Large Central Charge Research Articles

Toda theory in AdS2 and \U0001d4b2An-algebra structure of boundary correlators

We consider the conformal An Toda theory in AdS2. Due to the bulk full Virasoro symmetry, this system provides an instance of a non-gravitational AdS2/CFT1 correspondence where the 1d boundary theory enjoys enhanced ``frac{1}{2}- Virasoro" symmetry. General boundary correlators are expected to be captured by the restriction of chiral correlators in a suitable WAn Virasoro extension. At next-to-leading order in weak coupling expansion they have been conjectured to match the subleading terms in the large central charge expansion of the dual \U0001d4b2An correlators. We explicitly test this conjecture on the boundary four point functions of the Toda scalar fields dual to \U0001d4b2An generators with next-to-minimal spin 3 and 4. Our analysis is valid in the generic rank case and extends previous results for specific rank-2 Toda theories. On the AdS side, the extension is straightforward and requires the computation of a finite set of tree Witten diagrams. This is due to simple rank dependence and selection rules of cubic and quartic couplings. On the boundary, we exploit crossing symmetry and specific meromorphic properties of the W-algebra correlators at large central charge. We present the required 4-point functions in closed form for any rank and verify the bulk-boundary correspondence in full details.

Regenesis and quantum traversable wormholes

Recent gravity discussions of a traversable wormhole indicate that in holographic systems signals generated by a source could reappear long after they have dissipated, with the need of only performing some simple operations. In this paper we argue the phenomenon, to which we refer as “regenesis”, is universal in general quantum chaotic many-body systems, and elucidate its underlying physics. The essential elements behind the phenomenon are: (i) scrambling which in a chaotic system makes out-of-time-ordered correlation functions (OTOCs) vanish at large times; (ii) the entanglement structure of the state of the system. The latter aspect also implies that the regenesis phenomenon requires fine tuning of the initial state. Compared to other manifestations of quantum chaos such as the initial growth of OTOCs which deals with early times, and a random matrix-type energy spectrum which reflects very large time behavior, regenesis concerns with intermediate times, of order the scrambling time of a system. We also study the phenomenon in detail in general two-dimensional conformal field theories in the large central charge limit, and highlight some interesting features including a resonant enhancement of regenesis signals near the scrambling time and their oscillations in coupling. Finally, we discuss gravity implications of the phenomenon for systems with a gravity dual, arguing that there exist regimes for which traversability of a wormhole is quantum in nature, i.e. cannot be associated with a semi-classical spacetime causal structure.

Derivation of Holographic Negativity in AdS_{3}/CFT_{2}.

We present a derivation of the holographic dual of logarithmic negativity in AdS_{3}/CFT_{2} that was recently conjectured in Phys. Rev. D 99, 106014 (2019PRVDAQ2470-001010.1103/PhysRevD.99.106014). This is given by the area of an extremal cosmic brane that terminates on the boundary of the entanglement wedge. The derivation consists of relating the recently introduced Rényi reflected entropy to the logarithmic negativity in holographic conformal field theories. Furthermore, we clarify previously mysterious aspects of negativity at a large central charge seen in conformal blocks and comment on generalizations to generic dimensions, dynamical settings, and quantum corrections.

Generalized Eigenstate Thermalization Hypothesis in 2D Conformal Field Theories.

Infinite-dimensional conformal symmetry in two dimensions leads to integrability of 2D conformal field theories (CFTs) by giving rise to an infinite tower of local conserved quantum Korteweg-de Vries (qKdV) charges in involution. We discuss how the presence of conserved charges constrains equilibration in 2D CFTs. We propose that in the thermodynamic limit large central charge 2D CFTs satisfy generalized eigenstate thermalization, with the values of qKdV charges forming a complete set of thermodynamically relevant quantities, which unambiguously determine expectation values of all local observables from the vacuum family. Equivalence of ensembles further provides that local properties of an eigenstate can be described by the generalized Gibbs ensemble that includes only qKdV charges. In the case of a general initial state, upon equilibration, the emerging generalized Gibbs ensemble will necessarily include negative chemical potentials and holographically will be described by a quasiclassical black hole with quantum soft hair.

Non-abelian Toda theory on AdS2 and {mathrm{AdS}}_2/{mathrm{CFT}}_2^{raisebox{1ex}{1}!left/ !raisebox{-1ex}{2}right.} duality

It was recently observed that boundary correlators of the elementary scalar field of the Liouville theory on AdS2 background are the same (up to a non-trivial proportionality coefficient) as the correlators of the chiral stress tensor of the Liouville CFT on the complex plane restricted to the real line. The same relation generalizes to the conformal abelian Toda theory: boundary correlators of Toda scalars on AdS2 are directly related to the correlation functions of the chiral mathcal{W} -symmetry generators in the Toda CFT and thus are essentially controlled by the underlying infinite-dimensional symmetry. These may be viewed as examples of AdS2/CFT1 duality where the CFT1 is the chiral half of a 2d CFT; we shall to this as {mathrm{AdS}}_2/{mathrm{CFT}}_2^{raisebox{1ex}{1}!left/ !raisebox{-1ex}{2}right.} . In this paper we demonstrate that this duality applies also to the non-abelian Toda theory containing a Liouville scalar coupled to a 2d σ-model originating from the SL(2, ℝ)/U(1) gauged WZW model. Here the Liouville scalar is again dual to the chiral stress tensor T while the other two scalars are dual to the parafermionic operators V± of the non-abelian Toda CFT. We explicitly check the duality at the next-to-leading order in the large central charge expansion by matching the chiral CFT correlators of (T, V+, V−) (computed using a free field representation) with the boundary correlators of the three Toda scalars given by the tree-level and one-loop Witten diagrams in AdS2.

Universal lowest-twist in CFTs from holography

We probe the conformal block structure of a scalar four-point function in d ≥ 2 conformal field theories by including higher-order derivative terms in a bulk gravitational action. We consider a heavy-light four-point function as the boundary correlator at large central charge. Such a four-point function can be computed, on the gravity side, as a two-point function of the light operator in a black hole geometry created by the heavy operator. We consider analytically solving the corresponding scalar field equation in a near-boundary expansion and find that the multi-stress tensor conformal blocks are insensitive to the horizon boundary condition. The main result of this paper is that the lowest-twist operator product expansion (OPE) coefficients of the multi-stress tensor conformal blocks are universal: they are fixed by the dimension of the light operators and the ratio between the dimension of the heavy operator and the central charge CT. Neither supersymmetry nor unitary is assumed. Higher-twist coefficients, on the other hand, generally are not protected. A recursion relation allows us to efficiently compute universal lowest-twist coefficients. The universality result hints at the potential existence of a higher-dimensional Virasoro-like symmetry near the lightcone. While we largely focus on the planar black hole limit in this paper, we include some preliminary analysis of the spherical black hole case in an appendix.

R\xe9nyi entropy at large energy density in 2D CFT

We investigate the Rényi entropy and entanglement entropy of an interval with an arbitrary length in the canonical ensemble, microcanonical ensemble and primary excited states at large energy density in the thermodynamic limit of a two-dimensional large central charge c conformal field theory. As a generalization of the recent work [17], the main purpose of the paper is to see whether one can distinguish these various large energy density states by the Rényi entropies of an interval at different size scales, namely, short, medium and long. Collecting earlier results and performing new calculations in order to compare with and fill gaps in the literature, we give a more complete and detailed analysis of the problem. Especially, we find some corrections to the recent results for the holographic Rényi entropy of a medium size interval, which enlarge the validity region of the results. Based on the Rényi entropies of the three interval scales, we find that Rényi entropy cannot distinguish the canonical and microcanonical ensemble states for a short interval, but can do the job for both medium and long intervals. At the leading order of large c the entanglement entropy cannot distinguish the canonical and microcanonical ensemble states for all interval lengths, but the difference of entanglement entropy for a long interval between the two states would appear with 1/c corrections. We also discuss Rényi entropy and entanglement entropy differences between the thermal states and primary excited state. Overall, our work provide an up-to-date picture of distinguishing different thermal or primary states at various length scales of the subsystem.

Four-point conformal blocks with three heavy background operators

We study CFT2 Virasoro conformal blocks of the 4-point correlation function leftlangle {mathcal{O}}_L{mathcal{O}}_H{mathcal{O}}_H{mathcal{O}}_Hrightrangle with three background operators mathcal{O} H and one perturbative operator mathcal{O} L of dimensions ΔL/ΔH ≪ 1. The conformal block function is calculated in the large central charge limit using the monodromy method. From the holographic perspective, the background operators create AdS3 space with three conical singularities parameterized by dimensions ΔH, while the perturbative operator corresponds to the geodesic line stretched from the boundary to the bulk. The geodesic length calculates the perturbative conformal block. We propose how to address the block/length correspondence problem in the general case of higher-point correlation functions leftlangle {mathcal{O}}_Lcdots {mathcal{O}}_L{mathcal{O}}_Hcdots {mathcal{O}}_Hrightrangle with arbitrary numbers of background and perturbative operators.

Black holes, heavy states, phase shift and anomalous dimensions

We compute the phase shift of a highly energetic particle traveling in the background of an asymptotically AdS black hole. In the dual CFT, the phase shift is related to a four point function in the Regge limit. The black hole mass is translated to the ratio between the conformal dimension of a heavy operator and the central charge. This ratio serves as a useful expansion parameter; its power measures the number of stress tensors appearing in the intermediate channel. We compute the leading term in the phase shift in a holographic CFT of arbitrary dimensionality using Conformal Regge Theory and observe complete agreement with the gravity result. In a two-dimensional CFT with a large central charge the heavy-heavy-light-light Virasoro vacuum block reproduces the gravity phase shift to all orders in the expansion parameter. We show that the leading order phase shift is related to the anomalous dimensions of certain double trace operators and verify this agreement using known results for the latter. We also perform a separate gravity calculation of these anomalous dimensions to second order in the expansion parameter and compare with the phase shift expansion.

Covariant holographic entanglement negativity for disjoint intervals in AdS_3/CFT_2

We advance a construction for the covariant holographic entanglement negativity for time dependent mixed states of disjoint intervals in (1+1) dimensional conformal field theories (CFT_{1+1}) dual to bulk non static AdS_3 geometries. Application of our proposal to such mixed states in a CFT_{1+1} dual to bulk non extremal and extremal rotating BTZ black holes exactly reproduces the replica technique results in the large central charge limit. We also investigate the time dependent holographic entanglement negativity for such mixed states in a CFT_{1+1} dual to a bulk Vaidya-AdS_3 geometry in the context of their thermalization involving bulk black hole formation.

Exact generalized partition function of 2D CFTs at large central charge

We discuss generalized partition function of 2d CFTs on thermal cylinder decorated by higher qKdV charges. We propose that in the large central charge limit qKdV charges factorize such that generalized partition function can be rewritten in terms of auxiliary non-interacting bosons. The explicit expression for the generalized free energy is readily available in terms of the boson spectrum, which can be deduced from the conventional thermal expectation values of qKdV charges. In other words, the picture of the auxiliary non-interacting bosons allows extending thermal one-point functions to the full non-perturbative generalized partition function. We verify this conjecture for the first seven qKdV charges using recently obtained pertrubative results and find corresponding contributions to the auxiliary boson masses. We further extend these results by conjecturing the full spectrum of bosons and find an exact expression for the generalized partition function as a function of infinite tower of chemical potentials in the limit of large central charge.

Fast conformal bootstrap and constraints on 3d gravity

The crossing equations of a conformal field theory can be systematically truncated to a finite, closed system of polynomial equations. In certain cases, solutions of the truncated equations place strict bounds on the space of all unitary CFTs. We describe the conditions under which this holds, and use the results to develop a fast algorithm for modular bootstrap in 2d CFT. We then apply it to compute spectral gaps to very high precision, find scaling dimensions for over a thousand operators, and extend the numerical bootstrap to the regime of large central charge, relevant to holography. This leads to new bounds on the spectrum of black holes in three-dimensional gravity. We provide numerical evidence that the asymptotic bound on the spectral gap from spinless modular bootstrap, at large central charge c, is Δ1 ≲ c/9.1.

A supersymmetric enhancement of mathcal{N} = 1 holographic minimal model

By studying the mathcal{N} = 1 holographic minimal model at the “critical” level, we obtain the lowest mathcal{N} = 2 higher spin multiplet of spins left(frac{3}{2},2,2,frac{5}{2}right) in terms of two adjoint fermion types for generic N. We subsequently determine operator product expansions (between the lowest and second lowest ( mathcal{N} = 2) higher spin multiplet of spins left(3,frac{7}{2},frac{7}{2},4right) , and the corresponding Vasiliev’s oscillator formalism with matrix generalization on AdS3 higher spin theory in the extension of OSp(2|2) superconformal algebra. Under the large N limit (equivalent to large central charge) in the extension of mathcal{N} = 2 superconformal algebra in two dimensions, operator product expansions provide asymptotic symmetry algebra in AdS3 higher spin theory.

Entropy generation at the horizon diffuses the cosmological constant in 2D de Sitter space

We investigate a solution of the exactly renormalized Liouville action to foresee the fate of the two-dimensional de Sitter space. We work in the semiclassical region with a large matter central charge $c$. Instead of de Sitter expansion, it performs a slow-roll inflation with the parameters $\epsilon=(1/2)\eta =6/c$. An inflaton field is induced in the effective theory to describe quantum effects of the Liouville theory. The geometric entropy increases logarithmically with the Hubble radius. We propose that de Sitter entropy is carried by superhorizon modes of the metric. It can be directly estimated from the partition function as $S=\log Z$ in Liouville gravity. We formulate a gravitational Fokker-Planck equation to elucidate the Brownian process at the horizon: the superhorizon modes are constantly jolted by newcomers. We show that such a built-in entropy-generating process diffuses the cosmological constant. We evaluate von Neumann entropy associated with the distribution function of superhorizon modes. It always increases under the Fokker-Planck equation in a consistent way with semiclassical estimates. The maximum entropy principle operates in quantum gravity. An analogous entropy production mechanism at the horizon might have increased the Hubble radius much beyond the microscopic physics scale in the Universe.

Holographic entanglement negativity conjecture for adjacent intervals in AdS3/CFT2

We propose a holographic entanglement negativity conjecture involving the bulk geometry, for mixed states of adjacent intervals in (1+1)-dimensional dual conformal field theories through the AdS/CFT correspondence. The holographic entanglement negativity is obtained from a specific algebraic sum of the geodesics anchored on respective intervals on the boundary which reduces to the holographic mutual information between them. Utilizing our conjecture we obtain the entanglement negativity of adjacent intervals in zero and finite temperature (1+1)-dimensional holographic conformal field theories dual to the bulk AdS3 vacuum and the Euclidean BTZ black hole respectively. Our holographic conjecture exactly reproduces the conformal field theory results obtained through the replica technique, in the large central charge limit. We briefly elucidate the corresponding issue for the AdSd+1/CFTd scenario.

The most irrational rational theories

We propose a two-parameter family of modular invariant partition functions of two-dimensional conformal field theories (CFTs) holographically dual to pure three-dimensional gravity in anti de Sitter space. Our two parameters control the central charge, and the representation of SL(2, ℤ). At large central charge, the partition function has a gap to the first nontrivial primary state of frac{c}{24} . As the SL(2, ℤ) representation dimension gets large, the partition function exhibits some of the qualitative features of an irrational CFT. This, for instance, is captured in the behavior of the spectral form factor. As part of these analyses, we find similar behavior in the minimal model spectral form factor as c approaches 1.

Late time Wilson lines

In the AdS3/CFT2 correspondence, physical interest attaches to understanding Virasoro conformal blocks at large central charge and in a kinematical regime of large Lorentzian time separation, t ∼ c. However, almost no analytical information about this regime is presently available. By employing the Wilson line representation we derive new results on conformal blocks at late times, effectively resumming all dependence on t/c. This is achieved in the context of “light-light” blocks, as opposed to the richer, but much less tractable, “heavy-light” blocks. The results exhibit an initial decay, followed by erratic behavior and recurrences. We also connect this result to gravitational contributions to anomalous dimensions of double trace operators by using the Lorentzian inversion formula to extract the latter. Inverting the stress tensor block provides a pedagogical example of inversion formula machinery.

Entanglement and chaos in warped conformal field theories

Various aspects of warped conformal field theories (WCFTs) are studied including entanglement entropy on excited states, the Rényi entropy after a local quench, and out-of-time-order four-point functions. Assuming a large central charge and dominance of the vacuum block in the conformal block expansion, (i) we calculate the single-interval entanglement entropy on an excited state, matching previous finite temperature results by changing the ensemble; and (ii) we show that WCFTs are maximally chaotic, a result that is compatible with the existence of black holes in the holographic duals. Finally, we relax the aforementioned assumptions and study the time evolution of the Rényi entropy after a local quench. We find that the change in the Rényi entropy is topological, vanishing at early and late times, and nonvanishing in between only for charged states in spectrally-flowed WCFTs.

Quantum null energy condition and its (non)saturation in 2d CFTs

We consider the Quantum Null Energy Condition (QNEC) for holographic conformal field theories in two spacetime dimensions (CFT_22). We show that QNEC saturates for all states dual to vacuum solutions of AdS_33 Einstein gravity, including systems that are far from thermal equilibrium. If the Ryu-Takayanagi surface encounters bulk matter QNEC does not need to be saturated, whereby we give both analytical and numerical examples. In particular, for CFT_22 with a global quench dual to AdS_33-Vaidya geometries we find a curious half-saturation of QNEC for large entangling regions. We also address order one corrections from quantum backreactions of a scalar field in AdS_33 dual to a primary operator of dimension h in a large central charge expansion and explicitly compute both, the backreacted Ryu–Takayanagi surface part and the bulk entanglement contribution to EE and QNEC. At leading order for small entangling regions the contribution from bulk EE exactly cancels the contribution from the back-reacted Ryu-Takayanagi surface, but at higher orders in the size of the region the contributions are almost equal while QNEC is not saturated. For a half-space entangling region we find that QNEC is gapped by h/4h/4 in the large h expansion.

DS/dS and Toverline{T}

The Toverline{T} deformation of a conformal field theory has a dual description as a cutoff AdS3 spacetime, at least at the level of pure 3d gravity. We generalize this deformation in such a way that it builds up a patch of bulk dS3 spacetime instead. At each step along the trajectory in the space of 2d theories, the theory is deformed by a specific combination of Toverline{T} and the two-dimensional cosmological constant. This provides a concrete holographic dual for the warped throat on the gravity side of the dS/dS duality, at leading order in large central charge. We also analyze a sequence of excitations of this throat on both sides of the duality, as well as the entanglement entropy. Our results point toward a mechanism for obtaining de Sitter solutions starting from seed conformal field theories with AdS duals.