Modave lectures on bulk reconstruction in AdS/CFT

These lecture notes are based on a series of lectures given at the XIII Modave summer school in mathematical physics. We review the construction due to Hamilton, Kabat, Lifschytz and Lowe for reconstructing local bulk operators from CFT operators in the context of AdS/CFT and show how to recover bulk correlation functions from this definition. Building on the work of these authors, it has been noted that the bulk displays quantum error correcting properties. We will discuss tensor network toy models to exemplify these remarkable features. We will discuss the role of gauge invariance and of diffeomorphism symmetry in the reconstruction of bulk operators. Lastly, we provide another method of bulk reconstruction specified to AdS$_3$/CFT$_2$ in which bulk operators create cross-cap states in the CFT.Comment: 35 pages, 8 figures, lecture notes, v4: a few minor improvements upon the published proceedings version (version 3 of these lecture notes in arXiv) have been implemente

Increasing information feed in the process of structural steel design

Research initiatives throughout history have shown how a designer typically makes associations and references to a vast amount of knowledge based on experiences to make decisions. With the increasing usage of information systems in our everyday lives, one might imagine an information system that provides designers access to the ‘architectural memories’ of other architectural designers during the design process, in addition to their own physical architectural memory. In this paper, we discuss how the increased adoption of semantic web technologies might advance this idea. We investigate to what extent information can be described with these technologies in the context of structural steel design. This investigation indicates significant possibilities regarding information reuse in the process of structural steel design and, by extent, in other design contexts as well. However, important obstacles and question remarks can still be outlined as well

Heavy-Heavy-Light-Light correlators in Liouville theory

We compute four-point functions of two heavy and two "perturbatively heavy" operators in the semiclassical limit of Liouville theory on the sphere. We obtain these "Heavy-Heavy-Light-Light" (HHLL) correlators to leading order in the conformal weights of the light insertions in two ways: (a) via a path integral approach, combining different methods to evaluate correlation functions from complex solutions for the Liouville field, and (b) via the conformal block expansion. This latter approach identifies an integral over the continuum of normalizable states and a sum over an infinite tower of lighter discrete states, whose contribution we extract by analytically continuing standard results to our HHLL setting. The sum over this tower reproduces the sum over those complex saddlepoints of the path integral that contribute to the correlator. Our path integral computations reveal that when the two light operators are inserted at equal time in radial quantization, the leading-order HHLL correlator is independent of their separation, and more generally that at this order there is no short-distance singularity as the two light operators approach each other. The conformal block expansion likewise shows that in the discrete sum short-distance singularities are indeed absent for all intermediate states that contribute. In particular, the Virasoro vacuum block, which would have been singular at short distances, is not exchanged. The separation-independence of equal-time correlators is due to cancelations between the discrete contributions. These features lead to a Lorentzian singularity that, in conformal theories with anti-de Sitter (AdS) duals, would be associated to locality below the AdS scale.Comment: 40 pages, 1 figure; v2: clarifications added, minor typos corrected, published versio

Ontwikkeling van een Google SketchUp-plugin als ontwerpinstrument voor een energiezuinige architectuur

Sinds 1 januari 2006 is in Vlaanderen de energieprestatieregelgeving van kracht. Vlaamse architecten, ontwerpers en ingenieurs worstelen met deze nieuwe wetgeving en zijn nog steeds op zoek naar een instrument waarmee ze reeds van in de beginfase van het onwerp het peil van energieprestatie kunnen bepalen. Deze paper behandelt het onderzoek naar en de ontwikkeling van een Google SketchUp-plugin, dat kadert in de masterscriptie van Tine Jonckheere. Er werd onderzocht wat de specifieke noden zijn van de architecten in Vlaanderen en welke oplossingen er reeds voorhanden zijn. Vervolgens werd een prototype van de SketchUp-plugin ontwikkeld en getoetst aan de eisen en noden van de architect

Entanglement versus entwinement in symmetric product orbifolds

We study the entanglement entropy of gauged internal degrees of freedom in a two dimensional symmetric product orbifold CFT, whose configurations consist of $N$ strands sewn together into "long" strings, with wavefunctions symmetrized under permutations. In earlier work a related notion of "entwinement" was introduced. Here we treat this system analogously to a system of $N$ identical particles. From an algebraic point of view, we point out that the reduced density matrix on $k$ out of $N$ particles is not associated with a subalgebra of operators, but rather with a linear subspace, which we explain is sufficient. In the orbifold CFT, we compute the entropy of a single strand in states holographically dual in the D1/D5 system to a conical defect geometry or a massless BTZ black hole and find a result identical to entwinement. We also calculate the entropy of two strands in the state that represents the conical defect; the result differs from entwinement. In this case, matching entwinement would require finding a gauge-invariant way to impose continuity across strands.Comment: 21 pages, v2: short entwinement review section added, and prev. section 2 rewritten to increase clarity. Matches published versio