I review the reformulation of the AdS/CFT correspondence as

a quantum error correcting code, and explain how this gives a new

perspective on several mysterious features of the correspondence, in

particular the Ryu-Takayanagi formula.

In 4D CFT, the anomaly coefficient c is the coefficient of an anomaly in scale transformations. In this talk, we show how scale anomaly coefficients such as c can be computed in terms of CFT data, namely the operator spectrum and OPE coefficients. We obtain expressions for the anomaly coefficient using Euclidean position space and Minkowski momentum space. The latter gives the coefficient in terms of a positive-definite sum over states, but may suffer from UV and IR divergences. We confirm our expressions by explicit calculations for anomalies involving scalar operators. Along the way, we show how to do calculations in Minkowski momentum space using the state-operator correspondence, a technique which may have many other applications.

We have investigated the time evolution of (Renyi) entanglement entropies for a locally excited state defined by acting with a local operator. When the subsystem is given by a half of the total space, we have found that they approach finite constants in free field theories or solvable models. On the other hand, those entropies in holographic field theories logarithmically increase with t even at the late time. Thus they are expected to be a candidate which reveals the characteristic feature of holographic field theories. In the current works, we found that the entropies in free field theories are given by the probability with which the quasi-particle created by local operators located in the subsystem. We also introduce a toy model where a particle created by a local operator freely propagates. By using this model, we can reproduce the result which is obtained by the replica trick. I would like to explain the detail of our results as long as permitted.

We discuss information loss from black hole physics in AdS3, focusing on two sharp signatures infecting CFT2 correlators at large central charge c: 'forbidden singularities' arising from Euclidean-time periodicity due to the effective Hawking temperature, and late-time exponential decay in the Lorentzian region. We study an infinite class of examples where forbidden singularities can be resolved by non-perturbative effects at finite c, and we show that the resolution has certain universal features that also apply in the general case. Analytically continuing to the Lorentzian regime, we find that the non-perturbative effects that resolve forbidden singularities qualitatively change the behavior of correlators at times t∼SBH, the black hole entropy. This may resolve the exponential decay of correlators at late times in black hole backgrounds. By Borel resumming the 1/c expansion of exact examples, we explicitly identify 'information-restoring' effects from heavy states that should correspond to classical solutions in AdS3. Our results suggest a line of inquiry towards a more precise formulation of the gravitational path integral in AdS3.

A possible solution to the notorious sign problem for systems with non-zero chemical potential is to deform the integration region in the complex plane to a Lefschetz thimble. We introduce an easy to implement Monte Carlo algorithm to sample the dominant thimble, based on a contraction map on a thimble. We point out that manifolds other than Lefschetz thimble could be useful for numerical simulations. We describe a family of such manifolds, using the contraction map, that interpolate between the tangent space at one critical point (where the sign problem could be severe) and the union of relevant thimbles (where the sign problem is mild but a multimodal distribution function complicates the Monte Carlo sampling). We show how this works in a couple of models.

I will discuss nontrivial constraints on the spectrum and structure constants of 2D unitary CFTs from the associativity of OPE and modular invariance, and some implications for non-perturbative completions of gravity in AdS3.