Causal Wedge Reconstruction in Holography

• Causal wedge reconstruction is a framework that defines bulk regions via causal influence from designated boundary subdomains.
• It utilizes diamond regions and modular flows to mathematically reconstruct both local Euclidean and global de Groot dual topologies.
• This approach bridges holographic duality, quantum gravity, and algebraic quantum field theory by encoding geometric information purely from causal data.

Causal wedge reconstruction is a framework for recovering or characterizing bulk spacetime regions based solely on the causal structure and associated data accessible from specified subregions of a boundary field theory. This approach is fundamental to the geometric underpinnings of holographic duality, quantum gravity, and algebraic quantum field theory, where causal relations, modular dynamics, and operator algebras play primary roles in encoding both local and global spacetime topology. The causal wedge is typically constructed as the bulk region that can both influence and be influenced by a selected boundary domain, thereby serving as a minimal, causally defined reconstructable region. The evolution of this concept has illuminated how causality alone can, under certain conditions, reconstruct local topology, motivate new algebraic dualities, and situate wedge reconstruction amid modern developments in quantum error correction and modular theory.

1. Causal Wedge Construction: Formalism and Definition

Causal wedge reconstruction begins with the identification of a boundary subregion and its domain of dependence, denoted $\mathcal{D}[\mathcal{A}]$ in boundary field theory. The causal wedge $\mathcal{C}_{\mathcal{A}}$ in the bulk is then defined as: $$\mathcal{C}_{\mathcal{A}} = J^+[\mathcal{D}[\mathcal{A}]] \cap J^-[\mathcal{D}[\mathcal{A}]],$$ where $J^{\pm}[S]$ denotes the bulk future and past domains of influence of $S$. The boundary of $\mathcal{C}_{\mathcal{A}}$ is constructed from null surfaces generated by inward-oriented geodesics from the “top” and “bottom” of $\mathcal{D}[\mathcal{A}]$, with their intersection $$\Xi_{\mathcal{A}} = \partial J^+[\mathcal{D}[\mathcal{A}]] \cap \partial J^-[\mathcal{D}[\mathcal{A}]]$$ called the causal information surface (Hubeny et al., 2012).

In geometric terms, in Minkowski space the causal wedge can be built from a lattice of "diamonds,"

$D(p, q) = \{ x \in M : p < x < q \},$

where $p < x < q$ indicates that $x$ lies strictly between $p$ and $q$ in the causal order. The collection of such diamonds (with $p, q$ from a dense set, e.g., $\mathbb{Q}^4$) forms a base for the wedge structure and generates the "places" or regions underpinning the point-less causal site approach (Kovár, 2011).

The causal wedge is also defined in more general symmetric or curved spaces (e.g., AdS, symmetric coset spaces $M = G/H$), via modular flows and positivity criteria for vector fields generated by distinguished elements (Euler elements) in the Lie algebra (Neeb et al., 2021, Morinelli et al., 2023).

2. Algebraic and Topological Reconstruction from Causality

A key advance is the demonstration that spacetime topology—at least locally—can be entirely reconstructed from causal ordering data. The construction proceeds by considering "causal sites" consisting of regions organized by inclusion ($\subset$) and strict causal precedence ($<$). To such a causal site, one associates a framework $(P, \mathcal{T})$, where $P$ is the collection of all regions and $\mathcal{T}$ is the collection of "centered" overlaps possessing the finite intersection property. The dual framework is constructed by identifying maximal centered families,

$f(p) = \{ C \in P : p \in C \},$

mirroring the way ultrafilters characterize points in pointless topology. The topology generated on the space $X$ of maximal centered families uses the closed subbase $\{ T(A) : A \in P \}$, with $T(A) = \{ x \in X : A \in x \}$. The result is that $(X, \mathcal{T})$ is a compact $T_1$ space.

Crucially, when this framework is applied to Minkowski space with the diamond lattice, the reconstructed topology on each compact subset coincides with the Euclidean topology, while globally it yields the de Groot dual (co-compact) topology

$$T_E^* = \{ U \subseteq M : M \setminus U \text{ is compact in the Euclidean topology} \}$$

(Kovár, 2011). This demonstrates that causality encodes both the local and global topological structure—locally indistinguishable, globally compactified with non-Hausdorff features.

3. Causality as a Primary Structure and its Conceptual Consequences

The result that spacetime topology can be reconstructed purely from causal data strengthens the argument that causality is not merely an ordering of events but contains full topological (and potentially geometric) information.

The local equivalence between the reconstructed and Euclidean topology explains the empirical indistinguishability for local physics, whereas the global de Groot dual structure (co-compact topology) suggests a compact global universe without the need for additional points at infinity.

This causal-topological reconstruction fits conceptually with approaches to quantum gravity in which the fundamental structure is algebraic or "point-less," and smooth spacetime emerges only as a derived concept (Kovár, 2011). This is tightly related to the philosophy of "causal sites" (Christensen–Crane), the algebraic net approach of AQFT, and modern uses of wedge domains, operator algebras, and modular theory (Neeb et al., 2021, Morinelli et al., 2023).

4. Mathematical Structures and Key Formulas

Causal wedge reconstruction leverages several central mathematical objects:

Concept	Definition/Formula	Role
Diamond region	$D(p, q) = \{ x \in M : p < x < q \}$	Basic building block for sites/wedges
Maximal centered family	$f(p) = \{ C \in P : p \in C \}$	Encodes point-like data from regions
Co-compact topology	$$T_E^* = \{ U \subseteq M : M \setminus U \text{ is compact} \}$$	Global topology from causality
Causal wedge	$$\mathcal{C}_{\mathcal{A}} = J^+[\mathcal{D}[\mathcal{A}]] \cap J^-[\mathcal{D}[\mathcal{A}]]$$	Reconstructable bulk region
Causal information surface	$$\Xi_{\mathcal{A}} = \partial J^+[\mathcal{D}[\mathcal{A}]] \cap \partial J^-[\mathcal{D}[\mathcal{A}]]$$	Boundary of the causal wedge

This framework enables explicit construction of the reconstructive homeomorphism between points in Minkowski space (or more general backgrounds) and elements of the dual framework constructed from causal data.

5. Connections to Holography and Causal Wedge Reconstruction in AdS/CFT

The analogy between the causal-site–driven topology reconstruction and AdS/CFT causal wedge reconstruction is underscored by the parallel role played by causal structure.

In AdS/CFT, causal wedge reconstruction asserts that the reduced density matrix $\rho_{\mathcal{A}}$ for a boundary subregion $\mathcal{A}$ suffices to reconstruct physics in the bulk causal wedge $\mathcal{C}_{\mathcal{A}}$ (Hubeny et al., 2012). The process of reconstructing the co-compact topology from the causal site is thus a mathematical analogue to the holographic paradigm, where the causal relations among “wedges” (regions) reconstruct the global structure of spacetime.

In both approaches, causal information alone specifies the reconstructable domain, and both local and global features of spacetime are controlled by the causal, not metric, data. The wedge itself and its boundary (the causal information surface, or more generally, the relevant modular flow–generated domain in AQFT) acquire primary status (Neeb et al., 2021, Morinelli et al., 2023).

6. Implications, Limitations, and Extensions

The ability to reconstruct the global co-compact topology from causal order alone demonstrates that causality encodes all required structural information for local topological recovery. However, global distinctions, such as the non-Hausdorff nature of the de Groot dual, and the necessity of adopting compact saturated sets as a closed base rather than conventional open balls, can have significant implications for global physics and the organization of infinity.

This approach also naturally interfaces with quantum gravity perspectives, where the fundamental structure may be discrete or algebraic, and topology emerges from causal order and overlap properties. Modular flow and wedge duality formulations adapt these insights to algebraic QFT and holography, connecting causal wedge reconstruction to modern operator-algebraic understandings of subregion duality and bulk-boundary correspondence.

Consequently, causal wedge reconstruction is foundational for both mathematical and physical theories of spacetime—providing a bridge between order-theoretic models, topological recovery, algebraic duality, and the quantum information perspective on holography.