We present orthogonality relations for quasinormal modes of a wide class of asymptotically AdS black holes. The definition is obtained from a standard product, modified by a CPT operator and placed on a complex radial contour which avoids branch points of the modes. They are inspired by existing constructions for de Sitter and Kerr spacetimes. The CPT operator is needed to map right eigenfunctions of the Hamiltonian into left eigenfunctions. The radial contour connects two copies of the dual QFT on a thermal Schwinger-Keldysh contour, making contact with real-time holography and the double cone wormhole.

United Kingdom

Paolo Arnaudo

Javier Carballo

Benjamin Withers

QNM orthogonality relations for AdS black holes

We study the non-modal stability of black hole spacetimes under linear perturbations. We show that large-amplitude growth can occur at finite time, despite asymptotic decay of linear perturbations. In the example presented, the physical mechanism is a transient form of superradiance, and is qualitatively similar to the transition to turbulence in Navier-Stokes shear flows. As part of the construction we provide a theorem for the positivity of QNM energies, and introduce a truncated-Hamiltonian approach to black hole pseudospectra which does not suffer from convergence issues.

Ireland

Christiana Pantelidou

Non-modal effects in black hole perturbation theory: transient superradiance

<p>We obtain exact expressions for correlation functions of charged scalar operators at finite density and low temperature in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>CFT</mi></mrow><mn>4</mn></msub></math> dual to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>RN</mi><mtext>−</mtext><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow></mrow></math> black-brane. We use recent developments in the Heun connection problem in black hole perturbation theory arising from Liouville conformal field theory (CFT) and the Alday-Gaiotto-Tachikawa correspondence. The connection problem is solved perturbatively in an instanton counting parameter, which is controlled in a double-scaling limit where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ω</mi><mo>,</mo><mi>T</mi><mo stretchy="false">→</mo><mn>0</mn></math> holding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ω</mi><mo>/</mo><mi>T</mi></math> fixed. This provides analytic control over the emergence of the zero temperature branch cut as a confluent limit of the Heun equation. From the Green’s function we extract analytic results for the critical temperature of the holographic superconductor, as well as dispersion relations for both gapped and gapless low temperature quasinormal modes. We demonstrate precise agreement with numerics.</p>

Exact low-temperature Green’s functions in AdS/CFT: From the Heun equation to the confluent Heun equation

Quasinormal modes of spacetimes with event horizons are typically governed by a non-normal operator. This gives rise to spectral instabilities, a topic of recent interest in the black hole pseudospectrum programme. In this work we show that non-normality leads to the existence of arbitrarily long-lived sums of short-lived quasinormal modes, corresponding to localising packets of energy near the future horizon. There exist sums of M quasinormal modes whose lifetimes scale as log M. This transient behaviour results from large cancellations between non-orthogonal quasinormal modes. We provide simple closed-form examples for a massive scalar field in the static patch of dS d+1 and the BTZ black hole. We also provide numerical examples for scalar perturbations of Schwarzschild-AdS d+1, and gravitational perturbations of Schwarzschild in asymptotically flat spacetime, using hyperboloidal foliations. The existence of these perturbations is linked to certain properties of black hole pseudospectra. We comment on implications for thermalisation times in holographic plasmas.

Transient dynamics of quasinormal mode sums

We present a new formalism to solve the kinematical constraints due to Weyl invariance for CFTs in curved backgrounds and/or non-trivial states, and we apply it to thermal CFTs and to CFTs on squashed spheres. The ambient space formalism is based on constructing a class of geometric objects that are Weyl covariant and identifying them as natural building blocks of correlation functions. We construct (scalar) n-point functions and we illustrate the formalism with a detailed computation of 2-point functions. We compare our results for thermal 2-point functions with results that follow from thermal OPEs and holographic computations, finding exact agreement. In our holographic computation we also obtain the OPE coefficient of the leading double-twist contribution, and we discuss how the double-twist coefficients may be computed from the multi-energy-momentum contributions, given knowledge of the analytic structure of the correlator. The 2-point function for the CFT on squashed spheres is a new result. We also discuss the relation of our work to flat holography.

Enrico Parisini

Kostas Skenderis

The ambient space formalism

We prepare an excited finite temperature state in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM by means of a Euclidean path integral with a relevant deformation. The deformation explicitly breaks imaginary-time translations along the thermal circle whilst preserving its periodicity. We then study how the state relaxes to thermal equilibrium in real time. Computations are performed using real-time AdS/CFT, by constructing novel mixed-signature black holes in numerical relativity corresponding to Schwinger-Keldysh boundary conditions. These correspond to deformed cigar geometries in the Euclidean, glued to a pair of dynamical spacetimes in the Lorentzian.

Black hole excited states from broken translations in Euclidean time

We use causality to derive a number of simple and universal constraints on dispersion relations, which describe the location of singularities of retarded two-point functions in relativistic quantum field theories. We prove that all causal dissipative dispersion relations have a finite radius of convergence in cases where stochastic fluctuations are negligible. We then give two-sided bounds on all transport coefficients in units of this radius, including an upper bound on diffusivity.

Belgium

Michal P. Heller

Spain

Alexandre Serantes

Poland

Michał Spaliński

Rigorous Bounds on Transport from Causality

We compute fully retarded scalar three-point functions of holographic CFTs at finite temperature using real-time holography. They describe the nonlinear response of a holographic medium under scalar forcing, and display single and higher-order poles associated to resonant QNM excitations. This involves computing the bulk-to-bulk propagator on a piecewise mixed-signature spacetime, the dual of the Schwinger-Keldysh contour. We show this construction is equivalent to imposing ingoing boundary conditions on a single copy of a black hole spacetime, similar to the case of the two-point function. We also compute retarded scalar correlators with stress-tensor insertions in general CFTs by solving Ward identities on the Schwinger-Keldysh contour.

Thermal three-point functions from holographic Schwinger-Keldysh contours

We present a generalization of the embedding space formalism to conformal field theories (CFTs) on nontrivial states and curved backgrounds, based on the ambient metric of Fefferman and Graham. The ambient metric is a Lorentzian Ricci-flat metric in <math><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></math> dimensions and replaces the Minkowski metric of the embedding space. It is canonically associated with a <math><mi>d</mi></math>-dimensional conformal manifold, which is the physical spacetime where the <math><mrow><msub><mrow><mi>CFT</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></math> lives. We propose a construction of <math><mrow><msub><mrow><mi>CFT</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></math> <math><mi>n</mi></math>-point functions in nontrivial states and on curved backgrounds using appropriate geometric invariants of the ambient space as building blocks. This captures the contributions of nonvanishing one-point functions of multi-stress-energy tensors, at least in holographic CFTs. We apply the formalism to two-point functions of thermal CFT, finding exact agreement with a holographic computation and expectations based on thermal operator product expansions (OPEs), and to CFTs on squashed spheres where no prior results are known and existing methods are difficult to apply, demonstrating the utility of the method.

Embedding formalism for CFTs in general states on curved backgrounds

The gradient expansion is the fundamental organizing principle underlying relativistic hydrodynamics, yet understanding its convergence properties for general nonlinear flows has posed a major challenge. We introduce a simple method to address this question in a class of fluids modeled by Israel-Stewart–type relaxation equations. We apply it to (<math><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional flows and provide numerical evidence for factorially divergent gradient expansions. This generalizes results previously only obtained for (<math><mn>0</mn><mo>+</mo><mn>1</mn></math>)-dimensional comoving flows, notably Bjorken flow. We also demonstrate that the only known nontrivial case of a convergent hydrodynamic gradient expansion at the nonlinear level relies on Bjorken flow symmetries and becomes factorially divergent as soon as these are relaxed. Finally, we show that factorial divergence can be removed using a momentum space cutoff, which generalizes a result obtained earlier in the context of linear response.