Conformal Field Theories in Fractional Dimensions

We study the conformal bootstrap in fractional space-time dimensions, obtaining rigorous bounds on operator dimensions. Our results show strong evidence that there is a family of unitary CFTs connecting the 2D Ising model, the 3D Ising model, and the free scalar theory in 4D. We give numerical predictions for the leading operator dimensions and central charge in this family at different values of D and compare these to calculations of phi^4 theory in the epsilon-expansion.Comment: 11 pages, 4 figures - references updated - one affiliation modifie

Bootstrapping 3D Fermions with Global Symmetries

We study the conformal bootstrap for 4-point functions of fermions $\langle \psi_i \psi_j \psi_k \psi_{\ell} \rangle$ in parity-preserving 3d CFTs, where $\psi_i$ transforms as a vector under an $O(N)$ global symmetry. We compute bounds on scaling dimensions and central charges, finding features in our bounds that appear to coincide with the $O(N)$ symmetric Gross-Neveu-Yukawa fixed points. Our computations are in perfect agreement with the $1/N$ expansion at large $N$ and allow us to make nontrivial predictions at small $N$. For values of $N$ for which the Gross-Neveu-Yukawa universality classes are relevant to condensed-matter systems, we compare our results to previous analytic and numerical results.Comment: 29 pages, 7 figure

Gas sensing based on optical fibre coupled diode laser spectroscopy : a new approach to sensor systems for safety monitoring

We describe an entirely passive fibre optic network which senses, amongst other species, CH¬4¬ and CO¬¬2 , with sensitivity and selectivity compatible with safety sensing in the mine environment. The basic principle is that a single laser diode source targeted to a particular species addresses up to 200 sensing points which may be spread over an area of dimensions ten or more km. The detection and processing electronics is typically located with the laser source. Several laser sources can be introduced in parallel to enable monitoring multiple species. The network itself, entirely linked through optical fibre, is inherently intrinsically safe. It is self checking for faults at the sensing location and continuously self calibrating. In the methane sensing mode its sensitivity is sub 100ppm and it responds accurately up to 100% methane. It is therefore capable of detecting extremely hazardous gas pockets which are completely missed by other sensor technologies. The network has demonstrated stability with zero maintenance or recalibration over periods in excess of two years. We believe that this system offers unique benefits in the context of mine safety and ventilation system monitoring

Bootstrapping 3D Fermions

We study the conformal bootstrap for a 4-point function of fermions $\langle\psi\psi\psi\psi\rangle$ in 3D. We first introduce an embedding formalism for 3D spinors and compute the conformal blocks appearing in fermion 4-point functions. Using these results, we find general bounds on the dimensions of operators appearing in the $\psi \times \psi$ OPE, and also on the central charge $C_T$. We observe features in our bounds that coincide with scaling dimensions in the Gross-Neveu models at large $N$. We also speculate that other features could coincide with a fermionic CFT containing no relevant scalar operators.Comment: 45 pages, 8 figures; V2: added references and small clarifications to match JHEP versio

Fermion-Scalar Conformal Blocks

We compute the conformal blocks associated with scalar-scalar-fermion-fermion 4-point functions in 3D CFTs. Together with the known scalar conformal blocks, our result completes the task of determining the so-called `seed blocks' in three dimensions. Conformal blocks associated with 4-point functions of operators with arbitrary spins can now be determined from these seed blocks by using known differential operators.Comment: 25 pages; V2: added small clarifications to match JHEP versio

Chronic ketamine use and psychotic symptomatology.

This thesis examines the effects of chronic use of ketamine, a non-competitive N-methyl-D-aspartate (NMDA) receptor antagonist, on subjective experience and cognition. It is important to explore the chronic effects of ketamine as the number of individuals using the drug recreationally is increasing both in the UK and worldwide. There is a paucity of research exploring the chronic effects of ketamine. Many studies have shown that acutely ketamine induces psychotic-like symptomatology and specific cognitive dysfunction in healthy, drug-naive volunteers. For this reason, a ketamine model of the psychoses has been proposed. However the few studies of the effects of chronic ketamine have provided mixed findings. Part 1 of the thesis comprises a literature review, which investigates the psychotomimetic effects of ketamine, through the synthesis of current research findings, to determine whether ketamine is a useful model of the symptomatology characteristic of the psychoses. It presents an overview of ketamine and its association with the psychoses, before providing a detailed account of the functional psychoses and drug models of the psychoses (namely the dopamine hypothesis, the serotonin hypothesis and the glutamate hypothesis). The review then synthesises the acute and chronic ketamine studies to date, highlighting which states appear to be best modelled (i.e. the pre-psychotic, acute or chronic state experienced by individuals with idiopathic psychoses). Finally, the review briefly considers the treatment implications of the ketamine model of psychoses, and the risk chronic ketamine use poses to users in terms of developing fully-manifest psychotic symptomatology. In Part 2, an investigation of the chronic effects of ketamine on subjective experiences and cognitive functioning is reported, in order to determine whether chronic ketamine models symptomatology associated with the pre-psychotic state of idiopathic psychoses (where the term idiopathic refers to psychotic symptomatology of unknown aetiology, i.e. that which occurs in the majority of the general population and is not drug-induced). This investigation was part of a joint project conducted with 2 other trainees to investigate the chronic effects of ketamine, cannabis and cocaine on subjective experiences and cognitive functioning (See Appendix 1 for details of the contribution made by each trainee). The empirical paper reports a between subjects study which compared 21 frequent ketamine users (who used ketamine daily), 20 infrequent ketamine users (who used ketamine a maximum of once or twice a week) and 20 controls (who reported no illicit drug use). On a clinical index of symptomatology (SPI-A), a 'frequency' effect was observed: frequent ketamine users were found to be higher in psychotic-like symptomatology (i.e. basic symptoms) than infrequent users, who in turn were found to be higher in symptomatology than controls. Both groups of ketamine users were also found to be higher in psychosis proneness on a general population index of psychotic-like markers (OLIFE) compared with controls. Furthermore, both groups of ketamine users demonstrated impaired episodic memory and working memory compared to controls. Group differences were found in executive functioning. Part 3 comprises a critical appraisal of the research. It includes reflections on my experience of the research process and conducting research with the ketamine using population, as well as reflections on clinically relevant observations

Discrete complex analysis on planar quad-graphs

We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph yields more instructive proofs of discrete analogs of several classical theorems and even new results. We provide discrete counterparts of fundamental concepts in complex analysis such as holomorphic functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss discrete versions of important basic theorems such as Green's identities and Cauchy's integral formulae. For the first time, we discretize Green's first identity and Cauchy's integral formula for the derivative of a holomorphic function. In this paper, we focus on planar quad-graphs, but we would like to mention that many notions and theorems can be adapted to discrete Riemann surfaces in a straightforward way. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths, we construct a discrete Green's function and discrete Cauchy's kernels with asymptotics comparable to the smooth case. Further restricting to the integer lattice of a two-dimensional skew coordinate system yields appropriate discrete Cauchy's integral formulae for higher order derivatives.Comment: 49 pages, 8 figure

A discrete Laplace-Beltrami operator for simplicial surfaces

We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called ``cotan formula'') except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace-Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces.Comment: 18 pages, 6 vector graphics figures. v2: Section 2 on Delaunay triangulations of piecewise flat surfaces revised and expanded. References added. Some minor changes, typos corrected. v3: fixed inaccuracies in discussion of flip algorithm, corrected attributions, added references, some minor revision to improve expositio

Functional Maps Representation on Product Manifolds

We consider the tasks of representing, analyzing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace--Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.Comment: Accepted to Computer Graphics Foru