On the Complexified Spacetime Manifold Mapping of AdS to dS

TL;DR

This work introduces HUFT, a holomorphic unified field theory on a complex ambient manifold where AdS and dS emerge as two real slices connected by analytic continuation of the curvature scale. By encoding gravity and gauge interactions in a single holomorphic connection and using a Picard–Lefschetz contour, HUFT preserves bulk unitarity on the real slice and transports holographic structures like RT/HRT/QES across the AdS–dS bridge, while admitting a unitary AdS boundary theory but only a state-preparation holography for dS. The AdS to dS map is realized through a complex rotation $L\to i/H$ (and $z\to i\eta$), leaving curvature invariants finite and avoiding new singularities, with boundary terms and variational structures matched across the transition. The framework also demonstrates compatibility with Coleman–Mandula in the flat limit, provides a UV-finite regulator via entire functions, and yields a calculable pipeline for cosmology, black-hole thermodynamics, and holographic entanglement in a Universe with $\Lambda>0$. Overall, HUFT offers a coherent, calculable picture in which AdS and dS are two real faces of a single holomorphic theory, enabling cross-pollination of tools between holography, quantum gravity, and cosmology.

Abstract

In a complex manifold, one can bridge anti-de Sitter and de Sitter spacetimes via analytic continuation, preserving geometric invariants and regularity, avoiding singularities during the AdS-dS transition. It unifies gravitational and gauge interactions under a complexified symmetry group, maintaining bulk unitarity for both AdS and dS. Boundary unitarity is upheld in AdS but not in dS due to the spacelike conformal boundary. The theory uses holographic principles like the MacDowell-Mansouri and Quantum Extremal Surface prescriptions to align entanglement and black hole entropy with AdS/CFT and general relativity. HUFT provides insights into AdS and dS holography, the cosmological constant, and quantum gravity unitarity and entanglement.

Figures (1)

• Figure 1: Global embeddings and conformal boundaries of dS$_4$ and AdS$_4$. The panels show the two real forms SO$(4,1)$ and SO$(3,2)$ of the same holomorphic ambient symmetry and the analytic continuation $iR_{\mathrm{dS}}\!\mapsto\! R_{\mathrm{AdS}}$ ($\Lambda\!\mapsto\!-\Lambda$) Strominger2001dSCFT. The global embeddings and conformal boundaries of dS$_4$ and AdS$_4$. are $-t^{2}+x^{2}+y^{2}+z^{2}+u^{2}=R_{\mathrm{dS}}^{2}$ in $\mathbb{R}^{1,4}$ (group $\text{SO}(4,1)$) with spacelike boundaries $\Im^\pm \cong S^{3}$SpradlinStromingerVolovich2001. And $-t^{2}-u^{2}+x^{2}+y^{2}+z^{2}=-R_{\mathrm{AdS}}^{2}$ in $\mathbb{R}^{2,3}$ (group $SO(3,2)$) with timelike boundary $\partial\mathrm{AdS}_4 \cong S^{2}\times\mathbb{R}$.