Boundary Gravitons Research Articles

Floquet SYK wormholes

We study the non-equilibrium dynamics of two coupled SYK models, conjectured to be holographically dual to an eternal traversable wormhole in AdS2. We consider different periodic drivings of the parameters of the system. We analyze the energy flows in the wormhole and black hole phases of the model as a function of the driving frequency. Our numerical results show a series of resonant frequencies in which the energy absorption and heating are enhanced significantly and the transmission coefficients drop, signalling a closure of the wormhole. These frequencies correspond to part of the conformal tower of states and to the boundary graviton of the dual gravitational theory. Furthermore, we provide evidence supporting the existence of a hot wormhole phase between the black hole and wormhole phases. When driving the strength of the separate SYK terms we find that the transmission can be enhanced by suitably tuning the driving.

Eternal traversable wormholes in three dimensions

We consider three-dimensional gravity with negative cosmological constant coupled to a large number of light matter fields dual to relevant operators. By imposing suitable boundary conditions on the matter fields we find eternal traversable wormhole deformations of the BTZ black hole, leading to a three-dimensional analogue of the AdS2 eternal traversable wormhole found by Maldacena and Qi. We further identify the field theory of boundary gravitons in this setting, which we then use to compute the spectrum of gravitational fluctuations.

Refining the cutoff 3d gravity/ Toverline{T} correspondence

Pure gravity in AdS3 is a theory of boundary excitations, most simply expressed as a constrained free scalar with an improved stress tensor that is needed to match the Brown-Henneaux central charge. Excising a finite part of AdS gives rise to a static gauge Nambu-Goto action for the boundary graviton. We show that this is the Toverline{T} deformation of the infinite volume theory, as the effect of the improvement term on the deformed action can be absorbed into a field redefinition. The classical gravitational stress tensor is reproduced order by order by the Toverline{T} trace equation. We calculate the finite volume energy spectrum in static gauge and find that the trace equation imposes sufficient constraints on the ordering ambiguities to guarantee agreement with the light-cone gauge prediction. The correlation functions, however, are not completely fixed by the trace equation. We show how both the gravitational action and the Toverline{T} deformation allow for finite improvement terms, and we match these to the undetermined total derivative terms in Zamolodchikov’s point splitting definition of the Toverline{T} operator.

Field Theory of Interacting Boundary Gravitons

Pure three-dimensional gravity is a renormalizable theory with two free parameters labelled by GG and \LambdaΛ. As a consequence, correlation functions of the boundary stress tensor in AdS_33 are uniquely fixed in terms of one dimensionless parameter, which is the central charge of the Virasoro algebra. The same argument implies that AdS_33 gravity at a finite radial cutoff is a renormalizable theory, but now with one additional parameter corresponding to the cutoff location. This theory is conjecturally dual to a T\overline{T}TT¯-deformed CFT, assuming that such theories actually exist. To elucidate this, we study the quantum theory of boundary gravitons living on a cutoff planar boundary and the associated correlation functions of the boundary stress tensor. We compute stress tensor correlation functions to two-loop order (GG being the loop counting parameter), extending existing tree level results. This is made feasible by the fact that the boundary graviton action simplifies greatly upon making a judicious field redefinition, turning into the Nambu-Goto action. After imposing Lorentz invariance, the correlators at this order are found to be unambiguous up to a single undetermined renormalization parameter.

Charges in the extended BMS algebra: Definitions and applications

This is a review of selected topics from recent work on symmetry charges in asymptotically flat spacetime done by the author in collaboration with U. Kol and R. Javadinezhad. First we reinterpret the reality constraint on the boundary graviton as the gauge fixing of a new local symmetry, called dual supertranslations. This symmetry extends the BMS group and bears many similarities to the dual (magnetic) gauge symmetry of electrodynamics. We use this new gauge symmetry to propose a new description of the TAUB-NUT space that does not contain closed time-like curves. Next we summarize progress towards the definition of Lorentz and super-Lorentz charges that commute with supertranslations and with the soft graviton mode.

Holographic duality for Ising CFT with boundary

We extend the holographic duality between 3d pure gravity and 2d Ising CFT proposed in ref. [1] to CFTs with boundaries. Besides the usual asymptotic boundary, the dual bulk spacetime now has a real cutoff, on which live branes with finite tension, giving Neumann boundary condition on the metric tensor. The strongly coupled bulk theory requires that we dress the well-known semiclassical AdS/BCFT answer with boundary gravitons, turning the partition function into the form of Virasoro characters. Using this duality, we relate the brane tensions to the modular S-matrix elements of the dual BCFT and derive the transformation between gravitational solutions with different brane tensions under modular S action.

Excitation spectra of quantum matter without quasiparticles. II. Random t−J models

We present numerical solutions of the spectral functions of $t$-$J$ models with random and all-to-all exchange and global SU($M$) spin rotation symmetry. The solutions are obtained from the saddle-point equations of the large volume limit, followed by the large $M$ limit. These saddle point equations involve Green's functions for fractionalized spinons and holons carrying emergent U(1) gauge charges, obeying relations similar to those of the Sachdev-Ye-Kitaev (SYK) models. The low frequency spectral functions are compared with an analytic analysis of the operator scaling dimensions, with good agreement. We also compute the low frequency and temperature behavior of gauge-invariant observables: the electron Green's function, the local spin susceptibility and the optical conductivity; along with the temperature dependence of the d.c. resistivity. The time reparameterization soft mode (equivalent to the boundary graviton in holographically dual models of two-dimensional quantum gravity) makes important contributions to all observables, and provides a linear-in-temperature contribution to the d.c. resistivity.

Excitation spectra of quantum matter without quasiparticles. I. Sachdev-Ye-Kitaev models

We study the low frequency spectra of complex Sachdev-Ye-Kitaev (SYK) models at general densities. The analysis applies also to SU($M$) magnets with random exchange at large $M$. The spectral densities are computed by numerical analysis of the saddle point equations on the real frequency ($\omega$) axis at zero temperature ($T$). The asymptotic low $\omega$ behaviors are found to be in excellent agreement with the scaling dimensions of irrelevant operators which perturb the conformally invariant critical states. Of possible experimental interest is our computation of the universal spin spectral weight of the SU($M$) magnets at low $\omega$ and $T$: this includes a contribution from the time reparameterization mode, which is the boundary graviton of the holographic dual. This analysis is extended to a random $t$-$J$ model in a companion paper.

Free partition functions and an averaged holographic duality

We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

Liouville quantum gravity

We define a three-dimensional quantum theory of gravity as the holographic dual of the Liouville conformal field theory. The theory is consistent and unitary by definition. The corresponding theory of gravity with negative cosmological constant has peculiar properties. The quantum theory has no normalizable AdS3 vacuum. The model contains primary black holes with zero spin. All states can be interpreted as BTZ geometries dressed with boundary gravitons. There is a unique universal interaction between these states consistent with unitarity and the conformal symmetry of the model. This theory of gravity, though conceptually isolated from other models of quantum gravity, is worth scrutinizing.

Nearly AdS2 holography in quantum CGHS model

In light of recent developments in nearly AdS2 holography, we revisit the semi­classical version of two-dimensional dilaton gravity proposed by Callan, Giddings, Harvey, and Strominger (CGHS) [1] in the early 90’s. In distinction to the classical model, the quantum-corrected CGHS model has an AdS2 vacuum with a constant dilaton. By turning on a non-normalizable mode of the Liouville field, i.e. the conformal mode of the 2d gravity, the explicit breaking of the scale invariance renders the AdS2 vacuum nearly AdS2. As a consequence, there emerges an effective one-dimensional Schwarzian-type theory of pseudo Nambu-Goldstone mode-the boundary graviton-on the boundary of the nearly AdS2 space. We go beyond the linear order perturbation in non-normalizable fluctuations of the Liouville field and work up to the second order. As a main result of our analysis, we clarify the role of the boundary graviton in the holographic framework and show the Virasoro/Schwarzian correspondence, namely that the 2d bulk Virasoro constraints are equivalent to the graviton equation of motion of the 1d boundary theory, at least, on the SL(2, R) invariant vacuum.

A theory of reparameterizations for AdS3 gravity

We rewrite the Chern-Simons description of pure gravity on global AdS3 and on Euclidean BTZ black holes as a quantum field theory on the AdS boundary. The resulting theory is (two copies of) the path integral quantization of a certain coadjoint orbit of the Virasoro group, and it should be regarded as the quantum field theory of the boundary gravitons. This theory respects all of the conformal field theory axioms except one: it is not modular invariant. The coupling constant is 1/c with c the central charge, and perturbation theory in 1/c encodes loop contributions in the gravity dual. The QFT is a theory of reparametrizations analogous to the Schwarzian description of nearly AdS2 gravity, and has several features including: (i) it is ultraviolet-complete; (ii) the torus partition function is the vacuum Virasoro character, which is one-loop exact by a localization argument; (iii) it reduces to the Schwarzian theory upon compactification; (iv) it provides a powerful new tool for computing Virasoro blocks at large c via a diagrammatic expansion. We use the theory to compute several observables to one-loop order in the bulk, including the “heavy-light” limit of the identity block. We also work out some generalizations of this theory, including the boundary theory which describes fluctuations around two-sided eternal black holes.

Holographic description of boundary gravitons in (3+1) dimensions

Gravity is uniquely situated in between classical topological field theories and standard local field theories. This can be seen in the quasi-local nature of gravitational observables, but is nowhere more apparent than in gravity’s holographic formulation. Holography holds promise for simplifying computations in quantum gravity. While holographic descriptions of three-dimensional spacetimes and of spacetimes with a negative cosmological constant are well-developed, a complete boundary description of zero curvature, four-dimensional spacetime is not currently available. Building on previous work in three-dimensions, we provide a new route to four-dimensional holography and its boundary gravitons. Using Regge calculus linearized around a flat Euclidean background with the topology of a solid hyper-torus, we obtain the effective action for a dual boundary theory, which describes the dynamics of the boundary gravitons. Remarkably, in the continuum limit and at large radii this boundary theory is local and closely analogous to the corresponding result in three-dimensions. The boundary effective action has a degenerate kinetic term that leads to singularities in the one-loop partition function that are independent of the discretization. These results establish a rich boundary dynamics for four-dimensional flat holography.

Thomas precession for dressed particles

We consider a particle dressed with boundary gravitons in three-dimensional Minkowski space. The existence of BMS transformations implies that the particle’s wavefunction picks up a Berry phase when subjected to changes of reference frames that trace a closed path in the asymptotic symmetry group. We evaluate this phase and show that, for BMS superrotations, it provides a gravitational generalization of Thomas precession. In principle, such phases are observable signatures of asymptotic symmetries.

Berry phases on Virasoro orbits

We point out that unitary representations of the Virasoro algebra contain Berry phases obtained by acting on a primary state with conformal transformations that trace a closed path on a Virasoro coadjoint orbit. These phases can be computed exactly thanks to the Maurer-Cartan form on the Virasoro group, and they persist after combining left- and right-moving sectors. Thinking of Virasoro representations as particles in AdS3 dressed with boundary gravitons, the Berry phases associated with Brown-Henneaux diffeomorphisms provide a gravitational extension of Thomas precession.

Boundary conditions for General Relativity on AdS3 and the KdV hierarchy

It is shown that General Relativity with negative cosmological constant in three spacetime dimensions admits a new family of boundary conditions being labeled by a nonnegative integer $k$. Gravitational excitations are then described by "boundary gravitons" that fulfill the equations of the $k$-th element of the KdV hierarchy. In particular, $k=0$ corresponds to the Brown-Henneaux boundary conditions so that excitations are described by chiral movers. In the case of $k=1$, the boundary gravitons fulfill the KdV equation and the asymptotic symmetry algebra turns out to be infinite-dimensional, abelian and devoid of central extensions. The latter feature also holds for the remaining cases that describe the hierarchy ($k>1$). Our boundary conditions then provide a gravitational dual of two noninteracting left and right KdV movers, and hence, boundary gravitons possess anisotropic Lifshitz scaling with dynamical exponent $z=2k+1$. Remarkably, despite spacetimes solving the field equations are locally AdS, they possess anisotropic scaling being induced by the choice of boundary conditions. As an application, the entropy of a rotating BTZ black hole is precisely recovered from a suitable generalization of the Cardy formula that is compatible with the anisotropic scaling of the chiral KdV movers at the boundary, in which the energy of AdS spacetime with our boundary conditions depends on $z$ and plays the role of the central charge. The extension of our boundary conditions to the case of higher spin gravity and its link with different classes of integrable systems is also briefly addressed.

Symplectic and Killing symmetries of AdS3 gravity: holographic vs boundary gravitons

The set of solutions to the AdS$_3$ Einstein gravity with Brown-Henneaux boundary conditions is known to be a family of metrics labeled by two arbitrary periodic functions, respectively left and right-moving. It turns out that there exists an appropriate presymplectic form which vanishes on-shell. This promotes this set of metrics to a phase space in which the Brown-Henneaux asymptotic symmetries become symplectic symmetries in the bulk of spacetime. Moreover, any element in the phase space admits two global Killing vectors. We show that the conserved charges associated with these Killing vectors commute with the Virasoro symplectic symmetry algebra, extending the Virasoro symmetry algebra with two $U(1)$ generators. We discuss that any element in the phase space falls into the coadjoint orbits of the Virasoro algebras and that each orbit is labeled by the $U(1)$ Killing charges. Upon setting the right-moving function to zero and restricting the choice of orbits, one can take a near-horizon decoupling limit which preserves a chiral half of the symplectic symmetries. Here we show two distinct but equivalent ways in which the chiral Virasoro symplectic symmetries in the near-horizon geometry can be obtained as a limit of the bulk symplectic symmetries.

Poisson structure of the boundary gravitons in 3D gravity with negative Λ

We use the Hamiltonian formalism to study the asymptotic structure of three-dimensional gravity with a negative cosmological constant. We start by defining very general fall-off conditions for the canonical variables and study the implied Poisson structure of the boundary gravitons. From the allowed differentiable gauge transformations, we can extract all the possible boundary conditions on the lagrange multipliers and the associated boundary Hamiltonians. In the last section, we use this general framework to describe some of the previously known boundary conditions.

Long string dynamics in pure gravity on AdS3

We study the classical dynamics of a completion of pure AdS gravity in 3D, whose only degrees of freedom are boundary gravitons and long strings. We argue that the best regime for describing pure gravity is that of "heavy" strings, for which back-reaction effects on the metric must be taken into account. We show that once back-reaction is properly accounted for, regular finite-energy states are produced by heavy strings even in the infinite-tension limit. Such a process is similar to, but different from, nucleation of space out of a "bubble of nothing."

Quantization of conical spaces in 3D gravity

We discuss the quantization and holographic aspects of a class of conical spaces in 2+1 dimensional pure AdS gravity. These appear as topological solitons in the Chern-Simons formulation of the theory and are closely related to the recently studied conical solutions in higher spin gravity. We discuss the classical fluctuations around these solutions, which form exceptional coadjoint orbits of the asymptotic Virasoro group. We argue that the quantization of these solutions leads to nonunitary representations of the Virasoro algebra, on account of their having boundary graviton fluctuations which lower the energy. We propose a framework to quantize them in a semiclassical expansion in the inverse central charge, which we use to compute their one-loop corrected energies. Interestingly, the resulting Virasoro representations contain a null vector, thus providing an appearance of Kac's degenerate representations, which are nonunitary at large central charge, in the context of gravity. We match the computed quantum corrections in the bulk with the properties of a class of primaries in Kac's classification.