On Universality of Holographic Results for (2+1)-Dimensional CFTs on Curved Spacetimes

The behavior of holographic CFTs is constrained by the existence of a bulk dual geometry. For example, in (2+1)-dimensional holographic CFTs living on a static spacetime with compact spatial slices, the vacuum energy must be nonpositive, certain averaged energy densities must be nonpositive, and the spectrum of scalar operators is bounded from below by the Ricci scalar of the CFT geometry. Are these results special to holographic CFTs? Here we show that for perturbations about appropriate backgrounds, they are in fact universal to all CFTs, as they follow from the universal behavior of two- and three-point correlators of known operators. In the case of vacuum energy, we extend away from the perturbative regime and make global statements about its negativity properties on the space of spatial geometries. Finally, we comment on the implications for dynamics which are dissipative and driven by such a vacuum energy and we remark on similar results for the behavior of the Euclidean partition function on deformations of flat space or the round sphere.Comment: 35+4 pages, 1 figure. v2: corrected discussion of torus to deformed flat space; additional comments adde

Natural‐language processing applied to an ITS interface

The aim of this paper is to show that with a subset of a natural language, simple systems running on PCs can be developed that can nevertheless be an effective tool for interfacing purposes in the building of an Intelligent Tutoring System (ITS). After presenting the special characteristics of the Smalltalk/V language, which provides an appropriate environment for the development of an interface, the overall architecture of the interface module is discussed. We then show how sentences are parsed by the interface, and how interaction takes place with the user. The knowledge‐acquisition phase is subsequently described. Finally, some excerpts from a tutoring session concerned with elementary geometry are discussed, and some of the problems and limitations of the approach are illustrated

A Bound on Holographic Entanglement Entropy from Inverse Mean Curvature Flow

Entanglement entropies are notoriously difficult to compute. Large-N strongly-coupled holographic CFTs are an important exception, where the AdS/CFT dictionary gives the entanglement entropy of a CFT region in terms of the area of an extremal bulk surface anchored to the AdS boundary. Using this prescription, we show -- for quite general states of (2+1)-dimensional such CFTs -- that the renormalized entanglement entropy of any region of the CFT is bounded from above by a weighted local energy density. The key ingredient in this construction is the inverse mean curvature (IMC) flow, which we suitably generalize to flows of surfaces anchored to the AdS boundary. Our bound can then be thought of as a "subregion" Penrose inequality in asymptotically locally AdS spacetimes, similar to the Penrose inequalities obtained from IMC flows in asymptotically flat spacetimes. Combining the result with positivity of relative entropy, we argue that our bound is valid perturbatively in 1/N, and conjecture that a restricted version of it holds in any CFT.Comment: 33+7 pages, 7 figures. v2: addressed referee comment