Conformal Field Research Articles

Conformal Yang-Mills field in (A)dS space

Ordinary-derivative (second-derivative) Lagrangian formulation of classical conformal Yang-Mills field in the (A)dS space of six, eight, and ten dimensions is developed. For such conformal field, we develop two gauge invariant Lagrangian formulations which we refer to as generic formulation and decoupled formulation. In both formulations, the usual Yang-Mills field is accompanied by additional vector and scalar fields where the scalar fields are realized as Stueckelberg fields. In the generic formulation, the usual Yang-Mills field is realized as a primary field, while the additional vector fields are realized as auxiliary fields. In the decoupled formulation, the usual Yang-Mills field is realized as massless field, while the additional vector fields together with the Stueckelberg are realized as massive fields. Some massless/massive fields appear with the wrong sign of kinetic terms, hence demonstrating explicitly that the considered models are not unitary. The use of embedding space method allows us to treat the isometry symmetries of (A)dS space manifestly and obtain conformal transformations of fields in a relatively straightforward way. By accompanying each vector field by the respective gauge parameter, we introduce an extended gauge algebra. Levy-Maltsev decomposition of such algebra is noted. Use of the extended gauge algebra setup allows us to present concise form for the Lagrangian and gauge transformations of the conformal Yang-Mills field. Higher-derivative representation of the Lagrangian is also obtained.

4D Weyl anomaly and diversity of the interior structure of quantum black hole

We study the interior metric of 4D spherically symmetric static black holes by using the semi-classical Einstein equation and find a consistent class of geometries with large curvatures. We approximate the matter fields by conformal fields and consider the contribution of the 4D Weyl anomaly, giving a state-independent constraint. Combining this with an equation of state yields an equation that determines the interior geometry completely. We explore the solution space of the equation in a non-perturbative manner for ħ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar $$\\end{document}. First, we find four types of asymptotic behaviors and examine the general features of the solutions. Then, by imposing physical conditions, we obtain approximately a general class of interior geometries: various combinations of dilute and dense structures without a horizon or singularity. This represents the diversity of the interior structure. Finally, we show that the number of possible patterns of such interior geometries corresponds to the Bekenstein–Hawking entropy.

Proton Therapy for Ocular Melanoma: A Dosimetric Comparison of Three Different Delivery Systems

The purpose of this study is to compare the treatment plans for intraocular tumors with two types of Pencil Beam Scanning (PBS) machine configurations and compare them with the widely used uniform scanning (US)/scattering technique. Two virtual PBS machines were constructed for ocular planning. The first machine (P1) consisted of a standard spot size and a 7.5cm WET range shifter (RS) positioned 50cm upstream from the aperture. The second machine (P2) had a smaller spot size and used a 4cm WET RS but had no aperture support. These PBS delivery systems resemble currently available models. Ten patients with different locations and sizes of intraocular tumors were planned with US using 2-3 conformal fields following our institution's intraocular criteria. PBS spot patterns for each machine were optimized using the same beam geometry as the US plan to achieve identical tumor coverage. PBS distribution was calculated using Monte Carlo. All air gaps were minimized as small as clinically achievable. PBS plans passed robustness criteria with all scenarios meeting CTV D95>95% with 2mm translational offsets and 3.5% range uncertainty. The average conformity index (CI) of 95% reference isodose was better for both PBS techniques (P1: 1.43±0.22, P2:1.5±0.32) versus US (1.72±0.56) for all plans (p = 0.025, p = 0.034). The average CI of 50% reference isodose was worse for the P2 plans (10.4±3.3) compared to US (9.34±2.93). However, CI 50% for P1 plans was better than US (5.21±1.24, p<0.003). There were no significant improvements to OARs using P2. However, the mean dose to anterior structures ciliary body, lacrimal gland, and D0.03cc lens were significantly lower in the P1 plans than in the US plans (p<0.016). Small spot size PBS systems without apertures can achieve similar coverage to US plans but have higher OAR dose. PBS systems with a standard spot size and reconfigured RS and apertures are better for anterior OAR sparing and is non-inferior to US planning of intraocular targets.

Linearised conformal Einstein field equations

The linearisation of a second-order formulation of the conformal Einstein field equations (CEFEs) in generalised harmonic gauge (GHG), with trace-free matter is derived. The linearised equations are obtained for a general background and then particularised for the study linear perturbations around a flat background—the inversion (conformal) representation of the Minkowski spacetime—and the solutions discussed. We show that the generalised Lorenz gauge (defined as the linear analogue of the GHG-gauge) propagates. Moreover, the equation for the conformal factor can be trivialised with an appropriate choice for the gauge source functions; this permits a scri-fixing strategy using gauge source functions for the linearised wave-like CEFE-GHG, which can in principle be generalised to the nonlinear case. As a particular application of the linearised equations, the far-field and compact source approximation is employed to derive quadrupole-like formulae for various conformal fields such as the perturbation of the rescaled Weyl tensor.

A new method of generating the lump molecules and localized interaction solutions to the (2+1)-dimensional SK equation

The (2+1)-dimensional Sawada-Kotera (SK) equation is an important integrable model, which has wide applications in rivers, lakes, atmosphere, the conformal field and quantum gravity gauge field. In this paper, we introduce a new method for constructing the lump molecules in the SK equation. Then, by using this method and imposing the complexification restrictions, velocity resonant principle, we obtain some hybrid wave solutions, which include the interactions between a lump molecule and two separated solitons, between a lump-lump-soliton molecule and a single soliton, between a lump molecule and a soliton molecule, between a lump molecule and a breather, as well as the lump-lump-breather molecule. Dynamical behaviors of these solutions are analyzed theoretically and graphically. The method introduced can be effectively used to study the wave solutions of other nonlinear PDEs. The results obtained not only enrich the types of soliton molecules, but also can be helpful in the study of the propagation behaviors of nonlinear waves.

Dosimetric effects of embolization material artefacts in arteriovenous malformations stereotactic radiosurgery on treatment planning calculation.

Background and PurposeStereotactic Radiosurgery (SRS) is a specialized radiotherapy treatment technique for Arteriovenous Malformations (AVM) in which Computed Tomography (CT) images are used for dose calculations. The purpose of this study was to investigate CT image distortions caused by embolic agents and quantify the influence of these distortions on dose calculations.MethodsEight AVM patients administered embolic agents prior to SRS were included. Original plans were compared to new recalculated plans using two sets of images. The first set was created by masking the embolic material and artefacts, the second was the diagnostic CT images. In addition, treatment plans were created for an anthropomorphic phantom with water inserts, then with known volumes of embolic materials to study the dosimetric effect of each material.ResultsRelative to patients’ original plans, maximum Monitor Unit (MU) difference was −4.4% with whole brain masking, −1.3% with artefact masking, −4.1% with embolic masking, and −4.5% with artefact-free diagnostic images. Calculated dose differences were within ± 3.5% for all plans. In phantom, Gamma pass rate was 96% for both embolic agents with conformal fields and 99.9% with dynamic arcs. Dose and MU differences in phantom plans were negligible.ConclusionRelative dose differences between the original plans and the corrected ones were not clinically remarkable. We recommend evaluating the effect of embolic materials on individual patients’ plans. The whole brain corrected planning CT images or diagnostic CT images could be utilized to calculate the magnitude of dose reduction caused by embolic materials and correct it if necessary.

Evaluation of implicit solvent models in molecular dynamics simulation of α-Synuclein

We report conventional and accelerated molecular dynamics simulations of α-Synuclein, designed to assess performance of using different starting conformation, solvation environment and force field combination. Backbone and sidechain chemical shifts, radius of gyration, presence of β-hairpin structures in KTK(E/Q)GV repeats and secondary structure percentages were used to evaluate how variations in forcefield, solvation model and simulation protocol provide results that correlate with experimental findings. We show that with suitable choice of forcefield and solvent, ff03ws and OBC implicit model, respectively, acceptable reproduction of experimental data on size and secondary structure is obtained by both conventional and accelerated MD. In contrast to the implicit solvent model, simulations in explicit TIP4P/2005 solvent do not properly represent size or secondary structure of α-Synuclein. Communicated by Ramaswamy H. Sarma

Some characterizations of spheres by conformal vector fields

Some characterizations of spheres by conformal vector fields

Almost Kenmotsu (k,μ)′-manifolds of dimension three and conformal vector fields

In this paper, we prove that the only almost Kenmotsu [Formula: see text]-manifold of dimension three admitting a non-Killing conformal vector field is the product space [Formula: see text]. All conformal vector fields on this product are determined.

Conformal fields and the structure of the space of solutions of the Einstein constraint equations

The drift method, introduced by the second author, provides a new formulation of the Einstein constraint equations, either in vacuum or with matter fields. The natural of the geometry underlying this method compensates for its slightly greater analytic complexity over, say, the conformal or conformal thin sandwich methods. We review this theory here and apply it to the study of solutions of the constraint equations with non-constant mean curvature. We show that this method reproduces previously known existence results obtained by other methods, and does better in one important regard. Namely, it can be applied even when the underlying metric admits conformal Killing (but not true Killing) vector fields. We also prove that the absence of true Killing fields holds generically.

Heart atlas for retrospective cardiac dosimetry: a multi-institutional study on interobserver contouring variations and their dosimetric impact

PurposeCardiac effects after breast cancer radiation therapy potentially affect more patients as survival improves. The heart’s heterogeneous radiation exposure and composition of functional structures call for establishing individual relationships between structure dose and specific late effects. However, valid dosimetry requires reliable contouring which is challenging for small volumes based on older, lower-quality computed tomography imaging. We developed a heart atlas for robust heart contouring in retrospective epidemiologic studies.Methods and materialsThe atlas defined the complete heart and geometric surrogate volumes for six cardiac structures: aortic valve, pulmonary valve, all deeper structures combined, myocardium, left anterior myocardium, and right anterior myocardium. We collected treatment planning records from 16 patients from 4 hospitals including dose calculations for 3D conformal tangential field radiation therapy for left-sided breast cancer. Six observers each contoured all patients. We assessed spatial contouring agreement and corresponding dosimetric variability.ResultsContouring agreement for the complete heart was high with a mean Jaccard similarity coefficient (JSC) of 89%, a volume coefficient of variation (CV) of 5.2%, and a mean dose CV of 4.2%. The left (right) anterior myocardium had acceptable agreement with 63% (58%) JSC, 9.8% (11.5%) volume CV, and 11.9% (8.0%) mean dose CV. Dosimetric agreement for the deep structures and aortic valve was good despite higher spatial variation. Low spatial agreement for the pulmonary valve translated to poor dosimetric agreement.ConclusionsFor the purpose of retrospective dosimetry based on older imaging, geometric surrogate volumes for cardiac organs at risk can yield better contouring agreement than anatomical definitions, but retain limitations for small structures like the pulmonary valve.

Conformal vector fields of Bianchi type-I spacetimes

This paper aims to investigate Conformal Vector Fields (CVFs) of Bianchi type-I spacetimes. A set of 10-coupled Partial Differential Equations (PDEs) is obtained from the conformal Killing equations. These equations are solved by using direct integration techniques to explore the components of CVFs. Utilizing these components, we get a system of three integrability conditions. Finally, we achieve CVFs along with conformal factors for unique possibilities of unknown metric functions from the solution of these conditions. From our results, it is examined that Bianchi type-I spacetimes admit five or fifteen CVFs for specific choices of metric functions.

Analytic conformal bootstrap and Virasoro primary fields in the Ashkin-Teller model

We revisit the critical two-dimensional Ashkin–Teller model, i.e. the \mathbb{Z}_2ℤ2 orbifold of the compactified free boson CFT at c=1c=1. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at c=1c=1. We show that blocks with a degenerate channel field should be computed by taking limits in the central charge, rather than in the conformal dimension. In particular, Al. Zamolodchikov’s simple explicit expression for the blocks that appear in four-twist correlation functions is only valid in the non-degenerate case: degenerate blocks, starting with the identity block, are more complicated generalized theta functions.

Crossing-symmetric twist field correlators and entanglement negativity in minimal CFTs

We study conformal twist field four-point functions on a ℤN orbifold. We examine in detail the case N = 3 and analyze theories obtained by replicated N-times a minimal model with central charge c < 1. A fastly convergent expansion of the twist field correlation function in terms of sphere conformal blocks with central charge Nc is obtained by exploiting covering map techniques. We discuss extensive applications of the formalism to the entanglement of two disjoint intervals in CFT, in particular we propose a conformal block expansion for the partially transposed reduced density matrix. Finally, we refine the bounds on the structure constants of unitary CFTs determined previously by the genus two modular bootstrap.

Exploring conformal vector fields of Bianchi type-I perfect fluid solutions in f(T) gravity

In this paper, first, we deduce various classes of Bianchi type-I perfect fluid solutions in the [Formula: see text] gravity. In order to obtain the above-mentioned classes, we use some algebraic techniques that help to formulate 18 cases. As an application, we bifurcate the obtained classes according to their conformal vector fields (CVFs). Inspecting each class using the process of direct integration, we come to know that in one case, the space-time admits proper CVFs whereas in rest of the cases, obtained metrics either become conformally flat or admit underlying symmetries of the CVFs i.e. homothetic vector fields (HVFs) or Killing vector fields (KVFs). The overall dimension of CVFs for the Bianchi type-I perfect fluid space-times in [Formula: see text] gravity turns out to be three, four, five, six and fifteen.

High‐gain omnidirectional patch antenna for conformal application based on near‐zero‐index metamaterials

Achieving a high-gain omnidirectional patch antenna with low profile is a challenging task. On the basis of improving antennas by metamaterials, a broadband near-zero-index metamaterial (NZIM) is designed to enhance the horizontal gain of the omnidirectional patch antenna. The zero-index metamaterial is loaded on the substrate of three antennas with different working bands (4.99–5.50 GHz). The results show that the horizontal gain of antennas is enhanced by more than 0.6 dB at various frequencies. The roundness of antennas is also optimised. Therefore, using NZIM can increase the horizontal gain with a low profile (λ/30) structure size. The low profile, high-gain omnidirectional patch antenna has great potential in the conformal field.

The Clinical, Philosophical, Evolutionary and Mathematical Machinery of Consciousness: An Analytic Dissection of the Field Theories and a Consilience of Ideas.

The Cartesian model of mind-body dualism concurs with religious traditions. However, science has supplanted this idea with an energy-matter theory of consciousness, where matter is equivalent to the body and energy replaces the mind or soul. This equivalency is analogous to the concept of the interchange of mass and energy as expressed by Einstein’s famous equation E=mc^{2}. Immanuel Kant, in his Critique of Pure Reason, provided the intellectual and theoretical framework for a theory of mind or consciousness. Any theory of consciousness must include the fact that a conscious entity, as far as is known, is a wet biological medium (the brain), of stupendously high entropy. This organ or entity generates a field that must account for the "binding problem", which we will define. This proposed field, the conscious electro-magnetic information (CEMI) field, also has physical properties, which we will outline. We will also demonstrate the seamless transition of the Kantian philosophy of the a priori conception of space and time, the organs of perception and conception, into the CEMI field of consciousness. We will explore the concept of the CEMI field and its neurophysiological correlates, and in particular, synchronous and coherent gamma oscillations of various neuronal ensembles, as in William J Freeman's experiments in the early 1970s with olfactory perception in rabbits.The expansion of the temporo-parietal-occipital (TPO) cortex in hominid evolution epitomizes metaphorical and abstract thinking. This area of the cortex, with synchronous thalamo-cortical oscillations has the best fit for a minimal neural correlate of consciousness. Our field theory shifts consciousness from an abstract idea to a tangible energy with defined properties and a mathematical framework.Even further, it is not a coincidence that the cerebral cortex is very thin with respect to the diameter of the brain. This is in keeping with its fantastically high entropy, as we see in the event horizon of a black hole and the conformal field theory/anti-de Sitter (CFT/ADS) holographic model of the universe.We adumbrate the uniqueness of consciousness of an advanced biological system such as the human brain and draw insight from Avicenna’s gendanken, floating man thought experiment. The multi-system high volume afferentation of a biological wet system honed after millions of years of evolution, its high entropy, and the CEMI field variation inducing currents in motor output pathways are proposed to spark the seeds of consciousness.We will also review Karl Friston's free energy principle, the concept of belief-update in a Bayesian inference framework, the minimization of the divergence of prior and posterior probability distributions, and the entropy of the brain. We will streamline these highly technical papers, which view consciousness as a minimization principle akin to Hilbert's action in deriving Einstein's field equation or Feynman's sum of histories in quantum mechanics. Consciousness here is interpreted as flow of probability densities on a Riemmanian manifold, where the gradient of ascent on this manifold across contour lines determines the magnitude of perception or the degree of update of the belief-system in a Bayesian inference model.Finally, the science of consciousness has transcended metaphysics and its study is now rooted in the latest advances of neurophysiology, neuro-radiology under the aegis of mathematics.

The superconformal equation

Crossing symmetry provides a powerful tool to access the non-perturbative dynamics of conformal and superconformal field theories. Here we develop the mathematical formalism that allows to construct the crossing equations for arbitrary four-point functions in theories with superconformal symmetry of type I, including all superconformal field the- ories in d = 4 dimensions. Our advance relies on a supergroup theoretic construction of tensor structures that generalizes an approach which was put forward in [1] for bosonic theories. When combined with our recent construction of the relevant superblocks, we are able to derive the crossing symmetry constraint in particular for four-point functions of arbitrary long multiplets in all 4-dimensional superconformal field theories.

Critical point equation and closed conformal vector fields

AbstractIn this article, we study the critical points of the total scalar curvature functional restricted to the space of metrics with constant scalar curvature of unitary volume, for simplicity, CPE metrics. Here, we prove that a CPE metric admitting a non‐trivial closed conformal vector field must be isometric to a round sphere metric, which provides a partial answer to the CPE conjecture.

Conformal vector fields of some vacuum classes of static spherically symmetric space-times in f(T,B) gravity

The aim of this paper is to investigate the conformal vector fields (CVFs) for some vacuum classes of static spherically symmetric space-times in [Formula: see text] gravity. First, we have explored the space-times by solving the Einstein field equations in [Formula: see text] gravity. These solutions have been obtained by imposing various conditions on the space-time components and selecting separable form of the bivariate function [Formula: see text]. Second, we find the CVFs of the obtained space-times via direct integration approach. The overall study reveals that there exist 17 cases. From these 17 cases, the space-times in five cases admit proper CVFs whereas in rest of the 12 cases, CVFs become Killing vector fields (KVFs). We have also calculated the torsion scalar and boundary term for each of the obtained solutions.