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Abstract 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.

Abstract 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.

Abstract 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 $$c \rightarrow \infty $$ c → ∞ 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.

Nielsen's approach to quantum state complexity relates the minimal number of quantum gates required to prepare a state to the length of geodesics computed with a certain norm on the manifold of unitary transformations. For a bipartite system, we investigate binding complexity, which corresponds to norms in which gates acting on a single subsystem are free of cost. We reduce the problem to the study of geodesics on the manifold of Schmidt coefficients, equipped with an appropriate metric. Binding complexity is closely related to other quantities such as distributed computing and quantum communication complexity, and has a proposed holographic dual in the context of AdS/CFT. For finite dimensional systems with a Riemannian norm, we find an exact relation between binding complexity and the minimal Rényi entropy. We also find analytic results for the most commonly used non-Riemannian norm (the so-called $F_1$ norm) and provide lower bounds for the associated notion of state complexity ubiquitous in quantum computation and holography. We argue that our results are valid for a large class of penalty factors assigned to generators acting across the subsystems. We demonstrate that our results can be borrowed to study the usual complexity (not-binding) for a single spin for the case of the $F_1$ norm which was previously lacking from the literature. Finally, we derive bounds for multi-partite binding complexities and the related (continuous) circuit complexity where the circuit contains at most $2$-local interactions.

Abstract 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.

Abstract Spin Matrix theory describes near-BPS limits of N $$ \mathcal{N} $$ = 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 Q $$ \mathcal{Q} $$ which is cubic in terms of the letters, we construct the Hamiltonian of the largest Spin Matrix theory of N $$ \mathcal{N} $$ = 4 SYM, called the PSU(1, 2|3) Spin Matrix theory, as H = Q Q † $$ H=\left\{\mathcal{Q},{\mathcal{Q}}^{\dagger}\right\} $$ . 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 N $$ \mathcal{N} $$ = 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.

Abstract In 2+1 dimensions, we propose a renormalizable non-linear sigma model action which describes the N $$ \mathcal{N} $$ = 2 supersymmetric generalization of Galilean Electrodynamics. We first start with the simplest model obtained by null reduction of the relativistic Abelian N $$ \mathcal{N} $$ = 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.

Abstract 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.

Abstract The holographic complexity of formation for the AdS3 2-sided Randall-Sundrum model and the AdS3/BCFT2 models is logarithmically divergent according to the volume conjecture, while it is finite using the action proposal. One might be tempted to conclude that the UV divergences of the volume and action conjectures are always different for defects and boundaries in two-dimensional conformal field theories. We show that this is not the case. In fact, in Janus AdS3 we find that both volume and action proposals provide the same kind of logarithmic divergences.

Abstract We consider limits of N $$ \mathcal{N} $$ = 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 N $$ \mathcal{N} $$ = 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 1 8 $$ \frac{1}{8} $$ -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.

Abstract We investigate the complexity=volume proposal in the case of Janus AdS3 geometries, both at zero and finite temperature. The leading contribution coming from the Janus interface is a logarithmic divergence, whose coefficient is a function of the dilaton excursion. In the presence of the defect, complexity is no longer topological and becomes temperature-dependent. We also study the time evolution of the extremal volume for the time-dependent Janus BTZ black hole. This background is not dual to an interface but to a pair of entangled CFTs with different values of the couplings. At late times, when the equilibrium is restored, the couplings of the CFTs do not influence the complexity rate. On the contrary, the complexity rate for the out-of-equilibrium system is always smaller compared to the pure BTZ black hole background.

Abstract We consider limits of N $$ \mathcal{N} $$ = 4 super-Yang-Mills (SYM) theory that approach BPS bounds. These limits result in non-relativistic near-BPS 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 N $$ \mathcal{N} $$ = 4 SYM on a three-sphere and integrating out the fields that become non-dynamical in the limits. We perform the sphere reduction for the near-BPS limit with SU(1, 2|2) symmetry, which has several new features compared to the previously considered cases with SU(1) symmetry, including a dynamical gauge field. We discover a new structure in the classical limit of the interaction term. We show that the interaction term is built from certain blocks that comprise an irreducible representation of the SU(1, 2|2) algebra. Moreover, the full interaction term can be interpreted as a norm in the linear space of this representation, explaining its features including the positive definiteness. This means one can think of the interaction term as a distance squared from saturating the BPS bound. The SU(1, 1|1) near-BPS theory, and its subcases, is seen to inherit these features. These observations point to a way to solve the strong coupling dynamics of these near-BPS theories.

We construct two distinct actions for scalar fields that are invariant under local Carroll boosts and Weyl transformations. Conformal Carroll field theories were recently argued to be related to the celestial holography description of asymptotically flat spacetimes. However, only few explicit examples of such theories are known, and they lack local Carroll boost symmetry on a generic curved background. We derive two types of conformal Carroll scalar actions with boost symmetry on a curved background in any dimension and compute their energy-momentum tensors, which are traceless. In the first type of theories, time derivatives dominate and spatial derivatives are suppressed. In the second type, spatial derivatives dominate, and constraints are present to ensure local boost invariance. By integrating out these constraints, we show that the spatial conformal Carroll theories can be reduced to lower-dimensional Euclidean CFTs, which is reminiscent of the embedding space construction.

Abstract We analytically compute subsystem action complexity for a segment in the BTZ black hole background up to the finite term, and we find that it is equal to the sum of a linearly divergent term proportional to the size of the subregion and of a term proportional to the entanglement entropy. This elegant structure does not survive to more complicated geometries: in the case of a two segments subregion in AdS3, complexity has additional finite contributions. We give analytic results for the mutual action complexity of a two segments subregion.

Abstract We compute the ultraviolet divergences of holographic subregion complexity for the left and right factors of the thermofield double state in warped AdS3 black holes, both for the action and the volume conjectures. Besides the linear divergences, which are also present in the BTZ black hole, additional logarithmic divergences appear. For the action conjecture, these log divergences are not affected by the arbitrarity in the length scale associated with the counterterm needed to ensure reparameterization invariance. We find that the subregion action complexity obeys the superadditivity property for the thermofield double in warped AdS3, independently from the action counterterm coefficient. We study the temperature dependence of subregion complexity at constant angular momentum and we find that it is correlated with the sign of the specific heat.

Abstract We consider a Galilean N = 2 $$ \mathcal{N}=2 $$ supersymmetric theory with F-term couplings in 2 + 1 dimensions, obtained by null reduction of a relativistic Wess-Zumino model. We compute quantum corrections and we check that, as for the relativistic parent theory, the F-term does not receive quantum corrections. Even more, we find evidence that the causal structure of the non-relativistic dynamics together with particle number conservation constrain the theory to be one-loop exact.

Abstract The Complexity=Action conjecture is studied for black holes in Warped AdS3 space, realized as solutions of Einstein gravity plus matter. The time dependence of the action of the Wheeler-DeWitt patch is investigated, both for the non-rotating and the rotating case. The asymptotic growth rate is found to be equal to the Hawking temperature times the Bekenstein-Hawking entropy; this is in agreement with a previous calculation done using the Complexity=Volume conjecture.

Abstract We study the Complexity=Volume conjecture for Warped AdS3 black holes. We compute the spatial volume of the Einstein-Rosen bridge and we find that its growth rate is proportional to the Hawking temperature times the Bekenstein-Hawking entropy. This is consistent with expectations about computational complexity in the boundary theory.

Abstract We study the coupling of a 2 + 1 dimensional non-relativistic spin 1/2 fermion to a curved Newton-Cartan geometry, using null reduction from an extra-dimensional relativistic Dirac action in curved spacetime. We analyze Weyl invariance in detail: we show that at the classical level it is preserved in an arbitrary curved background, whereas at the quantum level it is broken by anomalies. We compute the trace anomaly using the Heat Kernel method and we show that the anomaly coefficients a, c are proportional to the relativistic ones for a Dirac fermion in 3 + 1 dimensions. As for the previously studied scalar case, these coefficents are proportional to 1/m, where m is the non-relativistic mass of the particle.