Erik Verlinde

Contribution to the Festschrift for Gerard ’t Hooft on the occasion of his 80th birthday.In the early 80’s Gerard ’t Hooft was the first to argue that the Bekenstein–Hawking entropy provides information about the microscopic density of states of the degrees of freedom associated to event horizons. He also envisaged the importance of gravitational shockwave interactions and discovered the holographic principle. Following Gerard’s insights and advise, I presented in December 2009 the idea that the inertial and gravitational force should be viewed as being emergent and have their origin in changes in the entropy of the underlying microscopic degrees of freedom. In this note, I explain why this point of view explains “why gravity knows about microscopic entropy”.

Quantum entanglement is monogamous: a system cannot be maximally entangled with multiple partners. Yet thought experiments in gravitational contexts, notably the Hayden–Preskill protocol, appear to challenge this rule. We argue that the illusion of polygamy arises from neglecting the operational cost of decoding. Redistributing entanglement requires teleportation-like operations so computationally demanding that they have gravitational consequences. Entanglement can only “cheat” through complex operations whose backreaction is unavoidable: they may displace the event horizon, render a wormhole traversable, or induce more drastic changes in the spacetime geometry.

A bstract: 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 𝒰 _q (𝔰𝔲(1, 1)) as a subalgebra of the chord algebra. Specifically, we construct the generators of 𝒰 _q (𝔰𝔲(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 𝒰 _q (𝔰𝔲(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 also discuss the relation between the 𝒰 _q (𝔰𝔲(1, 1)) structure uncovered here and the 𝒰 _q (𝔰𝔩(2 , ℝ)) algebra previously studied from the boundary perspective. Finally, we study the gravitational wavefunction with matter in the Schwarzian regime.

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 → 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 II₁ 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”.

We study the semi-classical thermodynamics of two-dimensional de Sitter space (dS₂) in Jackiw-Teitelboim (JT) gravity coupled to conformal matter. We extend the quasi-local formalism of Brown and York to dS₂, where a timelike boundary is introduced in the static patch to uniquely define conserved charges, including quasi-local energy. The boundary divides the static patch into two systems, a cosmological system and a black hole system, the former being unstable under thermal fluctuations while the latter is stable. A semi-classical quasi-local first law is derived, where the Gibbons–Hawking entropy is replaced by the generalized entropy. In the microcanonical ensemble the generalized entropy is stationary. Further, we show the on-shell Euclidean microcanonical action of a causal diamond in semi-classical JT gravity equals minus the generalized entropy of the diamond, hence extremization of the entropy follows from minimizing the action. Thus, we provide a first principles derivation of the island rule for U(1) symmetric dS₂ backgrounds, without invoking the replica trick. We discuss the implications of our findings for static patch de Sitter holography.

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.