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We study the "complexity equals volume" (CV) and "complexity equals action"(CA) conjectures by examining moments of of time symmetry for $\rm AdS_3$wormholes having $n$ asymptotic regions and arbitrary (orientable) internaltopology. For either prescription, the complexity relative to $n$ copies of the$M=0$ BTZ black hole takes the form $\Delta C = \alpha c \chi $, where $c$ isthe central charge and $\chi$ is the Euler character of the bulk time-symmetricsurface. The coefficients $\alpha_V = -4\pi/3$, $\alpha_A = 1/6 $ defined by CVand CA are independent of both temperature and any moduli controlling thegeometry inside the black hole. Comparing with the known structure of dual CFTstates in the hot wormhole limit, the temperature and moduli independence of$\alpha_V$, $\alpha_A$ implies that any CFT gate set defining either complexitycannot be local. In particular, the complexity of an efficient quantum circuitbuilding local thermofield-double-like entanglement of thermal-sized patchesdoes not depend on the separation of the patches so entangled. We also commenton implications of the (positive) sign found for $\alpha_A$, which requires theassociated complexity to decrease when handles are added to our wormhole.

The AdS/CFT correspondence relates quantum entanglement between boundary conformal field theories and geometric connections in the dual asymptotically anti-de Sitter spacetime. We consider entangled states in the n-fold tensor product of a 1 + 1 dimensional CFT Hilbert space defined by the Euclidean path integral over a Riemann surface with n holes. In one region of moduli space, the dual bulk state is a black hole with n asymptotically AdS3 regions connected by a common wormhole, while in other regions the bulk fragments into disconnected components. We study the entanglement structure and compute the wave function explicitly in the puncture limit of the Riemann surface in terms of CFT n-point functions. We also use AdS minimal surfaces to measure entanglement more generally. In some regions of the moduli space the entanglement is entirely multipartite, though not of the GHZ type. However, even when the bulk is completely connected, there are regions of the moduli space in which the entanglement is instead almost entirely bipartite: significant entanglement occurs only between pairs of CFTs. We develop new tools to analyze intrinsically n-partite entanglement, and use these to show that for some wormholes with n similar sized horizons there is intrinsic entanglement between all n parties, and that the distillable entanglement between the asymptotic regions is at least (n + 1)/2 partite.

We give a holographic description of global conformal blocks in two dimensional conformal field theory on the sphere and on the torus. We show that the conformal blocks for one-point functions on the torus can be written as Witten diagrams in thermal AdS. This is accomplished by deriving a general conformal Casimir equation for global conformal blocks, and showing that Witten diagrams obey the same equation. We study the semi-classical limit of n-point conformal blocks, and show that these equal the action of a network of bulk world-lines obeying appropriate geodesic equations. We give an alternate description in the Chern-Simons formulation of 3D gravity, where the conformal blocks are described by networks of Wilson lines, and argue that these formulations are equivalent. [ABSTRACT FROM AUTHOR]

The space of permutation orbifolds is a simple landscape of two-dimensional CFTs, generalizing the well-known symmetric orbifolds. We consider constraints which a permutation orbifold with large central charge must obey in order to be holographically dual to a weakly coupled (but possibly stringy) theory of gravity in AdS. We then construct explicit examples of permutation orbifolds which obey these constraints. In our constructions, the spectrum remains finite at large N, but differs qualitatively from that of symmetric orbifolds. We also discuss under what conditions the correlation functions factorize at large N and thus reduce to those of a generalized free field in AdS. We show that this happens not just for symmetric orbifolds, but also for permutation groups which act 'democratically' in a sense which we define. [ABSTRACT FROM AUTHOR]

We construct a new class of entanglement measures by extending the usual definition of Rényi entropy to include a chemical potential. These charged Rényi entropies measure the degree of entanglement in different charge sectors of the theory and are given by Euclidean path integrals with the insertion of a Wilson line encircling the entangling surface. We compute these entropies for a spherical entangling surface in CFT’s with holographic duals, where they are related to entropies of charged black holes with hyperbolic horizons. We also compute charged Rényi entropies in free field theories. [ABSTRACT FROM AUTHOR]

We consider Rényi entropies $ {S_n}=\frac{1}{1-n } \log\;\mathrm{Tr}{\rho^n} $ of conformal field theories in flat space, with the entangling surface being a sphere. The AdS/CFT correspondence relates this Rényi entropy to that of a black hole with hyperbolic horizon; as the Rényi parameter n increases the temperature of the black hole decreases. If the CFT possesses a sufficiently low dimension scalar operator the black hole will be unstable to the development of hair. Thus, as n is varied, the Rényi entropies will exhibit a phase transition at a critical value of n. The location of the phase transition, along with the spectrum of the reduced density matrix ρ, depends on the dimension of the lowest dimension non-trivial scalar operator in the theory. [ABSTRACT FROM AUTHOR]

We study scattering coefficients in black hole spacetimes using analytic properties of complexified wave equations. For a concrete example, we analyse the singularities of the Teukolsky equation and relate the corresponding monodromies to scattering data. These techniques, valid in full generality, provide insights into complex-analytic properties of greybody factors and quasinormal modes. This leads to new perturbative and numerical methods which are in good agreement with previous results. [ABSTRACT FROM AUTHOR]

We consider CFTs conjectured to be dual to higher spin theories of gravity in AdS3 and AdS4. Two-dimensional CFTs with WN symmetry are considered in the λ = 0 (k → ∞) limit where they are conjectured to be described by continuous orbifolds. The torus partition function is computed, using reasonable assumptions, and equals that of a free-field theory. We find no phase transition at temperatures of order 1; the usual Hawking-Page phase transition is removed by the highly degenerate light states associated with conical defect states in the bulk. Three-dimensional Chern-Simons matter CFTs with vector-like matter are considered on T³, where the dynamics is described by an effective theory for the eigenvalues of the holonomies. Likewise, we find no evidence for a Hawking-Page phase transition at a large level k. [ABSTRACT FROM AUTHOR]

Extreme and very-near-extreme spin J Kerr black holes have been conjectured to be holographically dual to two-dimensional (2D) conformal field theories (CFTs) with left and right central charges cL=cR=12J. In this paper it is observed that the 2D conformal symmetry of the scalar wave equation at low frequencies persists for generic nonextreme values of the mass M≠J√. Interestingly, this conformal symmetry is not derived from a conformal symmetry of the spacetime geometry except in the extreme limit. The 2π periodic identification of the azimuthal angle ϕ is shown to correspond to a spontaneous breaking of the conformal symmetry by left and right temperatures TL=M2/2πJ and TR=√M4−J2 /2πJ. The well-known low-frequency scalar-Kerr scattering amplitudes coincide with correlators of a 2D CFT at these temperatures. Moreover, the CFT microstate degeneracy inferred from the Cardy formula agrees exactly with the Bekenstein-Hawking area law for all M and J. These observations provide evidence for the conjecture that the Kerr black hole is dual to a cL=cR=12J 2D CFT at temperatures (TL,TR) for every value of M and J., Physics

We show that the linearization of all exact solutions of classical chiral gravity around the AdS3 vacuum have positive energy. Non-chiral and negative-energy solutions of the linearized equations are infrared divergent at second order, and so are removed from the spectrum. In other words, chirality is confined and the equations of motion have linearization instabilities. We prove that the only stationary, axially symmetric solutions of chiral gravity are BTZ black holes, which have positive energy. It is further shown that classical log gravity – the theory with logarithmically relaxed boundary conditions – has finite asymptotic symmetry generators but is not chiral and hence may be dual at the quantum level to a logarithmic CFT. Moreover we show that log gravity contains chiral gravity within it as a decoupled charge superselection sector. We formally evaluate the Euclidean sum over geometries of chiral gravity and show that it gives precisely the holomorphic extremal CFT partition function. The modular invariance and integrality of the expansion coefficients of this partition function are consistent with the existence of an exact quantum theory of chiral gravity. We argue that the problem of quantizing chiral gravity is the holographic dual of the problem of constructing an extremal CFT, while quantizing log gravity is dual to the problem of constructing a logarithmic extremal CFT., Physics

We consider the one-loop partition function of free quantum field theory in locally Anti-de Sitter space-times. In three dimensions, the one loop determinants for scalar, gauge and graviton excitations are computed explicitly using heat kernel techniques. We obtain precisely the result anticipated by Brown and Henneaux: the partition function includes a sum over "boundary excitations" of AdS3, which are the Virasoro descendants of empty Anti-de Sitter space. This result also allows us to compute the one-loop corrections to the Euclidean action of the BTZ black hole as well its higher genus generalizations., Physics

We obtain an asymptotic formula for the average value of the operator product expansion coefficients of any unitary, compact two dimensional CFT with c > 1. This formula is valid when one or more of the operators has large dimension or — in the presence of a twist gap — has large spin. Our formula is universal in the sense that it depends only on the central charge and not on any other details of the theory. This result unifies all previous asymptotic formulas for CFT2 structure constants, including those derived from crossing symmetry of four point functions, modular covariance of torus correlation functions, and higher genus modular invariance. We determine this formula at finite central charge by deriving crossing kernels for higher genus crossing equations, which give analytic control over the structure constants even in the absence of exact knowledge of the conformal blocks. The higher genus modular kernels are obtained by sewing together the elementary kernels for four-point crossing and modular transforms of torus one-point functions. Our asymptotic formula is related to the DOZZ formula for the structure constants of Liouville theory, and makes precise the sense in which Liouville theory governs the universal dynamics of heavy operators in any CFT. The large central charge limit provides a link with 3D gravity, where the averaging over heavy states corresponds to a coarse-graining over black hole microstates in holographic theories. Our formula also provides an improved understanding of the Eigenstate Thermalization Hypothesis (ETH) in CFT2, and suggests that ETH can be generalized to other kinematic regimes in two dimensional CFTs. [ABSTRACT FROM AUTHOR]

We obtain an asymptotic formula for the average value of the operator product expansion coefficients of any unitary, compact two dimensional CFT with c > 1. This formula is valid when one or more of the operators has large dimension or — in the presence of a twist gap — has large spin. Our formula is universal in the sense that it depends only on the central charge and not on any other details of the theory. This result unifies all previous asymptotic formulas for CFT2 structure constants, including those derived from crossing symmetry of four point functions, modular covariance of torus correlation functions, and higher genus modular invariance. We determine this formula at finite central charge by deriving crossing kernels for higher genus crossing equations, which give analytic control over the structure constants even in the absence of exact knowledge of the conformal blocks. The higher genus modular kernels are obtained by sewing together the elementary kernels for four-point crossing and modular transforms of torus one-point functions. Our asymptotic formula is related to the DOZZ formula for the structure constants of Liouville theory, and makes precise the sense in which Liouville theory governs the universal dynamics of heavy operators in any CFT. The large central charge limit provides a link with 3D gravity, where the averaging over heavy states corresponds to a coarse-graining over black hole microstates in holographic theories. Our formula also provides an improved understanding of the Eigenstate Thermalization Hypothesis (ETH) in CFT2, and suggests that ETH can be generalized to other kinematic regimes in two dimensional CFTs. [ABSTRACT FROM AUTHOR]

We derive constraints on two-dimensional conformal field theories with higher spin symmetry due to unitarity, modular invariance, and causality. We focus on CFTs with WN symmetry in the “irrational” regime, where c > N − 1 and the theories have an infinite number of higher-spin primaries. The most powerful constraints come from positivity of the Kac matrix, which (unlike the Virasoro case) is non-trivial even when c > N − 1. This places a lower bound on the dimension of any non-vacuum higher-spin primary state, which is linear in the central charge. At large c, this implies that the dual holographic theories of gravity in AdS3, if they exist, have no local, perturbative degrees of freedom in the semi-classical limit. [ABSTRACT FROM AUTHOR]

We argue that higher-curvature terms in the gravitational Lagrangian lead, via non-relativistic gauge-gravity duality, to finite renormalization of the dynamical exponent of the dual conformal field theory. Our argument includes a proof of the non-renormalization of the Schrödinger and Lifshitz metrics beyond rescalings of their parameters, directly generalizing the AdS case. We use this effect to construct string-theory duals of non-relativistic critical systems with non-integer dynamical exponents, then use these duals to predict the viscosity/entropy ratios of these systems. The predicted values weakly violate the KSS bound. [ABSTRACT FROM AUTHOR]

We investigate the geometry of four dimensional black hole solutions in the presence of stringy higher curvature corrections to the low energy effective action. For certain supersymmetric two charge black holes these corrections drastically alter the causal structure of the solution, converting seemingly pathological null singularities into timelike singularities hidden behind a finite area horizon. We establish, analytically and numerically, that the string-corrected two-charge black hole metric has the same Penrose diagram as the extremal four-charge black hole. The higher derivative terms lead to another dramatic effect — the gravitational force exerted by a black hole on an inertial observer is no longer purely attractive! The magnitude of this effect is related to the size of the compactification manifold. [ABSTRACT FROM AUTHOR]

We evaluate the partition function of three-dimensional theories of gravity in the quantum regime, where the anti-de Sitter (AdS) radius is Planck scale and the central charge is of order one. The contribution from the AdS vacuum sector can--with certain assumptions--be computed and equals the vacuum character of a minimal model conformai field theory. The torus partition function is given by a sum over geometries which is finite and computable. For generic values of Newton's constant G and the AdS radius ℓ the result has no Hilbert space interpretation, but in certain cases it agrees with the partition function of a known conformai field theory. For example, the partition function of pure Einstein gravity with G = 3ℓ equals that of the Ising model, providing evidence that these theories are dual. We also present somewhat weaker evidence that the three-state and tricritical Potts models are dual to pure higher spin theories of gravity based on SL(3) and E6, respectively. [ABSTRACT FROM AUTHOR]

We derive constraints on two-dimensional conformal field theories with higher spin symmetry due to unitarity, modular invariance, and causality. We focus on CFTs with WN symmetry in the “irrational” regime, where c > N − 1 and the theories have an infinite number of higher-spin primaries. The most powerful constraints come from positivity of the Kac matrix, which (unlike the Virasoro case) is non-trivial even when c > N − 1. This places a lower bound on the dimension of any non-vacuum higher-spin primary state, which is linear in the central charge. At large c, this implies that the dual holographic theories of gravity in AdS3, if they exist, have no local, perturbative degrees of freedom in the semi-classical limit. [ABSTRACT FROM AUTHOR]