We realize an explicit conformal mapping between the state and operator pictures in a class of (2 + 1)-dimensional non-Lorentzian field theories with SU(1, 2) × U(1) conformal symmetry. The state picture arises from null reducing four-dimensional relativistic conformal field theories on a three-sphere, yielding a non-Lorentzian geometry with the conformal Killing symmetry group SU(1, 2). This is complementary to the operator picture recently studied by Lambert et al. [1], where the geometry acquires an Ω-deformation. We then use the geometric mapping between the two pictures to derive a correspondence between the generators. This provides a concrete realization of the state-operator correspondence in non-Lorentzian conformal field theories.

Italy

Stefano Baiguera

Sweden

Denmark

Troels Harmark

China

Yang Lei

Ziqi Yan

Conformal mapping of non-Lorentzian geometries in SU(1, 2) Conformal Field Theory

Recent developments in static patch holography proposed that quantum gravity in de Sitter space admits a dual description in terms of a quantum mechanical theory living on a timelike surface near the cosmological horizon. In parallel, geometric observables associated with the Einstein-Rosen bridge of a black hole background were suggested to compute the computational complexity of the state dual to a gravitational theory. In this work, we pursue the study of the complexity=volume and complexity=action conjectures in a Schwarzschild-de Sitter geometry perturbed by the insertion of a shockwave at finite boundary times. This analysis extends previous studies that focused either on the complexity=volume 2.0 conjecture, or on the case of a shockwave inserted along the cosmological horizon. We show that the switchback effect, describing the delay in the evolution of complexity in reaction to a perturbation, is a universal feature of the complexity proposals in asymptotically de Sitter space. The geometric origin of this phenomenon is related to the causal connection between the static patches of de Sitter space when a positive pulse of null energy is inserted in the geometry.

Israel

Rotem Berman

The cosmological switchback effect. Part II

According to static patch holography, de Sitter space admits a unitary quantum description in terms of a dual theory living on the stretched horizon, that is a timelike surface close to the cosmological horizon. In this manuscript, we compute several holographic complexity conjectures in a periodic extension of the Schwarzschild-de Sitter black hole. We consider multiple configurations of the stretched horizons to which geometric objects are anchored. The holographic complexity proposals admit a hyperfast growth when the gravitational observables only lie in the cosmological patch, except for a class of complexity=anything observables that admit a linear growth. All the complexity conjectures present a linear increase when restricted to the black hole patch, similar to the AdS case. When both the black hole and the cosmological regions are probed, codimension-zero proposals are time-independent, while codimension-one proposals can have non-trivial evolution with linear increase at late times. As a byproduct of our analysis, we find that codimension-one spacelike surfaces are highly constrained in Schwarzschild-de Sitter space. Therefore, different locations of the stretched horizon give rise to different behaviours of the complexity conjectures.

Belgium

Sergio Aguilar-Gutierrez

Japan

Nicolò Zenoni

Holographic complexity of the extended Schwarzschild-de Sitter space

Non-relativistic quantum field theory is a framework that describes systems where the velocities are much smaller than the speed of light. A large class of those obey Schrödinger invariance, which is the equivalent of the conformal symmetry in the relativistic world. In this review, we pedagogically introduce the main theoretical tools used to study non-relativistic physics: null reduction and  $$c \rightarrow \infty $$  <math> <mrow> <mi>c</mi> <mo>→</mo> <mi>∞</mi> </mrow> </math>   limits, where c is the speed of light. We present a historical overview of non-relativistic wave equations, Jackiw–Pi vortices, the Aharonov–Bohm scattering, and the trace anomaly for a Schrödinger scalar. We then review modern developments, including fermions at unitarity, the quantum Hall effect, off-shell actions, and a systematic classification of the trace anomaly. The last part of this review is dedicated to current research topics. We define non-relativistic supersymmetry and a corresponding superspace to covariantly deal with quantum corrections. Finally, we define the Spin Matrix Theory limit of the AdS/CFT correspondence, which is a non-relativistic sector of the duality obtained via a decoupling limit, where a precise matching of the two sides can be achieved.

Aspects of non-relativistic quantum field theories

The volume behind the black hole horizon was suggested as a holographic dual for the quantum computational complexity of the boundary state in AdS/CFT. This identification is strongly motivated by the switchback effect: a characteristic delay of complexity growth in reaction to an inserted perturbation, modelled as a shockwave in the bulk. Recent proposals of de Sitter (dS) holography suggest that a dual theory could be living on a stretched horizon near the cosmological horizon. We study how the spacetime volume behind the cosmological horizon in Schwarzschild-dS space reacts to the insertion of shockwaves in an attempt to characterize the properties of this dual theory. We demonstrate that a switchback effect can be observed in dS space. That is, the growth of complexity is delayed in reaction to a perturbation. This delay is longer for earlier shocks and depends on a scrambling time which is logarithmic in the strength of the shockwave and proportional to the inverse temperature of the cosmological dS horizon. This behavior is very similar to what happens for AdS black holes, albeit the geometric origin of the effect is somewhat different.

Shira Chapman

Canada

Robert Myers

The cosmological switchback effect

Universal properties of two-dimensional conformal interfaces are encoded by the flux of energy transmitted and reflected during a scattering process. We develop an innovative method that allows us to use results for the energy transmission in thin-brane holographic models to find the energy transmission for general smooth domain-wall solutions of three-dimensional gravity. Our method is based on treating the continuous geometry as a discrete set of branes. As an application, we compute the transmission coefficient of a Janus interface in terms of its deformation parameter.

France

Constantin Bachas

Giuseppe Policastro

Tal Schwartzman

Energy Transport for Thick Holographic Branes

Spin Matrix theory describes near-BPS limits of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM theory, which enables us to probe finite N effects like D-branes and black hole physics. In previous works, we have developed the spherical reduction and spin chain methods to construct Spin Matrix theory for various limits. In this paper, by considering a supercharge   <math> <mi>Q</mi> </math>  $$ \mathcal{Q} $$  which is cubic in terms of the letters, we construct the Hamiltonian of the largest Spin Matrix theory of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM, called the PSU(1, 2|3) Spin Matrix theory, as   <math> <mi>H</mi> <mo>=</mo> <mfenced> <mi>Q</mi> <msup> <mi>Q</mi> <mo>†</mo> </msup> </mfenced> </math>  $$ H=\left\{\mathcal{Q},{\mathcal{Q}}^{\dagger}\right\} $$ . We show the resulting Hamiltonian is automatically positive definite and manifestly invariant under supersymmetry. The Hamiltonian is made of basic blocks which transform as supermultiplets. A novel feature of this Hamiltonian is its division into D-terms and F-terms that are separately invariant under PSU(1, 2|3) symmetry and positive definite. As all the other Spin Matrix theories arising from   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM can be acquired by turning off certain letters in the theory, we consider our work as revealing the “Panorama” of Spin Matrix theory.

The Panorama of Spin Matrix theory

In 2+1 dimensions, we propose a renormalizable non-linear sigma model action which describes the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supersymmetric generalization of Galilean Electrodynamics. We first start with the simplest model obtained by null reduction of the relativistic Abelian   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetric QED in 3+1 dimensions and study its renormalization properties directly in non-relativistic superspace. Despite the existence of a non-renormalization theorem induced by the causal structure of the non-relativistic dynamics, we find that the theory is non-renormalizable. Infinite dimensionless, supersymmetric and gauge-invariant terms, which combine into an analytic function, are generated at quantum level. Renormalizability is then restored by generalizing the theory to a non-linear sigma model where the usual minimal coupling between gauge and matter is complemented by infinitely many marginal couplings driven by a dimensionless gauge scalar and its fermionic superpartner. Superconformal invariance is preserved in correspondence of a non-trivial conformal manifold of fixed points where the theory is gauge-invariant and interacting.

Lorenzo Cederle

Silvia Penati

Supersymmetric Galilean Electrodynamics

Charged and symmetry-resolved Rényi entropies are entanglement measures quantifying the degree of entanglement within different charge sectors of a theory with a conserved global charge. We use holography to determine the dependence of charged Rényi entropies on small shape deformations away from a spherical or planar entangling surface in general dimensions. This dependence is completely characterized by a single coefficient appearing in the two point function of the displacement operator associated with the Rényi defect. We extract this coefficient using its relation to the one point function of the stress tensor in the presence of a deformed entangling surface. This is mapped to a holographic calculation in the background of a deformed charged black hole with hyperbolic horizon. We obtain numerical solutions for different values of the chemical potential and replica number n in various spacetime dimensions, as well as analytic expressions for small chemical potential near n = 1. When the Rényi defect becomes supersymmetric, we demonstrate a conjectured relation between the two point function of the displacement operator and the conformal weight of the twist operator.

Lorenzo Bianchi

United Kingdom

Damián Galante

Shape deformations of charged Rényi entropies from holography

We consider limits of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-Yang-Mills (SYM) theory that approach BPS bounds. These limits result in non-relativistic theories that describe the effective dynamics near the BPS bounds and upon quantization are known as Spin Matrix Theories. The near-BPS theories can be obtained by reducing   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM on a three-sphere and integrating out the fields that become non-dynamical in the limits. In the previous works [1–3] we have considered various SU(1,1) and SU(1,2) types of subsectors in this limit. In the current work, we will construct the remaining Spin Matrix Theories defined near the   <math> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </math>  $$ \frac{1}{8} $$ -BPS subsectors, which include the PSU(1,1|2) and SU(2|3) cases. We derive the Hamiltonians by applying the spherical reduction algorithm and show that they match with the spin chain result, coming from the loop corrections to the dilatation operator. In the PSU(1,1|2) case, we prove the positivity of the spectrum by constructing cubic supercharges using the enhanced PSU(1|1)2 symmetry and show that they close to the interacting Hamiltonian. We finally analyse the symmetry structure of the sectors in view of an interpretation of the interactions in terms of fundamental blocks.

Spin Matrix Theory in near   <math> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </math>  $$ \frac{1}{8} $$ -BPS corners of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-Yang-Mills