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

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.

💡 Research Summary

These lecture notes provide a pedagogical yet comprehensive overview of the Anti‑de Sitter/Conformal Field Theory (AdS/CFT) correspondence and its recent extensions to de Sitter (dS) and flat spacetimes. The material is organized into six main sections, mirroring a series of six 90‑minute lectures delivered in 2025.

The introductory part motivates holography as a tool for probing strongly coupled quantum field theories (QFTs) that are inaccessible to ordinary perturbation theory. It emphasizes that the AdS/CFT duality, first proposed by Maldacena in 1997, maps the strongly coupled regime of a d‑dimensional conformal field theory (most famously N = 4 supersymmetric Yang‑Mills in four dimensions) to a weakly coupled gravitational theory in a (d + 1)‑dimensional AdS bulk (type IIB supergravity on AdS₅ × S⁵). This mapping is made precise through a dictionary that relates bulk fields to boundary operators, identifies the ’t Hooft coupling λ with the AdS radius L (L⁴ ∝ λ α′²), and equates the generating functional of connected correlators in the CFT with the on‑shell bulk action (the GKPW prescription).

Section 2 reviews the essential structure of conformal field theory. For dimensions d ≥ 3 the authors derive the infinitesimal conformal transformations by solving the Killing‑type equation ∂μ ε_ν + ∂ν ε_μ = (2/d)(∂·ε) η{μν}. They show that ε_μ can be at most quadratic in the coordinates, leading to four independent types of generators: translations (P_μ), Lorentz rotations (L{μν}), dilatations (D), and special conformal transformations (K_μ). The corresponding algebra closes into the SO(d,2) (Lorentzian) or SO(d+1,1) (Euclidean) conformal group. Scaling dimensions Δ are introduced via the transformation law Φ(λx) = λ^{‑Δ} Φ(x); for a free massless scalar Δ = (d‑2)/2. Using these symmetries the two‑point function is fixed to ⟨𝒪₁(x₁)𝒪₂(x₂)⟩ ∝ |x₁‑x₂|^{‑(Δ₁+Δ₂)}.

Section 3 introduces the geometry of AdS space, both in Lorentzian and Euclidean signatures. The global coordinates, Poincaré patch, and the conformal compactification (the “Penrose diagram”) are discussed, emphasizing how the AdS boundary is a d‑dimensional Minkowski (or Euclidean) space. The authors also present the Poincaré coordinates that make the bulk‑boundary relation explicit: ds² = (L²/z²)(dz² + η_{μν}dx^μdx^ν).

Section 4 is devoted to the core of the correspondence. After a brief review of D‑branes and supersymmetry, Maldacena’s conjecture is stated, and several consistency checks are described: matching of global symmetries, agreement of correlation functions (via bulk‑to‑boundary propagators), and the thermodynamic correspondence between AdS black holes and finite‑temperature CFTs. The finite‑temperature holography subsection explains how the AdS‑Schwarzschild geometry encodes the thermal ensemble of the dual CFT, reproducing the expected entropy density S ∝ N²T³ for N = 4 SYM.

Section 5 surveys three vibrant research directions that have emerged in the past decade.

1. Holographic Entanglement Entropy (HEE): The Ryu‑Takayanagi proposal (S_A = Area(γ_A)/4G_N) is presented for AdS₃/CFT₂, illustrating how minimal surfaces anchored on a boundary region compute the entanglement entropy of that region. Extensions to time‑dependent settings (the Hubeny‑Rangamani‑Takayanagi covariant prescription) are mentioned.
2. Holographic Complexity: Two competing definitions are discussed – the “Complexity‑Volume” conjecture (complexity proportional to the maximal bulk volume behind the horizon) and the “Complexity‑Action” conjecture (complexity proportional to the on‑shell action evaluated on the Wheeler‑DeWitt patch). Both aim to capture the growth of quantum circuit complexity in the dual CFT.
3. Partition Functions and Correlators from the Bulk: By evaluating the Euclidean on‑shell action with appropriate boundary conditions, one reproduces the CFT partition function Z_CFT = e^{‑S_bulk} and derives n‑point functions via functional differentiation. The authors illustrate this for a scalar field, showing the emergence of the expected power‑law behavior.

Section 6 explores generalizations beyond AdS.

• dS/CFT: The authors note that de Sitter space has a positive cosmological constant and two spacelike conformal boundaries (future I⁺ and past I⁻). They discuss proposals that relate quantum gravity in dS_{d+1} to a Euclidean CFT living on these boundaries, emphasizing the role of the SO(d,1) isometry group and the challenges posed by the lack of a globally timelike Killing vector. Recent frameworks such as double holography (where a higher‑dimensional bulk contains a lower‑dimensional gravitating brane) and wedge holography (where the bulk is restricted to a causal wedge) are presented as ways to make the dS/CFT correspondence more concrete.
• Flat Space Holography: By slicing Minkowski space into hyperbolic or de Sitter slices, one can apply wedge holography to flat space. The resulting asymptotic symmetry group is the Bondi‑Metzner‑Sachs (BMS) group, and the authors outline how BMS‑invariant field theories on the celestial sphere can serve as holographic duals. The discussion includes how entanglement entropy and complexity might be defined in this context, though many open questions remain.

The notes conclude with a set of exercises that reinforce the derivations of generators, correlation functions, and holographic formulas, and a curated bibliography (references