Conformal Field Theories, Graphs and Quantum Algebras

This article reviews some recent progress in our understanding of the structure of Rational Conformal Field Theories, based on ideas that originate for a large part in the work of A. Ocneanu. The consistency conditions that generalize modular invariance for a given RCFT in the presence of various types of boundary conditions --open, twisted-- are encoded in a system of integer multiplicities that form matrix representations of fusion-like algebras. These multiplicities are also the combinatorial data that enable one to construct an abstract ``quantum'' algebra, whose $6j$- and $3j$-symbols contain essential information on the Operator Product Algebra of the RCFT and are part of a cell system, subject to pentagonal identities. It looks quite plausible that the classification of a wide class of RCFT amounts to a classification of ``Weak $C^*$- Hopf algebras''.Comment: 23 pages, 12 figures, LateX. To appear in MATHPHYS ODYSSEY 2001 --Integrable Models and Beyond, ed. M. Kashiwara and T. Miwa, Progress in Math., Birkhauser. References and comments adde

Hyaluronidase Coated Molecular Envelope Technology Nanoparticles Enhance Drug Absorption via the Subcutaneous Route

Parenteral chemotherapy is usually administered intravenously, although patient preference and health economics suggest the subcutaneous (sc) route could be an attractive alternative. However, due to the low aqueous solubility of hydrophobic drugs and injection volume limitations, the total amount of drug that can be administered in a single sc injection is frequently insufficient. We have developed hyaluronidase coated nanoparticles (NPs) that efficiently encapsulate such drugs, thus addressing both issues and allowing sufficient amounts of hydrophobic drug to be administered and absorbed effectively. CUDC-101, a poorly water-soluble multitargeted anticancer drug that simultaneously inhibits the receptor tyrosine kinases (RTKs) EGFR and HER2, as well as histone deacetylase (HDAC), was encapsulated in polymeric Molecular Envelope Technology (MET) NPs. The role of polymer chemistry, formulation parameters, and coating with hyaluronidase (HYD) on MET-CUDC-101 NP formulations was examined and optimized to yield high drug loading and colloidal stability, and, after freeze-drying, stable storage at room temperature for up to 90 days. The pharmacokinetic studies in healthy rats showed that plasma AUC0-24h after sc administration correlates tightly with formulation physical chemistry, specifically in vitro colloidal stability. Compared to uncoated NPs, the HYD-coating doubled the drug plasma exposure. In a murine A431 xenograft model, the coated HYD-MET-CUDC-101 NPs at a dose equivalent to 90 mg kg-1 CUDC-101 increased the survival time from 15 days (control animals treated with hyaluronidase alone) to 43 days. Polymer MET nanoparticles coated with hyaluronidase enabled the subcutaneous delivery of a hydrophobic drug with favorable therapeutic outcomes

Determination of quantum symmetries for higher ADE systems from the modular T matrix

We show that the Ocneanu algebra of quantum symmetries, for an ADE diagram (or for higher Coxeter-Dynkin systems, like the Di Francesco - Zuber system) is, in most cases, deduced from the structure of the modular T matrix in the A series. We recover in this way the (known) quantum symmetries of su(2) diagrams and illustrate our method by studying those associated with the three genuine exceptional diagrams of type su(3), namely E5, E9 and E21. This also provides the shortest way to the determination of twisted partition functions in boundary conformal field theory with defect lines.Comment: 30 pages, 16 figures. Several misprints have been corrected. We added several references and the appendix has been enlarged (one section on essential paths and one section devoted to open problems). This article will appear in the Journal of Mathematical Physic

Comments about quantum symmetries of SU(3) graphs

For the SU(3) system of graphs generalizing the ADE Dynkin digrams in the classification of modular invariant partition functions in CFT, we present a general collection of algebraic objects and relations that describe fusion properties and quantum symmetries associated with the corresponding Ocneanu quantum groupo\"{i}ds. We also summarize the properties of the individual members of this system.Comment: 36 page

Spectrum of a duality-twisted Ising quantum chain

The Ising quantum chain with a peculiar twisted boundary condition is considered. This boundary condition, first introduced in the framework of the spin-1/2 XXZ Heisenberg quantum chain, is related to the duality transformation, which becomes a symmetry of the model at the critical point. Thus, at the critical point, the Ising quantum chain with the duality-twisted boundary is translationally invariant, similar as in the case of the usual periodic or antiperiodic boundary conditions. The complete energy spectrum of the Ising quantum chain is calculated analytically for finite systems, and the conformal properties of the scaling limit are investigated. This provides an explicit example of a conformal twisted boundary condition and a corresponding generalised twisted partition function.Comment: LaTeX, 7 pages, using IOP style

From modular invariants to graphs: the modular splitting method

We start with a given modular invariant M of a two dimensional su(n)_k conformal field theory (CFT) and present a general method for solving the Ocneanu modular splitting equation and then determine, in a step-by-step explicit construction, 1) the generalized partition functions corresponding to the introduction of boundary conditions and defect lines; 2) the quantum symmetries of the higher ADE graph G associated to the initial modular invariant M. Notice that one does not suppose here that the graph G is already known, since it appears as a by-product of the calculations. We analyze several su(3)_k exceptional cases at levels 5 and 9.Comment: 28 pages, 7 figures. Version 2: updated references. Typos corrected. su(2) example has been removed to shorten the paper. Dual annular matrices for the rejected exceptional su(3) diagram are determine

A PCA-based automated finder for galaxy-scale strong lenses

We present an algorithm using Principal Component Analysis (PCA) to subtract galaxies from imaging data, and also two algorithms to find strong, galaxy-scale gravitational lenses in the resulting residual image. The combined method is optimized to find full or partial Einstein rings. Starting from a pre-selection of potential massive galaxies, we first perform a PCA to build a set of basis vectors. The galaxy images are reconstructed using the PCA basis and subtracted from the data. We then filter the residual image with two different methods. The first uses a curvelet (curved wavelets) filter of the residual images to enhance any curved/ring feature. The resulting image is transformed in polar coordinates, centered on the lens galaxy center. In these coordinates, a ring is turned into a line, allowing us to detect very faint rings by taking advantage of the integrated signal-to-noise in the ring (a line in polar coordinates). The second way of analysing the PCA-subtracted images identifies structures in the residual images and assesses whether they are lensed images according to their orientation, multiplicity and elongation. We apply the two methods to a sample of simulated Einstein rings, as they would be observed with the ESA Euclid satellite in the VIS band. The polar coordinates transform allows us to reach a completeness of 90% and a purity of 86%, as soon as the signal-to-noise integrated in the ring is higher than 30, and almost independent of the size of the Einstein ring. Finally, we show with real data that our PCA-based galaxy subtraction scheme performs better than traditional subtraction based on model fitting to the data. Our algorithm can be developed and improved further using machine learning and dictionary learning methods, which would extend the capabilities of the method to more complex and diverse galaxy shapes

From conformal embeddings to quantum symmetries: an exceptional SU(4) example

We briefly discuss several algebraic tools that are used to describe the quantum symmetries of Boundary Conformal Field Theories on a torus. The starting point is a fusion category, together with an action on another category described by a quantum graph. For known examples, the corresponding modular invariant partition function, which is sometimes associated with a conformal embedding, provides enough information to recover the whole structure. We illustrate these notions with the example of the conformal embedding of SU(4) at level 4 into Spin(15) at level 1, leading to the exceptional quantum graph E4(SU(4)).Comment: 22 pages, 3 color figures. Version 2: We changed the color of figures (ps files) in such a way that they are still understood when converted to gray levels. Version 3: Several references have been adde

Ocular Manifestations In COVID-19: Clinical Case Reports And A Literature (Review)

Coronavirus disease 2019 (COVID-19) has affected more than 130 million people as of April 2021. During the current pandemic, SARS-CoV-2 has been found to cause conjunctivitis with anecdotal evidence of a wide range of manifestations from scleritis to retinitis, occlusive vasculitis and optic neuritis. The purpose of this study is to raise awareness about possible COVID-19 related ocular manifestations, including papillophlebitis, anterior ischemic optic neuropathy, uveitis and neuroretinitis. Further detailed analyses will be needed to elucidate the link between SARS-CoV-2 and ocular pathology

Owning an overweight or underweight body: distinguishing the physical, experienced and virtual body

Our bodies are the most intimately familiar objects we encounter in our perceptual environment. Virtual reality provides a unique method to allow us to experience having a very different body from our own, thereby providing a valuable method to explore the plasticity of body representation. In this paper, we show that women can experience ownership over a whole virtual body that is considerably smaller or larger than their physical body. In order to gain a better understanding of the mechanisms underlying body ownership, we use an embodiment questionnaire, and introduce two new behavioral response measures: an affordance estimation task (indirect measure of body size) and a body size estimation task (direct measure of body size). Interestingly, after viewing the virtual body from first person perspective, both the affordance and the body size estimation tasks indicate a change in the perception of the size of the participant’s experienced body. The change is biased by the size of the virtual body (overweight or underweight). Another novel aspect of our study is that we distinguish between the physical, experienced and virtual bodies, by asking participants to provide affordance and body size estimations for each of the three bodies separately. This methodological point is important for virtual reality experiments investigating body ownership of a virtual body, because it offers a better understanding of which cues (e.g. visual, proprioceptive, memory, or a combination thereof) influence body perception, and whether the impact of these cues can vary between different setups