A key issue in both the field of quantum chaos and quantum gravity is an effective description of chaotic conformal field theories (CFTs), that is CFTs that have a quantum ergodic limit. We develop a framework incorporating the constraints of conformal symmetry and locality, allowing the definition of ensembles of ‘CFT data’. These ensembles take on the same role as the ensembles of random Hamiltonians in more conventional quantum ergodic phases of many-body quantum systems. To describe individual members of the ensembles, we introduce the notion of approximate CFT, defined as a collection of ‘CFT data’ satisfying the usual CFT constraints approximately, i.e. up to small deviations. We show that they generically exist by providing concrete examples. Ensembles of approximate CFTs are very natural in holography, as every member of the ensemble is indistinguishable from a true CFT for low-energy probes that only have access to information from semi-classical gravity. To specify these ensembles, we impose successively higher moments of the CFT constraints. Lastly, we propose a theory of pure gravity in AdS3 as a random matrix/tensor model implementing approximate CFT constraints. This tensor model is the maximum ignorance ensemble compatible with conformal symmetry, crossing invariance, and a primary gap to the black-hole threshold. The resulting theory is a random matrix/tensor model governed by the Virasoro 6j-symbol.

Italy

Switzerland

Alexandre Belin

Netherlands

Jan Boer

United States

Daniel Jafferis

CERN

Pranjal Nayak

Julian Sonner

Approximate CFTs and random tensor models

The spectral form factor is a powerful probe of quantum chaos that diagnoses the statistics of energy levels, but is blind to other features of a theory such as matrix elements of operators or OPE coefficients in conformal field theories. In this paper, we introduce generalized spectral form factors: new probes of quantum chaos sensitive to the dynamical data of a theory. These quantities can be studied using random matrix theory and an effective theory of quantum chaos. We focus our attention on a particular combination of heavy-heavy-heavy OPE coefficients that generalizes the genus-2 partition function of two-dimensional CFTs, for which we define a form factor. Assuming that random matrix theory applies to chaotic CFTs, we probe heavy-heavy-heavy OPE coefficients and find statistical correlations that agree with the OPE Randomness Hypothesis: these coefficients have a random tensor component. The EFT of quantum chaos predicts that the genus-2 form factor displays a ramp and a plateau. Our results suggest that this is a common property of generalized spectral form factors.

Generalized spectral form factors and the statistics of heavy operators

Correlation functions of the higher-spin current operators in large Chern-Simons theories are important to understand approximate higher-spin symmetries in these theories. Moreover, they also provide stronger checks for conjectured dualities in these theories. In this paper, we compute the two, three and four-point functions of the operators in the spin zero multiplet of $$ \mathcal{N} $$ = 2 Supersymmetric vector matter Chern-Simons theory at large to all orders of ’t Hooft coupling. While the two- and three-point functions are computed by solving the Schwinger-Dyson equation, this method becomes intractable for the computation of the four-point functions. Thereby, we use bootstrap method to evaluate four-point function of scalar operator $$ {J}_0^f=\overline{\psi}\psi $$ and $$ {J}_0^b=\overline{\phi}\phi . $$ Interestingly, because $$ \left\langle {J}_0^f{J}_0^f{J}_0^b\right\rangle $$ is a contact term, the four point function of $$ {J}_0^f $$ operator resembles the corresponding correlation function in the free theory, up to overall coupling constant dependent factors and up to some ‘bulk AdS’ contact terms. On the other hand the $$ {J}_0^b $$ four-point function receives an additional contribution compared to the free theory expression due to the $$ {J}_0^f $$ exchange. We find that the double discontinuity of this single trace operator $$ {J}_0^f $$ vanishes and hence it only contributes to AdS-contact term.

Israel

Karthik Inbasekar

India

Sachin Jain

Vinay Malvimat

Abhishek Mehta

South Africa

Tarun Sharma

Correlation functions in

In this paper we provide a universal description of the behavior of the basic operators of the Schwarzian theory in pure states. When the pure states are energy eigenstates, expectation values of non-extensive operators are thermal. On the other hand, in coherent pure states, these same operators can exhibit ergodic or non-ergodic behavior, which is characterized by elliptic, parabolic or hyperbolic monodromy of an auxiliary equation; or equivalently, which coadjoint Virasoro orbit the state lies on. These results allow us to establish an extended version of the eigenstate thermalization hypothesis (ETH) in theories with a Schwarzian sector. We also elucidate the role of FZZT-type boundary conditions in the Schwarzian theory, shedding light on the physics of microstates associated with ZZ branes and FZZT branes in low dimensional holography.

Manuel Vielma

Extended eigenstate thermalization and the role of FZZT branes in the Schwarzian theory

The Sachdev-Ye-Kitaev (SYK) model provides an uncommon example of a chaotic theory that can be analysed analytically. In the deep infrared limit, the original model has an emergent conformal (reparametrisation) symmetry that is broken both spontaneously and explicitly. The explicit breaking of this symmetry comes about due to pseudo-Nambu-Goldstone modes that are not exact zero-modes of the model. In this paper, we study a version of the model which preserves the reparametrisation symmetry at all length scales. We study the heavy-light correlation functions of the operators in the conformal spectrum of the theory. The three point functions of such operators allow us to demonstrate that matrix elements of primaries   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$   n  of the CFT1 take the form postulated by the Eigenstate Thermalisation Hypothesis. We also discuss the implications of these results for the states in AdS2 gravity dual.

Eigenstate thermalisation in the conformal Sachdev-Ye-Kitaev model: an analytic approach

Dual conformal symmetry and Yangian symmetry are symmetries of amplitudes that have aided the study of scattering amplitudes in highly supersymmetric theories like   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM and ABJM. However, in general such symmetries are absent from the theories with lesser or no supersymmetry. In this paper, we show that the tree level 2 → 2 scattering amplitude in the 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 Chern-Simons theory coupled to a fundamental chiral multiplet is dual superconformal invariant. In the ’t Hooft large N limit, the 2 → 2 scattering amplitude in this theory has been shown to be tree-level exact in non-anyonic channels, while having only an overall multiplicative coupling dependent renormalisation in the anyonic channel. Therefore, the dual superconformal symmetry that we demonstrate in this paper is all loop exact. This is unlike the previously studied highly supersymmetric theories where dual superconformal symmetry is anomalous at loop levels.

Sucheta Majumdar

Turmoli Neogi

Ritam Sinha

V Umesh

Dual superconformal symmetry of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 Chern-Simons theory with fundamental matter at large N

We show that Britto-Cachazo-Feng-Witten (BCFW) recursion relations can be used to compute all tree level scattering amplitudes in terms of <math><mn>2</mn><mo>→</mo><mn>2</mn></math> scattering amplitude in <math><mrow><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></math> <math><mi>N</mi><mo>=</mo><mn>2</mn></math> Chern-Simons (CS) theory coupled to matter in the fundamental representation. As a by-product, we also obtain a recursion relation for the CS theory coupled to regular fermions, even though in this case standard BCFW deformations do not have a good asymptotic behavior. Moreover, at large <math><mi>N</mi></math>, <math><mn>2</mn><mo>→</mo><mn>2</mn></math> scattering can be computed exactly to all orders in ’t Hooft coupling as was done in earlier works by some of the authors. In particular, for <math><mi>N</mi><mo>=</mo><mn>2</mn></math> theory, it was shown that <math><mn>2</mn><mo>→</mo><mn>2</mn></math> scattering is tree level exact to all orders except in the anyonic channel [K. Inbasekar et al., J. High Energy Phys. 10 (2015) 176], where it gets renormalized by a simple function of ’t Hooft coupling. This suggests that it may be possible to compute the all loop exact result for arbitrary higher-point scattering amplitudes at large <math><mi>N</mi></math>.

V. Umesh

All Tree Level Scattering Amplitudes in Chern-Simons Theories with Fundamental Matter

We analyse the dynamics of near-extremal Reissner-Nordström black holes in asymptotically four-dimensional Anti de Sitter space (AdS4). We work in the spherically symmetric approximation and study the thermodynamics and the response to a probe scalar field. We find that the behaviour of the system, at low energies and to leading order in our approximations, is well described by the Jackiw-Teitelboim (JT) model of gravity. In fact, this behaviour can be understood from symmetry considerations and arises due to the breaking of time reparametrisation invariance. The JT model has been analysed in considerable detail recently and related to the behaviour of the SYK model. Our results indicate that features in these models which arise from symmetry considerations alone are more general and present quite universally in near-extremal black holes.

Canada

Ashish Shukla

Ronak Soni

Sandip Trivedi

V. Vishal

On the dynamics of near-extremal black holes

The Nambu-Goldstone (NG) bosons of the SYK model are described by a coset space Diff/SL(2, ℝ), where Diff, or Virasoro group, is the group of diffeomorphisms of the time coordinate valued on the real line or a circle. It is known that the coadjoint orbit action of Diff naturally turns out to be the two-dimensional quantum gravity action of Polyakov without cosmological constant, in a certain gauge, in an asymptotically flat spacetime. Motivated by this observation, we explore Polyakov action with cosmological constant and boundary terms, and study the possibility of such a two-dimensional quantum gravity model being the AdS dual to the low energy (NG) sector of the SYK model. We find strong evidences for this duality: (a) the bulk action admits an exact family of asymptotically AdS2 spacetimes, parameterized by Diff/SL(2, ℝ), in addition to a fixed conformal factor of a simple functional form; (b) the bulk path integral reduces to a path integral over Diff/SL(2, ℝ) with a Schwarzian action; (c) the low temperature free energy qualitatively agrees with that of the SYK model. We show, up to quadratic order, how to couple an infinite series of bulk scalars to the Polyakov model and show that it reproduces the coupling of the higher modes of the SYK model with the NG bosons.

Gautam Mandal

Spenta Wadia

Coadjoint orbit action of Virasoro group and two-dimensional quantum gravity dual to SYK/tensor models

We revisit AdS/CFT at a finite radial cut-off, specifically in the context of double trace perturbations, O n = O x ∂ 2 n O x $$ {\mathbb{O}}_n=\mathcal{O}(x){\left({\partial}^2\right)}^n\mathcal{O}(x) $$ , with arbitrary powers n . As well-known, the standard GKPW prescription, applied to a finite radial cut-off, leads to contact terms in correlators. de Haro et al. [1] introduced bulk counterterms to remove these. This prescription, however, yields additional terms in the correlator corresponding to spurious double trace deformations. Further, if we view the GKPW prescription coupled with the prescription in [1], in terms of a boundary wavefunction, we find that it is incompatible with radial Schrödinger evolution (in the spirit of holographic Wilsonian RG). We consider a more general wavefunction satisfying the Schrödinger equation, and find that generically such wavefunctions generate both (a) double trace deformations and (b) contact terms. However, we find that there exist special choices of these wavefunctions, amounting to a new AdS/CFT prescription at a finite cut-off, so that both (a) and (b) are removed and we obtain a pure power law behaviour for the correlator. We compare these special wavefunctions with a specific RG scheme in field theory. We give a geometric interpretation of these wavefunctions; these correspond to some specific smearing of boundary points in the Witten diagrams. We present a comprehensive calculation of exact double-trace beta-functions for all couplings O n $$ {\mathbb{O}}_n $$ and match with a holographic computation using the method described above. The matching works with a mapping between the field theory and bulk couplings; such a map is highly constrained because the beta-functions are quadratic and exact on both sides. Our discussions include a generalization of the standard double-trace Wilson-Fisher flow to the space of the infinite number of couplings.

Revisiting AdS/CFT at a finite radial cut-off