Abstract We study the two point correlation function of a local operator on an n-sheeted replica manifold corresponding to the half-space in the vacuum state of a conformal field theory. In analogy with the inverse Laplace transform, we define the Renyi transform of this correlation function, which is a function of one complex variable w, dual to the Renyi parameter n. Inspired by the inversion formula of Caron-Huot, we argue that if the Renyi transform f (w) has bounded behavior at infinity in the complex w plane, the discontinuity of the Renyi transform disc f (w) provides the unique analytic continuation in n of the original replica correlation function. We check our formula by explicitly calculating the Renyi transform of a particular replica correlator in a large N holographic CFTd in dimensions d > 2. We also discover that the discontinuity of the Renyi transform is related to the matrix element of local operators between two distinct eigenstates of the modular Hamiltonian. We calculate the Renyi transform in 2d conformal field theories, and use it to extract the off-diagonal elements of (modular) ETH. We argue that in 2d, this is equivalent to the off-diagonal OPE coefficients of a CFT and show that our technique exactly reproduces recent results in the literature.
We study the two point correlation function of a local operator on an
n-sheeted replica manifold corresponding to the half-space in the vacuum state of a conformal field theory. In analogy with the inverse Laplace transform, we define the Renyi transform of this correlation function, which is a function of one complex variable
w, dual to the Renyi parameter
n. Inspired by the inversion formula of Caron-Huot, we argue that if the Renyi transform
f(w) has bounded behavior at infinity in the complex
w plane, the discontinuity of the Renyi transform disc
f(w) provides the unique analytic continuation in
n of the original replica correlation function. We check our formula by explicitly calculating the Renyi transform of a particular replica correlator in a large
N holographic CFT
d in dimensions
d>2. We also discover that the discontinuity of the Renyi transform is related to the matrix element of local operators between two distinct eigenstates of the modular Hamiltonian. We calculate the Renyi transform in
2d conformal field theories, and use it to extract the off-diagonal elements of (modular) ETH. We argue that in
2d, this is equivalent to the off-diagonal OPE coefficients of a CFT and show that our technique exactly reproduces recent results in the literature.
We derive an all-orders long-distance expansion of vacuum mutual information between two disjoint balls in generic
d-dimensional CFTs. The derivation uses a novel radial sewing expansion of the joint vacuum density matrix, together with a modular response formula for relative entropy and the geometric modular flow of a single ball, without analytic continuation in replica number. The expansion coefficients are modular integrals of modular-flowed vacuum correlators. Our formula provides a systematic treatment of arbitrary multi-point sectors, and recovers known results on two-point sector derived from replica methods. The full two-point sector resums into a modular integral of the thermal partition function on
Sd−1, with an effective temperature depending on ball configuration and modular time. We compute the expansion explicitly in CFTs. In two-dimensional Ising CFT, using analytic reductions and multidimensional numerical integration, we obtain the first ten radial orders in the long-distance expansion, which involve sphere correlators with up to twenty spin insertions. The truncated expansion agrees well with lattice data at finite separation, and approximately satisfies crossing symmetry. Analytic calculations also reproduce known expansion coefficients for 2d free compact boson at self-dual radius and its current sector, as well as for free scalars in higher dimensions.
We present a novel replica trick that computes the relative entropy of two arbitrary states in conformal field theory. Our replica trick is based on the analytic continuation of partition functions that break the Z_{n} replica symmetry. It provides a method for computing arbitrary matrix elements of the modular Hamiltonian corresponding to excited states in terms of correlation functions. We show that the quantum Fisher information in vacuum can be expressed in terms of two-point functions on the replica geometry. We perform sample calculations in two-dimensional conformal field theories.
Abstract 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.
A bstract We develop a perturbative understanding of the modular Hamiltonian for a 2D CFT, divided into left and right half-spaces, with a weak local perturbation inserted in the future wedge. A formal perturbation series for the modular Hamiltonian is available, but must be properly interpreted in quantum field theory. We work inside correlation functions with spectator operators, and introduce a prescription for defining complex modular flow via analytic continuation to properly resolve singularities. From the correlators, we extract an operator expression for the modular Hamiltonian. It takes the form of a local operator in the future wedge plus contact terms with an unconventional singularity structure. Thanks to this structure the KMS conditions are satisfied, which independently establishes the validity of the results. Similar techniques apply to perturbations inserted in the past wedge. We mention various future directions, including an all-orders speculation for the excited state modular Hamiltonian.
We give a recipe for computing correlation functions of the displacement operator localized on a spherical or planar higher dimensional twist defect using
AdS/CFT. Such twist operators are typically used to construct the
nth Renyi entropies of spatial entanglement in conformal field theories and are holographically dual to black holes with hyperbolic horizons. The displacement operator then tells us how the Renyi entropies change under small shape deformations of the entangling surface. We explicitly construct the bulk to boundary propagator for the displacement operator insertion as a linearized metric fluctuation of the hyperbolic black hole and use this to extract the coefficient of the displacement operator two point function
CD in any dimension. The $n\ensuremath{\rightarrow}1$ limit of the twist displacement operator gives the same bulk response as the insertion of a null energy operator in vacuum, which is consistent with recent results on the shape dependence of entanglement entropy and modular energy.