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.ArXiv ePrint: 1310.4180 arXiv:1310.4180v2 [hep-th] An interesting exercise to gain better intuition for this background gauge field (2.15) is to expand the coordinates near the spherical entangling surface: t E = ρ sin θ and r = R + ρ cos θ with ρ R. To leading order in ρ/R, one then finds that the potential reduces to B µ E 2π dθ.

We study the boundary description of the volume of maximal Cauchy slices using the recently derived equivalence between bulk and boundary symplectic forms. The volume of constant mean curvature slices is known to be canonically conjugate to "York time". We use this to construct the boundary deformation that is conjugate to the volume in a handful of examples, such as empty AdS, a backreacting scalar condensate, or the thermofield double at infinite time. We propose a possible natural boundary interpretation for this deformation and use it to motivate a concrete version of the complexity=volume conjecture, where the boundary complexity is defined as the energy of geodesics in the Kähler geometry of half sided sources. We check this conjecture for Bañados geometries and a mini-superspace version of the thermofield double state. Finally, we show that the precise dual of the quantum information metric for marginal scalars is given by a particularly simple symplectic flux, instead of the volume as previously conjectured.Here, we are denoting linearized deformations by |δΨ λ ≈ |Ψ λ+δλ − |Ψ λ , the bulk field φ is dual to the operator sourced by λ, its conjugate momentum is π, and Σ is a bulk initial value surface. This is the main result of [21] and it gives a

We propose an ansatz for OPE coefficients in chaotic conformal field theories which generalizes the eigenstate thermalization hypothesis and describes any OPE coefficient involving heavy operators as a random variable with a Gaussian distribution. In two dimensions this ansatz enables us to compute higher moments of the OPE coefficients and analyse two and four-point functions of OPE coefficients, which we relate to genus-2 partition functions and their squares. We compare the results of our ansatz to solutions of Einstein gravity in AdS 3 , including a Euclidean wormhole that connects two genus-2 surfaces. Our ansatz reproduces the non-perturbative correction of the wormhole, giving it a physical interpretation in terms of OPE statistics. We propose that calculations performed within the semi-classical low-energy gravitational theory are only sensitive to the random nature of OPE coefficients, which explains the apparent lack of factorization in products of partition functions.

We compute the bulk entanglement entropy across the Ryu-Takayanagi surface for a one-particle state in a scalar field theory in AdS 3 . We work directly within the bulk Hilbert space and include the spatial spread of the scalar wavefunction. We give closed form expressions in the limit of small interval sizes and compare the result to a CFT computation of entanglement entropy in an excited primary state at large c. Including the contribution from the backreacted minimal area, we find agreement between the CFT result and the FLM and JLMS formulas for quantum corrections to holographic entanglement entropy. This provides a non-trivial check in a state where the answer is not dictated by symmetry. Along the way, we provide closed-form expressions for the scalar field Bogoliubov coefficients that relate the global and Rindler slicings of AdS 3 .

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We consider Rényi entropies S n = 1 1−n log Tr ρ 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.

The hypothesis that every theory of quantum gravity in AdS 3 is a dimensional reduction of string/M-theory leads to a natural conjecture for the density of states of two dimensional CFTs with a large central charge limit. We prove this conjecture for 2D CFTs which are orbifolds by permutation groups. In particular, we characterize those permutation groups which give CFTs with well-defined large N limits and can thus serve as holographic duals to bulk gravity theories in AdS 3 . We then show that the holographic dual of a permutation orbifold will have a Hagedorn spectrum in the large N limit. This is evidence that, within this landscape, every theory of quantum gravity with a semi-classical limit is a string theory.

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