$AdS/CFT$ to $dS/CFT$: Some Recent Developments

TL;DR

These notes provide a cohesive, pedagogical tour of the AdS/CFT correspondence and its extensions to de Sitter and flat spacetimes, beginning with conformal field theory basics and AdS geometry and culminating in modern holographic developments. They detail the GKPW dictionary, holographic entanglement entropy via the RT/HRT prescriptions, and holographic complexity, and then explore generalized holography including dS/CFT, AdS/BCFT, and wedge/double holography approaches. By connecting bulk gravity calculations to boundary CFT data, the text highlights how strongly coupled quantum fields, thermal physics, and quantum information concepts are encoded holographically, even in non-AdS backgrounds. Overall, the material illuminates a broad program for understanding holography in cosmological and flat spacetimes, with concrete computational tools and theoretical frameworks to compare bulk and boundary observables.

Abstract

These lecture notes aim to provide a pedagogical introduction to the AdS/CFT correspondence and its extensions to spacetimes with positive (de Sitter spacetime) and zero (flat spacetime) cosmological constant. We begin by explaining the physical motivation for holography and the significance of the AdS/CFT correspondence. We then review the basic ingredients of conformal field theory (CFT) and anti de Sitter (AdS) spacetime required to formulate the duality. Building on these foundations, we discuss the formulation of the AdS/CFT correspondence and discuss several consistency checks that support it. We conclude with a brief discussion of holography in de Sitter and flat spacetimes.

Figures (14)

• Figure 1: Global AdS spacetime. $\rho$ is along the horizontal direction in this picture and $S^{d-1}$ is located at top and bottom. This picture is taken from https://en.wikipedia.org/wiki/Anti-de_Sitter_space.
• Figure 2: Poincaré patch of $AdS_{d+1}$ spacetime. This figure is taken from Bayona:2005nq. To make it consistent with the current discussion, we need to replace $n$ with $d$ in this figure.
• Figure 3: In this figure, we have $Dp$-brane located at $X^a=\overline{x}_a$.
• Figure 4: In this figure, we have a cartoon picture of open strings attached between two $Dp$-branes.
• Figure 5: (a) (a) Scattering of a closed string from branes in flat spacetime. (b) Propagation of a closed string in a curved geometric background. Figure adapted from Penedones:2016voo.