arXiv subjects

Jeremy van der Heijden

Publications and source records attributed to Jeremy van der Heijden.

13 recordsLinked to original sources

arXiv2026

Algebras for generalized entanglement wedges

In asymptotically AdS spacetimes, the mathematical structure of the set of entanglement wedges reflects the algebraic structure of the underlying holographic description. For more general spacetimes, Bousso and Penington (BP) have recently proposed a generalization of entanglement wedges sharing many of the same properties as usual entanglement wedges. In this paper, we explore the hypothesis that each generalized entanglement wedge can be associated with an algebra in the (generally unknown) fundamental description (in a semiclassical limit). We postulate features of the map from entanglement wedges to algebras that provide a natural algebraic interpretation for some of the basic mathematical properties of the set of entanglement wedges. Quantitatively, we suggest a possible generalization of the Ryu-Takayanagi formula that associates the gravitational entropy of a generalized entanglement wedge with an entropic quantity for the associated algebra. Through this assignment, inclusion monotonicity and strong-subadditivity properties shown by BP for generalized entanglement wedges would follow from various inequalities satisfied by algebraic entropies. We include a detailed appendix reviewing relevant algebraic background, including a discussion of algebraic entropies and their inequalities.

hep-th↗

arXiv2026

Factor Code Networks

For a quantum code defined by an isometry $T:H_{\rm code} \to H_{\rm phys}$, an operator map is a *-homomorphism $φ: B(H_{\rm code}) \to B(H_{\rm phys})$ from the space of operators on the logical Hilbert space to the space of operators on the physical Hilbert space satisfying $φ({\cal O}) T = T {\cal O}$ for every operator ${\cal O}$ on $H_{\rm code}$. This can additionally be taken to be unital $φ(I_{code}) = I_{phys}$ if and only if $\dim H_{\rm code}$ divides $\dim H_{\rm phys}$. In this case, $φ(B(H_{\rm code})$ is a von Neumann factor subalgebra of $B(H_{\rm phys})$ and we call the pair $(T,φ)$ a {\it factor code}. A code network is a family of Hilbert spaces $\{{\cal H}_i\}$ indexed by a partially ordered set with isometries $T_{ji}:{\cal H}_i \to {\cal H}_j$ for $i < j$ satisfying $T_{kj} T_{ji} = T_{ki}$ for $i < j < k$. In this paper, we ask when this can be promoted to a {\it factor code network} by defining unital operator maps $φ_{ji}$ so that $(T_{ji}, φ_{ji})$ is a factor code and $φ_{kj} φ_{ji} = φ_{ki}$. We show that the divisibility conditions on Hilbert space dimensions are sufficient for any code network with a tree structure but not in general. For a diamond-shaped network, we show that promotion is possible if and only if the principle angles between $T_{42} H_2$ and $T_{43} H_3$ all have multiplicity divisible by $H_1$. For incomparable $i_1,i_2 < j$, we characterize when $φ_{j i_1} (B(H_{i_1}))$ and $φ_{j i_2} (B(H_{i_2}))$ commute, when they generate all of $B(H_j)$, and when they intersect only on multiples of the identity. For $i < j_1,j_2 < k$, we characterize when $φ_{k j_1} (B(H_{j_1})) \cap φ_{k j_2} (B(H_{j_2})) = φ_{k i} (B(H_{i}))$ and when these algebras together with $B(H_k)$ form a nondegenerate commuting square.

quant-ph↗

arXiv2026

Algebraic structure in holographic tensor networks

In a previous work 2511.21852, we explored the hypothesis that in general theories of quantum gravity, spacetime regions known as generalized entanglement wedges (introduced by Bousso and Penington) have an associated subalgebra of observables in a semiclassical limit. We defined an entropy associated with these subalgebras that obeys the same monotonicity and strong subadditivity properties as the generalized gravitational entropy of generalized entanglement wedges. Here, we test aspects of these ideas making use of tensor network toy models for holography. In this context, the analog of a generalized entanglement wedge is a region for which no region containing it has a smaller number of boundary legs. In the limit of large bond dimension (analogous to the semiclassical limit), we argue that each such region has a canonically associated subalgebra of the full algebra of observables on the boundary Hilbert space, as well as a family of unitarily related von Neumann factor subalgebras. We investigate in detail how various algebraic operations on these subalgebras (e.g. their intersections, joins, and commutants) are related to corresponding geometrical operations on the tensor network regions. We study to what extent the monotonicity and strong subadditivity properties of generalized entropy for generalized entanglement wedges follow from properties of our algebraic entropy. We also introduce an alternative entropic quantity $S(ω|| ω\circ E_{\mathcal{A}'})$ associated with a subalgebra $\mathcal{A}$ that also obeys monotonicity and strong subadditivity properties with suitable conditions. Finally, we explain how to define a type $II_1$ von Neumann algebra based on an infinite family of random tensor networks with increasing bond dimension such that the regions corresponding to generalized entanglement wedges can be associated with exact subalgebras of this full algebra.

hep-th↗

arXiv2026

Algebraic Complexity and Black Hole Complementarity

We develop an operator algebraic generalization of the Yoshida-Kitaev information recovery protocol that applies to von Neumann algebras of arbitrary type. The construction is based on finite-index inclusions, with the Jones basic construction and canonical endomorphisms providing the central algebraic tools. Unlike the finite-dimensional qubit description, the infinite-dimensional theory exhibits new structural phenomena that play an essential role in information recovery. In particular, the diary information is represented non-locally by a choice of Pimsner-Popa basis associated with the inclusion. Exploiting the Temperley-Lieb relations satisfied by the Jones projections, we introduce an algebraic notion of computational complexity and show that it is naturally measured by the Jones-Kosaki index. For irreducible depth-two inclusions, we demonstrate that the information transfer is implemented by an algebraic Fourier transform. Finally, we discuss a potential spacetime interpretation of the construction, including the emergence of an island algebra in terms of (non-)isometric embeddings and an operator algebraic realization of black hole complementarity.

hep-th↗

arXiv2026

The universality class of the first levels in low-dimensional gravity

We investigate the physics of a small group of quantum states defined above the sharply defined ground state of a chaotic ensemble. This `universality class of the first levels' (UFL) is realized in the majority of `synthetic' random matrix models but, for all we know, in only one microscopically defined system: low-dimensional gravity. We discuss the physical properties of these states, notably their exceptional rigidity against external perturbations, as quantified by the so-called quantum state fidelity. Examining these structures through the lenses of random matrix and string theory, we highlight their relevance to the physics of low-dimensional holographic principles.

hep-th↗

arXiv2026

Medicine show: A Calogero model with principal series states

The Calogero model is an interacting, $N$-particle, $\mathfrak{sl}(2,\mathbb R)$-invariant quantum mechanics, whose Hilbert space is furnished by a tower of discrete series modules. The system enjoys both classical and quantum integrability at any $N$ and at any value of the coupling; this is guaranteed by the existence of $N$ mutually-commuting currents, one of them being the Hamiltonian. In this paper, we alter the Calogero model so that it may accommodate states in the unitary principal series irreducible representation of $\mathfrak{sl}(2,\mathbb R)$. Doing so requires changing the domain of the quantum operators--a procedure which succeeds in preserving unitarity and $\mathfrak{sl}(2,\mathbb R)$-invariance, but alters the integrability properties of the theory. We explicitly solve the deformed model for $N=2,3$ and outline a procedure for solving the model at general $N$. We expect this deformed model to provide us with general lessons that carry over to other systems with states in the principal series, for example, interacting massive quantum field theories on de Sitter space.

hep-th↗

arXiv2025

Quantum Symmetry and Geometry in Double-Scaled SYK

The emergence of the quantum $R$-matrix in the double-scaled SYK model points to an underlying quantum group structure. In this work, we identify the quantum group $\mathcal{U}_q(\mathfrak{su}(1,1))$ as a subalgebra of the chord algebra. Specifically, we construct the generators of $\mathcal{U}_q(\mathfrak{s} \mathfrak{u}(1,1))$ from combinations of operators within the chord algebra and show that the one-particle chord Hilbert space decomposes into the positive discrete series representations of $\mathcal{U}_q(\mathfrak{s} \mathfrak{u}(1,1))$. Using the coproduct structure of the quantum group, we build the multi-particle Hilbert space and establish its equivalence with previous results defined by the chord rules. In particular, we show that the quantum $R$-matrix acts as a swapping operator that reverses the ordering of open chords in each fusion channel while incorporating the corresponding $q$-weighted penalty factors. This action enables an explicit derivation of the chord Yang-Baxter relation. We further explore a realization of the quantum group generators on the quantum disk, and present a novel factorization formula for the bulk gravitational wavefunction in the presence of matter. We further discuss the relation between the $\mathcal{U}_q(\mathfrak{s} \mathfrak{u}(1,1))$ structure uncovered here and the $\mathcal{U}_q(\mathfrak{s} \mathfrak{l}(2, \mathbb{R}))$ algebra previously studied from the boundary perspective. Finally, we study the gravitational wavefunction with matter in the Schwarzian regime.

hep-th↗

arXiv2025

Modular chaos, operator algebras, and the Berry phase

Modular Berry transport associates a geometric phase to a zero mode ambiguity in a family of modular operators. In holographic settings, this phase was shown to encode nontrivial information about the emergent spacetime geometry. We reformulate modular Berry transport for arbitrary von Neumann algebras, including giving a precise definition of the zero mode projection in terms of a conditional expectation. For a certain class of state perturbations, we demonstrate that the modular Berry phase gives rise to an emergent symplectic form in the large $N$ limit, extending related results in the context of subregion/subalgebra duality. We also show that the vanishing of the Berry curvature for modular scrambling modes signals the emergence of a local Poincaré algebra, which plays a key role in the quantum ergodic hierarchy. These results provide an intriguing relation between geometric phases, modular chaos and the local structure of spacetime.

hep-th↗

arXiv2024

An Operator Algebraic Approach To Black Hole Information

We present an operator algebraic perspective on the black hole information problem. For a black hole after Page time that is entangled with the early radiation we formulate a version of the information puzzle that is well-posed in the $G\to 0$ limit. We then give a description of the information recovery protocol in terms of von Neumann algebras using elements of the Jones index theory of type II$_1$ subfactors. The subsequent evaporation and recovery steps are represented by Jones's basic construction, and an operation called the canonical shift. A central element in our description is the Jones projection, which leads to an entanglement swap and implements an operator algebraic version of a quantum teleportation protocol. These aspects are further elaborated on in a microscopic model based on type I algebras. Finally, we argue that in the emergent type III algebra the canonical shift may be interpreted as a spacetime translation and, hence, that at the microscopic level "translation = teleportation".

hep-th↗

arXiv2023

Changing states in holography: From modular Berry curvature to the bulk symplectic form

We present a new perspective on bulk reconstruction using Berry phases in the boundary CFT. Our parallel transport of modular Hamiltonians is associated to a trajectory in the space of states, which we obtain from the insertion of a source in the Euclidean path integral. Using a modular version of the extrapolate dictionary and the equivalence between modular flow in the boundary and the bulk, we show that the expectation value of the modular Berry curvature on the boundary agrees with an appropriately defined bulk symplectic form associated to the entanglement wedge. In addition, we derive a quantum information metric on the space of density matrices from the Berry curvature, which is related to the canonical energy in the bulk. We also explore the case where a state change reduces to a shape change, uncovering the coadjoint orbit structure of kinematic space in higher dimensions.

hep-th↗

arXiv2022

Quantum chaos in 2D gravity

We present a quantitative and fully non-perturbative description of the ergodic phase of quantum chaos in the setting of two-dimensional gravity. To this end we describe the doubly non-perturbative completion of semiclassical 2D gravity in terms of its associated universe field theory. The guiding principle of our analysis is a flavor-matrix theory (fMT) description of the ergodic phase of holographic gravity, which exhibits $\mathrm{U}(n|n)$ causal symmetry breaking and restoration. JT gravity and its 2D-gravity cousins alone do not realize an action principle with causal symmetry, however we demonstrate that their {\it universe field theory}, the Kodaira-Spencer (KS) theory of gravity, does. After directly deriving the fMT from brane-antibrane correlators in KS theory, we show that causal symmetry breaking and restoration can be understood geometrically in terms of different (topological) D-brane vacua. We interpret our results in terms of an open-closed string duality between holomorphic Chern-Simons theory and its closed-string equivalent, the KS theory of gravity. Emphasis will be put on relating these geometric principles to the characteristic spectral correlations of the quantum ergodic phase.

hep-th↗

arXiv2022

Virasoro Entanglement Berry Phases

We study the parallel transport of modular Hamiltonians encoding entanglement properties of a state. In the case of 2d CFT, we consider a change of state through action with a suitable diffeomorphism on the circle: one that diagonalizes the adjoint action of the modular Hamiltonian. These vector fields exhibit kinks at the interval boundary, thus together with their central extension they differ from usual elements of the Virasoro algebra. The Berry curvature associated to state-changing parallel transport is the Kirillov-Kostant symplectic form on an associated coadjoint orbit, one which differs appreciably from known Virasoro orbits. We find that the boundary parallel transport process computes a bulk symplectic form for a Euclidean geometry obtained from the backreaction of a cosmic brane, with Dirichlet boundary conditions at the location of the brane. We propose that this gives a reasonable definition for the symplectic form on an entanglement wedge.

hep-th↗

arXiv2022

A universe field theory for JT gravity

We present a field theory description for the non-perturbative splitting and joining of baby universes in Euclidean Jackiw-Teitelboim (JT) gravity. We show how the gravitational path integral, defined as a sum over topologies, can be reproduced from the perturbative expansion of a Kodaira-Spencer (KS) field theory for the complex structure deformations of the spectral curve. We use that the Schwinger-Dyson equations for the KS theory can be mapped to the topological recursion relations. We refer to this dual description of JT gravity as a `universe field theory'. By introducing non-compact D-branes in the target space geometry, we can probe non-perturbative aspects of JT gravity. The relevant operators are obtained through a modification of the JT path integral with Neumann boundary conditions. The KS/JT identification suggests that the ensemble average for JT gravity can be understood in terms of a more standard open/closed duality in topological string theory.

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