatial slice; this is due to the plethora of states of the Chern-Simons theory that is needed to impose the singlet constraint. Various mechanisms are considered which might ameliorate this growth, but none seems completely satisfactory. We also study the topology dependence in Einstein gravity by computing the action of complex instantons; the wave function then depends on a choice of contour through the space of metrics. The most natural contour prescription leads to a growth with genus similar to the one found in Vasiliev theory, albeit with a different power of Newton's constant.