The emergence of the quantum $\textit{R}$-matrix in the double-scaled SYK model points to an underlying quantum group structure. In this work, we identify the quantum group 𝒰 (𝔰𝔲(1, 1)) as a subalgebra of the chord algebra. Specifically, we construct the generators of 𝒰 (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 (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 $\textit{R}$-matrix acts as a swapping operator that reverses the ordering of open chords in each fusion channel while incorporating the corresponding $\textit{q}$-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 (𝔰𝔲(1, 1)) structure uncovered here and the 𝒰 (𝔰𝔩(2$\textit{,}$ ℝ)) algebra previously studied from the boundary perspective. Finally, we study the gravitational wavefunction with matter in the Schwarzian regime.

Canada

Jeremy Heijden

Netherlands

Erik Verlinde

United States

Jiuci Xu

Quantum symmetry and geometry in double-scaled SYK

United Kingdom

Andrew Svesko

Evita Verheijden

Switzerland

Manus Visser

Erratum to: Quasi-local energy and microcanonical entropy in two-dimensional nearly de Sitter gravity

We use the Virasoro TQFT to derive an integral identity that we view as a non-rational generalization of the Verlinde formula for the Virasoro algebra with central charge $\textit{c}$ ≥ 25. The identity expresses the Virasoro fusion kernel as an integral over a ratio of modular S-kernels on the (punctured) torus. In particular, it shows that the one-point S-kernel diagonalizes the Virasoro 6$\textit{j}$ symbol. After carefully studying the analytic properties of this ‘Virasoro-Verlinde formula’, we present three applications. In boundary Liouville CFT, the formula ensures the open-closed duality of the boundary one-point function on the annulus. In pure 3d gravity, it provides an essential step in computing the partition function on hyperbolic 3-manifolds that fiber over the circle. Lastly, in AdS /CFT , the formula computes a three-boundary torus wormhole, which leads to a prediction for the statistical correlation between the density of states and two OPE coefficients in the dual large-$\textit{c}$ CFT ensemble. We conclude by discussing the implications of our result for the fusion rules in generic non-rational 2d CFTs.

Boris Post

France

Ioannis Tsiares

A non-rational Verlinde formula from Virasoro TQFT

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 $\textit{G}$ → 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 II 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”.

An operator algebraic approach to black hole information

Cardy-Verlinde (CV) formula relates the entropy of a strongly coupled conformal field theory (CFT) possessing AdS dual, at finite temperature, to its total energy (with an appropriate insertion of additional internal energy for charged systems) and Casimir energy. While the CV formula has been checked for various CFTs, in the present work, we intend to verify it directly, by exploiting a phenomenological matrix model, which is believed to qualitatively capture features of strongly coupled <math><mi>N</mi><mo>=</mo><mn>4</mn><mo>,</mo><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> Super Yang-Mills theory on <math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math>, at finite temperature and chemical potential at large N.

India

Pavan Kumar Yerra

Chandrasekhar Bhamidipati

Sudipta Mukherji

Cardy-Verlinde formula from boundary matrix model

We remind the definition and main properties of the Krichever quasiclassical tau-function, and turn to the application of these formulas for recent studies of two-dimensional quantum gravity. We show, that in the case of minimal gravity it turns to be directly related with the Verlinde formula for minimal models, giving in particular case its one more direct proof. Generalizations for continuous “complex Liouville” theory are also briefly discussed.

Russia

A. Marshakov

On Krichever tau-function and Verlinde formula

We investigate the composite systems consisting of topological orders separated by gapped domain walls. We derive a pair of domain-wall Verlinde formulae, that elucidate the connection between the braiding of interdomain excitations labeled by pairs of anyons in different domains and quasiparticles in the gapped domain wall with their respective fusion rules. Through explicit non-Abelian examples, we showcase the calculation of such braiding and fusion, revealing that the fusion rules for interdomain excitations are generally fractional or irrational. By investigating the correspondence between composite systems and anyon condensation, we unveil the reason for designating these fusion rules as symmetry fractionalized (irrationalized) fusion rules. Our findings hold promise for applications across various fields, such as topological quantum computation, topological field theory, conformal field theory, and parton physics.

China

Yu Zhao

Hongyu Wang

Yuting Hu

Yidun Wan

Symmetry fractionalized (irrationalized) fusion rules and two domain-wall Verlinde formulae

In this study, charged AdS black holes with a string cloud are taken into consideration. We identify three distinct black hole solutions within a specific temperature range, which subsequently merge into a single black hole beyond this range in the case of spherical space with low chemical potential and string density. However, in the case of large chemical potential and string density, only one black hole solution exists for all temperatures. Notably, we observe that for flat and hyperbolic space, there is only one black hole solution irrespective of the chemical potential and string density. By subtracting the contribution of the extremal black hole from the on-shell Euclidean action, the thermodynamical quantities associated with these black holes are calculated. We also investigate the stability of these black holes and find that medium-sized black hole is unstable. In the dual boundary theory side we verify the existence of bound state of quark-antiquark pair. Finally, we establish the Cardy-Verlinde formula for the boundary theory and we observed an additional energy term arising because of the presence of string cloud.

Rishi Pokhrel

Tanay K. Dey

Charged AdS black holes in presence of string cloud and Cardy-Verlinde formula

The topological Rényi and entanglement entropies depend on the bipartition of the manifold and the choice of the ground states. However, these entanglement quantities remain invariant under a coordinate transformation when the bipartition also undergoes the identical transformation. In topological quantum field theories (TQFTs), these coordinate transformations reduce to representations of the mapping class group on the manifold of the Hilbert space. We employ this invariant property of the Rényi and entanglement entropies under coordinate transformations for TQFTs in (2 + 1) dimensions on a torus with various bipartitions. By utilizing the replica trick and the surgery method to compute the topological Rényi and entanglement entropies, the invariant property results in Verlinde-like formulas. Furthermore, for the bipartition with interfaces as two non-intersecting torus knots, an SL(2, ℤ) transformation can untwist the torus knots, leading to a simple bipartition with an effective ground state. This invariant property allows us to demonstrate that the topological entanglement entropy has a lower bound −2 ln D, where D is the total quantum dimensions of the system.

Taiwan

Chih-Yu Lo

Po-Yao Chang

Topological entanglement entropy for torus-knot bipartitions and the Verlinde-like formulas

Modular fluctuations have previously been shown to obey an area law <math><mo>⟨</mo><mi>Δ</mi><msup><mi>K</mi><mn>2</mn></msup><mo>⟩</mo><mo>=</mo><mo>⟨</mo><mi>K</mi><mo>⟩</mo><mo>=</mo><mi>A</mi><mo>/</mo><mrow><mn>4</mn><msub><mi>G</mi><mi>N</mi></msub></mrow></math>. Furthermore, modular fluctuations generate fluctuations in the spacetime geometry of empty causal diamonds. Here we demonstrate the physical origin of these fluctuations, showing that the modular area law, in <math><mi>d</mi></math>-dimensionsal Minkowski space, can be reproduced from shockwaves arising from vacuum fluctuations. The size of the vacuum fluctuations is fixed by commutation relations in light-ray operators, of the same form postulated by ’t Hooft in the context of black hole horizons.

Kathryn M. Zurek

Modular fluctuations from shockwave geometries