<p>We demonstrate a precise relation between the rate of complexity of quantum states excited by local operators in two-dimensional conformal field theories and the radial momentum of particles in three-dimensional anti–de Sitter spacetimes. Similar relations have been anticipated based on qualitative models for operator growth. Here, we make this correspondence sharp with two key ingredients: the precise definition of quantum complexity given by the spread complexity of states, and the match of its growth rate to the bulk momentum measured in the proper radial distance coordinate.</p>

Poland

Sweden

Japan

Pawel Caputa

China

Bowen Chen

United Kingdom

Ross W. McDonald

Joan Simón

Benjamin Strittmatter

Spread complexity rate as proper momentum

We revisit modular invariance in two-dimensional Carrollian conformal field theories from a geometric perspective. Focusing on the characters of the induced and highest-weight representations of the theory, we show that there are regions of parameter space where the vacuum character dominates in the dual channel. We use this property to zoom into different subsectors of the Carrollian theory. One of them is reminiscent of the Schwarzian sector of a relativistic CFT and has a flat space holographic interpretation as $\textit{O}$-plane orbifold. It exists only for the highest-weight representation. We prove that for all sectors with vacuum dominance in the dual channel, the specific heat is negative, concurrent with the holographic interpretation of the negative specific heat of asymptotically flat spacetimes with horizons.

Austria

Ankit Aggarwal

India

Arjun Bagchi

Belgium

Stephane Detournay

Daniel Grumiller

Max Riegler

Universal sectors of two-dimensional Carrollian CFTs

We derive the modular S and T matrices that generate the modular transformations on the characters of two dimensional Carrollian and warped CFT algebras. We verify that they satisfy some of their defining properties.

Modular matrices of 2d Carrollian and warped CFTs

We consider the geometric action formulation for 3d pure gravity with vanishing cosmological constant. We use fermionic localization to compute the exact torus partition function for a constant representative coadjoint orbit of $$ {\hat{\textrm{BMS}}}_3 $$ . This allows us to discuss its 1-loop exactness.

Boyang Yu

Recent progress in AdS/CFT has provided a good understanding of how the bulk spacetime is encoded in the entanglement structure of the boundary CFT. However, little is known about how spacetime emerges directly from the bulk quantum theory. We address this question in an effective 3d quantum theory of pure gravity, which describes the high temperature regime of a holographic CFT. This theory can be viewed as a q-deformation and dimensional uplift of JT gravity. Using this model, we show that the Bekenstein-Hawking entropy of a two-sided black hole equals the bulk entanglement entropy of gravitational edge modes. In the conventional Chern-Simons description, these black holes correspond to Wilson lines in representations of PSL(2, ℝ) ⨂ PSL(2, ℝ). We show that the correct calculation of gravitational entropy suggests we should interpret the bulk theory as an extended topological quantum field theory associated to the quantum semi-group   <math> <msubsup> <mi>SL</mi> <mi>q</mi> <mo>+</mo> </msubsup> <mfenced> <mn>2</mn> <mi>ℝ</mi> </mfenced> <mo>⨂</mo> <msubsup> <mi>SL</mi> <mi>q</mi> <mo>+</mo> </msubsup> <mfenced> <mn>2</mn> <mi>ℝ</mi> </mfenced> </math>  $$ {\textrm{SL}}_q^{+}\left(2,\mathbb{R}\right)\bigotimes {\textrm{SL}}_q^{+}\left(2,\mathbb{R}\right) $$ . Our calculation suggests an effective description of bulk microstates in terms of collective, anyonic degrees of freedom whose entanglement leads to the emergence of the bulk spacetime.

Thomas Mertens

United States

Gabriel Wong

A proposal for 3d quantum gravity and its bulk factorization

We revisit the spectrum of linear axisymmetric gravitational perturbations of the (near-)extreme Kerr black hole. Our aim is to characterise those perturbations that are responsible for the deviations away from extremality, and to contrast them with the linearized perturbations treated in the Newman-Penrose formalism. For the near horizon region of the (near-)extreme Kerr solution, i.e. the (near-)NHEK background, we provide a complete characterisation of axisymmetric modes. This involves an infinite tower of propagating modes together with the much subtler low-lying mode sectors that contain the deformations driving the black hole away from extremality. Our analysis includes their effects on the line element, their contributions to Iyer-Wald charges around the NHEK geometry, and how to reconstitute them as gravitational perturbations on Kerr. We present in detail how regularity conditions along the angular variables modify the dynamical properties of the low-lying sector, and in particular their role in the new developments of nearly-AdS2 holography.

Netherlands

Alejandra Castro

Victor Godet

Wei Song

Gravitational perturbations from NHEK to Kerr

We consider operator growth for generic large-N gauge theories at finite temperature. Our analysis is performed in terms of Fourier modes, which do not mix with other operators as time evolves, and whose correlation functions are determined by their two-point functions alone, at leading order in the large-N limit. The algebra of these modes allows for a simple analysis of the operators with whom the initial operator mixes over time, and guarantees the existence of boundary CFT operators closing the bulk Poincaré algebra, describing the experience of infalling observers. We discuss several existing approaches to operator growth, such as number operators, proper energies, the many-body recursion method, quantum circuit complexity, and comment on its relation to classical chaos in black hole dynamics. The analysis evades the bulk vs boundary dichotomy and shows that all such approaches are the same at both sides of the holographic duality, a statement that simply rests on the equality between operator evolution itself. In the way, we show all these approaches have a natural formulation in terms of the Gelfand-Naimark-Segal (GNS) construction, which maps operator evolution to a more conventional quantum state evolution, and provides an extension of the notion of operator growth to QFT.

Argentina

Javier Magán

On operator growth and emergent Poincaré symmetries

We construct both local states and scattering states with finite energy in global AdS by inserting properly regularized operators in the CFT of arbitrary conformal dimension (<math><mi>Δ</mi></math>) at an instant of time. We give the state fixed angular momentum <math><mo>(</mo><mo>ℓ</mo><mo>)</mo></math> by integrating the result over a sphere with appropriate spherical harmonics. The energy of the states and their angular resolution are computed with CFT operator methods and are independent of having an AdS interpretation. In the semiclassical limit of large conformal dimension operators, these correspond to single particles localized within sub-AdS scales with width <math><mn>1</mn><mo>/</mo><msqrt><mi>Δ</mi></msqrt></math> in AdS units, whose subsequent evolution is controlled by bulk geodesics. Our construction allows us to place a particle in any desired geodesic. For radial geodesics, we show that the amplitude to produce the desired state can be thought of as a regularized tunneling amplitude from the boundary to the radial turning point of the radial geodesic, while for other geodesics we argue that the insertion is at the outermost radial turning point of the corresponding geodesic.

David Berenstein

Localized states in global AdS space

We investigate the variation of holographic complexity for two nearby target states. Based on Nielsen’s geometric approach, we find the variation only depends on the end point of the optimal trajectory, a result which we designate the first law of complexity. As an example, we examine the <math><mrow><mtext>complexity</mtext><mo>=</mo><mtext>action</mtext></mrow></math> conjecture when the anti–de Sitter vacuum is perturbed by a scalar field excitation, which corresponds to a coherent state. Remarkably, the gravitational contributions completely cancel and the final variation reduces to a boundary term coming entirely from the scalar field action. Hence, the null boundary of Wheeler-DeWitt patch appears to act like the “end of the quantum circuit”.

Canada

Italy

Alice Bernamonti

Federico Galli

Juan Hernandez

Robert C. Myers

Shan-Ming Ruan

First Law of Holographic Complexity

The holographic relation between quantum correlations and connectivity of spacetime is explored for single R-charged AdS5 black holes and their half-BPS limits (superstars). In a two boundary set-up, the wormhole between both universes reduces to a designable and computable quantum mechanical correlation between the dual microscopic degrees of freedom in the BPS limit. This quantum connectivity is seen as a naked singularity by a single sided observer. In a single boundary set-up, as a small step towards the description of entangled black holes, we describe quantum teleportation between two labs in different locations of the transverse 5-sphere using entangled gravitons in a reference state that provides a classical channel between both labs.

Correlations vs connectivity in R-charge