Generalised twisted partition functions

We consider the set of partition functions that result from the insertion of twist operators compatible with conformal invariance in a given 2D Conformal Field Theory (CFT). A consistency equation, which gives a classification of twists, is written and solved in particular cases. This generalises old results on twisted torus boundary conditions, gives a physical interpretation of Ocneanu's algebraic construction, and might offer a new route to the study of properties of CFT.Comment: 12 pages, harvmac, 1 Table, 1 Figure . Minor typos corrected, the figure which had vanished reappears

Conformal Boundary Conditions and what they teach us

The question of boundary conditions in conformal field theories is discussed, in the light of recent progress. Two kinds of boundary conditions are examined, along open boundaries of the system, or along closed curves or ``seams''. Solving consistency conditions known as Cardy equation is shown to amount to the algebraic problem of finding integer valued representations of (one or two copies of) the fusion algebra. Graphs encode these boundary conditions in a natural way, but are also relevant in several aspects of physics ``in the bulk''. Quantum algebras attached to these graphs contain information on structure constants of the operator algebra, on the Boltzmann weights of the corresponding integrable lattice models etc. Thus the study of boundary conditions in Conformal Field Theory offers a new perspective on several old physical problems and offers an explicit realisation of recent mathematical concepts.Comment: Expanded version of lectures given at the Summer School and Conference Nonperturbative Quantum Field Theoretic Methods and their Applications, August 2000, Budapest, Hungary. 35 page

Sparse dimensionality reduction approaches in Mendelian randomization with highly correlated exposures.

Multivariable Mendelian randomization (MVMR) is an instrumental variable technique that generalizes the MR framework for multiple exposures. Framed as a linear regression problem, it is subject to the pitfall of multi-collinearity. The bias and efficiency of MVMR estimates thus depends heavily on the correlation of exposures. Dimensionality reduction techniques such as principal component analysis (PCA) provide transformations of all the included variables that are effectively uncorrelated. We propose the use of sparse PCA (sPCA) algorithms that create principal components of subsets of the exposures with the aim of providing more interpretable and reliable MR estimates. The approach consists of three steps. We first apply a sparse dimension reduction method and transform the variant-exposure summary statistics to principal components. We then choose a subset of the principal components based on data-driven cutoffs, and estimate their strength as instruments with an adjusted F-statistic. Finally, we perform MR with these transformed exposures. This pipeline is demonstrated in a simulation study of highly correlated exposures and an applied example using summary data from a genome-wide association study of 97 highly correlated lipid metabolites. As a positive control, we tested the causal associations of the transformed exposures on CHD. Compared to the conventional inverse-variance weighted MVMR method and a weak-instrument robust MVMR method (MR GRAPPLE), sparse component analysis achieved a superior balance of sparsity and biologically insightful grouping of the lipid traits

Identifying and ranking causal biochemical biomarkers for breast cancer: a Mendelian randomisation study

Background: Only a few of the 34 biochemical biomarkers measured in the UK Biobank (UKB) have been associated with breast cancer, with many associations suffering from possible confounding and reverse causation. This study aimed to screen and rank all UKB biochemical biomarkers for possible causal relationships with breast cancer. Methods: We conducted two-sample Mendelian randomisation (MR) analyses on ~420,000 women by leveraging summary-level genetic exposure associations from the UKB study (n = 194,174) and summary-level genetic outcome associations from the Breast Cancer Association Consortium (n = 228,951). Our exposures included all 34 biochemical biomarkers in the UKB, and our outcomes were overall, oestrogen-positive, and oestrogen-negative breast cancer. We performed inverse-variance weighted MR, weighted median MR, MR-Egger, and MR-PRESSO for 30 biomarkers for which we found multiple instrumental variables. We additionally performed multivariable MR to adjust for known risk factors, bidirectional MR to investigate reverse causation, and MR Bayesian model averaging to rank the significant biomarkers by their genetic evidence. Results: Increased genetic liability to overall breast cancer was robustly associated with the following biomarkers by decreasing importance: testosterone (odds ratio (OR): 1.12, 95% confidence interval (CI): 1.04–1.21), high-density lipoprotein (HDL) cholesterol (OR: 1.08, 95% CI: 1.04–1.13), insulin-like growth factor 1 (OR: 1.08, 95% CI: 1.02–1.13), and alkaline phosphatase (ALP) (OR: 0.93, 95% CI: 0.89–0.98). Conclusions Our findings support a likely causal role of genetically predicted levels of testosterone, HDL cholesterol, and IGF-1, as well as a novel potential role of ALP in breast cancer aetiology. Further studies are needed to understand full disease pathways that may inform breast cancer prevention

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 new approach to the inverse problem for current mapping in thin-film superconductors

A novel mathematical approach has been developed to complete the inversion of the Biot-Savart law in one- and two-dimensional cases from measurements of the perpendicular component of the magnetic field using the well-developed Magneto-Optical Imaging technique. Our approach, especially in the 2D case, is provided in great detail to allow a straightforward implementation as opposed to those found in the literature. Our new approach also refines our previous results for the 1D case [Johansen et al., Phys. Rev. B 54, 16264 (1996)], and streamlines the method developed by Jooss et al. [Physica C 299, 215 (1998)] deemed as the most accurate if compared to that of Roth et al. [J. Appl. Phys. 65, 361 (1989)]. We also verify and streamline the iterative technique, which was developed following Laviano et al. [Supercond. Sci. Technol. 16, 71 (2002)] to account for in-plane magnetic fields caused by the bending of the applied magnetic field due to the demagnetising effect. After testing on magneto-optical images of a high quality YBa2Cu3O7 superconducting thin film, we show that the procedure employed is effective