Conformal Structure Research Articles

Conformal classes of Lorentzian surfaces with Killing fields

AbstractWe study the conformal classes of two‐dimensional Lorentzian tori with (nonzero) Killing fields. We define a map that associate to such a class a vector field on the circle (up to a scalar factor). This map is not injective but has finite‐dimensional fiber. It allows us to characterize the conformal classes of tori with Killing field satisfying a condition related to the existence of conjugate points given by Mehidi.

Non-invertible symmetry in Calabi-Yau conformal field theories

We construct examples of non-invertible global symmetries in two-dimensional superconformal field theories described by sigma models into Calabi-Yau target spaces. Our construction provides some of the first examples of non-invertible symmetry in irrational conformal field theories. Our approach begins at a Gepner point in the conformal manifold where the sigma model specializes to a rational conformal field theory and we can identify all supersymmetric topological Verlinde lines. By deforming away from this special locus using exactly marginal operators, we then identify submanifolds in moduli space where some non-invertible symmetry persists. For instance, along ten-dimensional loci in the complex structure moduli space of quintic Calabi-Yau threefolds there is a symmetry characterized by a Fibonacci fusion category. The symmetries we identify provide new constraints on spectra and correlation functions. As an application we show how they constrain conformal perturbation theory, consistent with recent results about scaling dimensions in the K3 sigma model near its Gepner point.

Bandwidth enhancement of resonating absorber using a lossy dielectric layer for RCS reduction in X-band

Abstract In this paper, a resonating absorber is proposed for radar cross section reduction in the X-band, with the inclusion of a high loss dielectric layer. The absorber consists of a pair of co-centered resonating Jerusalem cross printed on a FR4 substrate cascaded with a high-loss dielectric layer and a metallic back plane. The designed absorber offers 90 % absorption from 7.8–12 GHz with a fractional bandwidth of 42.42 %. The designed absorber is 0.106λ L thick with a compact square unit cell of size 0.076λ L (wavelength at the lowest frequency). An equivalent circuit model is developed to analyze the proposed absorber. The absorber is angularly stable for varying angle of incidences up to 40°–60° for TE and TM polarizations respectively. The planar as well as the conformal structure of the proposed absorber offer a minimum 10 dB radar cross section reduction in the X-band. The conventional approach which involves complex fabrication process of soldering thousands of lumped resistors to increase the bandwidth, whereas the proposed absorber is realized to offer wide absorption using simple PCB etching technique. To validate the design prototype is fabricated and the measurement is carried out.

Wideband Circularly Polarized Conformal Dielectric Resonator Antenna for High Gain Applications

ABSTRACTA compact and circularly polarized convex dielectric resonator antenna (CDRA) is investigated in this communication. Eigenmode method is used to analyze electromagnetic field propagation and resonant frequency of the antenna. Benefits of conformal structure over a planar surface are mentioned. A pair of vertical slots are introduced on the ground plane to produce orthogonal degenerate modes. Finally, the proposed curved DRA provides a fractional impedance bandwidth of 22%, axial ratio bandwidth of 11.27% and a stable gain of more than 5 dBi over the desired passband with a peak gain of 7 dBi. To validate the simulated results, a prototype is fabricated and measured.

Meat yields and primal cut weights from beef carcasses can be predicted with similar accuracies using in-abattoir 3D measurements or EUROP classification grade.

Three-dimensional (3D) measurements extracted from beef carcass images were used to predict the weight of four saleable meat yield (SMY) traits (total SMY and the SMY of the forequarter, flank, and hindquarter) and four primal cuts (sirloin, ribeye, topside and rump). Data were collected at two UK abattoirs using time-of-flight cameras and manual bone out methods. Predictions were made for 484 carcasses, using multiple linear regression (MLR) or machine learning (ML) techniques. Model inputs included breed type, sex, and abattoir as fixed effects, and cold carcass weight, visually assessed EUROP fat and conformation classes, and 3D measurements as covariates. Machine learning techniques were only used for models including 3D measurements. The CCW and fixed effects resulted in high accuracy (SMY R2=0.72-0.90, RMSE=2.12-3.96kg, primal R2=0.56-0.67, RMSE=0.36-0.91kg), and including the EUROP covariates increased accuracies (SMY R2=0.75-0.96, RMSE=2.00-3.11kg, primal R2=0.58-0.79, RMSE=0.36-0.79kg). The 3D measurement covariates and abattoir resulted in moderate accuracy (SMY MLR R2=0.39-0.58, RMSE=3.26-10.31kg, primal MLR R2=0.33-0.52, RMSE=0.44-1.14kg) and high accuracy when combined with CCW and all fixed effects (SMY MLR R2=0.72-0.95, RMSE=1.81-3.42kg, primal MLR R2=0.52-0.74, RMSE=0.40-0.81kg). The best ML models resulted in similar accuracies to the MLR models. Models including 3D measurements produced similar accuracies to models built using conventional data recorded at the abattoir, indicting the potential for automated prediction.

Tensor categories of weight modules of sl̂2$\widehat{\mathfrak {sl}}_2$ at admissible level

AbstractThe category of weight modules of the simple affine vertex algebra of at an admissible level is neither finite nor semisimple and modules are usually not lower‐bounded and have infinite‐dimensional conformal weight subspaces. However, this vertex algebra enjoys a duality with , the simple principal ‐algebra of at level (the super conformal algebra) where the levels are related via . Every weight module of is lower‐bounded and has finite‐dimensional conformal weight spaces. The main technical result is that every weight module of is ‐cofinite. The existence of a vertex tensor category follows and the theory of vertex superalgebra extensions implies the existence of vertex tensor category structure on for any admissible level . As applications, the fusion rules of ordinary modules with any weight module are computed, and it is shown that is a ribbon category if and only if is, in particular, it follows that for admissible levels and and , the category is a ribbon category.

SLE Partition Functions via Conformal Welding of Random Surfaces

Abstract Schramm–Loewner evolution (SLE) curves describe the scaling limit of interfaces from many 2D lattice models. Heuristically speaking, the SLE partition function is the continuum counterpart of the partition function of the corresponding discrete model. It is well known that conformally welding of Liouville quantum gravity (LQG) surfaces gives SLE curves as the interfaces. In this paper, we demonstrate in several settings how the SLE partition function arises from conformal welding of LQG surfaces. The common theme is that we conformally weld a collection of canonical LQG surfaces which produces a topological configuration with a random conformal structure. Conditioning on the conformal modulus, the surface after welding is described by Liouville conformal field theory, and the density of the random modulus contains the SLE partition function for the interfaces as a multiplicative factor. The settings we treat includes the multiple SLE for $\kappa \in (0,4)$, the flow lines of imaginary geometry on the disk with boundary marked points, and the boundary Green function. These results demonstrate an alternative approach to construct and study the SLE partition function, which complements the traditional method based on stochastic calculus and differential equation.

General Three-Body Problem in Conformal-Euclidean Space: New Properties of a Low-Dimensional Dynamical System

Despite the huge number of studies of the three-body problem in physics and mathematics, the study of this problem remains relevant due to both its wide practical application and taking into account its fundamental importance for the theory of dynamical systems. In addition, one often has to answer the cognitive question: is irreversibility fundamental for the description of the classical world? To answer this question, we considered a reference classical dynamical system, the general three-body problem, formulating it in conformal Euclidean space and rigorously proving its equivalence to the Newtonian three-body problem. It has been proven that a curved configuration space with a local coordinate system reveals new hidden symmetries of the internal motion of a dynamical system, which makes it possible to reduce the problem to a sixth-order system instead of the eighth order. An important consequence of the developed representation is that the chronologizing parameter of the motion of a system of bodies, which we call internal time, differs significantly from ordinary time in its properties. In particular, it more accurately describes the irreversible nature of multichannel scattering in a three-body system and other chaotic properties of a dynamical system. The paper derives an equation describing the evolution of the flow of geodesic trajectories, with the help of which the entropy of the system is constructed. New criteria for assessing the complexity of a low-dimensional dynamical system and the dimension of stochastic fractal structures arising in three-dimensional space are obtained. An effective mathematical algorithm is developed for the numerical simulation of the general three-body problem, which is traditionally a difficult-to-solve system of stiff ordinary differential equations.

Charting the conformal manifold of holographic CFT$_2$'s

We construct new continuous families of AdS_3× S^3× T^4AdS3×S3×T4 and AdS_3× S^3× S^3× S^1AdS3×S3×S3×S1 solutions in heterotic and type II supergravities. These families are found in three-dimensional consistent truncations and controlled by 17 parameters, which include TsT \betaβ deformations and encompass several supersymmetric sub-families. The different uplifts are constructed in a unified fashion by means of Exceptional Field Theory (ExFT). This allows the computation of the Kaluza-Klein spectra around the deformations, to test the stability of the solutions, and to interpret them holographically and as worldsheet models. To achieve this, we describe how the half-maximal S(8,8)S(8,8) ExFT can be embedded into E_{8(8)}E8(8) ExFT.

On Locally Conformal Kaehler Submersions

We study locally conformal Kaehler submersions, i.e., almost Hermitian submersions whose total manifolds are locally conformal Kaehler. We prove that the vertical distribution of a locally conformal Kaehler submersion is totally geodesic iff the Lee vector field of total manifold is vertical. We also obtain the O'Neill tensors $\tilde{\mathcal{A}}$ and $\tilde{\mathcal{T}}$ with respect to the Weyl connection of a locally conformal Kaehler submersion. Then, we proved that the horizontal distribution of such a submersion is integrable iff $\tilde{\mathcal{A}} \equiv 0$. Finally, we get Chen-Ricci inequalities for locally conformal Kaehler space form submersions and Hopf space form submersions.

Self-dual cosmology

We construct cosmological spacetimes with a self-dual Weyl tensor whose dynamics are described by conformally coupled scalars with only cubic self-interactions. Similar to the previously discovered cases in flat and (Anti) de Sitter backgrounds, the interactions are characterized by a bracket that encodes a kinematic algebra. We discuss how the color-kinematics duality and double copy are realized in these cosmological backgrounds. If we further impose that the Ricci scalar is that of an FLRW spacetime, we find two new self-dual metrics corresponding to radiation-dominated and coasting (non-accelerating) FLRW backgrounds. Relaxing this requirement, we find an infinite family of solutions given by three different conformal classes of cosmological self-dual metrics. These solutions approximate those of FLRW as long as we impose a simple additional constraint on the scalar theory.

Galilean fluids from non-relativistic gravity

The 1/c-expansion of general relativity appropriately sourced by matter can be used to derive an action principle for Newtonian gravity. The gravitational part of this action is known as non-relativistic gravity (NRG). It is possible to source NRG differently and in such a way that one can construct solutions that are not described by Newtonian gravity (as they do not admit a notion of absolute time). It is possible to include a negative cosmological constant such that NRG admits a non-relativistic AdS solution. This non-relativistic AdS vacuum has Killing vectors that form the Galilean conformal algebra and a boundary that admits a conformal class of Newton-Cartan geometries. This begs the question of whether there exists an analogue of the fluid/gravity correspondence for NRG. In this paper we derive a non-relativistic AdS brane solution of NRG and confirm that it corresponds to the 1/c2-expansion of the AdS black brane geometry. We perform a Galilean boost of the non-relativistic AdS brane and derive the associated boundary energy-momentum tensor. We then show that this is the energy-momentum tensor of a massless Galilean fluid and explain how this is linked to the conformal isometries of the boundary. Along the way, we also present several new results for the theory of non-relativistic gravity itself. In particular we present a rewriting that greatly shortens and simplifies the equations of motion of the NRG action.

Blackening S-folds

We construct the universal AdS4 black hole that asymptotes to the (φ, χ)-family of type IIB S-fold backgrounds dual to the conformal manifold of N = 2 S-fold CFT’s. We present the explicit type IIB embedding of such a universal black hole for two particular asymptotics: the N = 2 S-fold with U(2) symmetry at (φ, χ) = (0, 0) and the N = 4 S-fold with SO(4) symmetry at (φ, χ) = (1, 0). As a byproduct, we also present a novel two-parameter family of AdS2 × M8 supersymmetric S-fold backgrounds with M8 = ℍ2 × S5 × S1 that features a parametrically-controlled scale separation involving some of the internal directions.

Dirac Eigenvalue Optimisation and Harmonic Maps to Complex Projective Spaces

Abstract Consider a Dirac operator on an oriented compact surface endowed with a Riemannian metric and spin structure. Provided the area and the conformal class are fixed, how small can the $k$-th positive Dirac eigenvalue be? This problem mirrors the maximization problem for the eigenvalues of the Laplacian, which is related to the study of harmonic maps into spheres. We uncover the connection between the critical metrics for Dirac eigenvalues and harmonic maps into complex projective spaces. Using this approach we show that for many conformal classes on a torus the first nonzero Dirac eigenvalue is minimised by the flat metric. We also present a new geometric proof of Bär’s theorem stating that the first nonzero Dirac eigenvalue on the sphere is minimised by the standard round metric.

The Effects of Mechanical Loading on Resonant Response of a Conformal Load-Bearing Antenna System.

Glass fibre-reinforced polymer (GFRP) is a suitable substrate material for constructing a Conformal Load-Bearing Antenna Structure (CLAS). The relative permittivity of the CLAS substrate, which determines its resonant frequency, is affected by damage sustained by GFRP. This paper investigates the effects of damage (induced by mechanically loading the substrate) on the resonant response of the CLAS. Decoupling the antenna from the substrate was essential to evaluate the CLAS's true response to the induced damage. This paper details a systematic investigation examining how the frequency response of a "pristine" antenna and a surface-mounted antenna respond to a substrate subjected to quasi-statically induced mechanical damage and cyclic fatigue loading. The results demonstrate that the resonant frequency of the CLAS varies as a function of the substrate's mechanical damage. The prepared CLAS is tolerant to a certain degree of mechanical loading and related damage with its resonant frequency remaining within an acceptable bandwidth.

Compact Locally Conformally Pseudo-Kähler Manifolds with Essential Conformal Transformations

A conformal transformation of a semi-Riemannian manifold is essential if there is no conformally equivalent metric for which it is an isometry. For Riemannian manifolds the existence of an essential conformal transformation forces the manifold to be conformally flat. This is false for pseudo-Riemannian manifolds, however compact examples of conformally curved manifolds with essential conformal transformation are scarce. Here we give examples of compact conformal manifolds in signature $(4n+2k,4n+2\ell)$ with essential conformal transformations that are locally conformally pseudo-Kähler and not conformally flat, where $n\ge 1$, $k, \ell \ge 0$. The corresponding local pseudo-Kähler metrics obtained by a local conformal rescaling are Ricci-flat.

Flexible Tactile Sensors with Gradient Conformal Dome Structures.

The trade-off between high sensitivity and wide detection range remains a challenge for flexible capacitive pressure sensors. Gradient structure can provide continuous deformation and lead to a wide sensing range. However, it simultaneously augments the distance between two electrodes, which diminishes the variation in the relative distance and results in a decreased sensitivity. Herein, a conformal design is introduced into the gradient structure to construct a flexible capacitive pressure sensor. The gradient conformal dome structure is fabricated by a simple reverse dome adsorption process. Taking advantage of the progressive deformation behavior of the gradient dielectric, and the significant improvement of relative distance variation between two electrodes from the conformal design, the sensor achieves a sensitivity of 0.214 kPa-1 in an ultrabroad linear range up to 200 kPa. It maintains high-pressure resolution under the preload of 10 and 100 kPa. Benefiting from the rapid response and excellent repeatability, the sensor can be used for physiological monitor and human motion detection, including arterial pulse, joint bending, and motion state. The gradient conformal design strategy may pave a promising avenue to develop pressure sensors with high sensitivity and wide linear range.

Conformal structure of singularities in some varying fundamental constants bimetric cosmologies

In this paper, we explore the conformal structure of singularities arising from varying fundamental constants using the method of Penrose diagrams. We employ a specific type of bimetric model featuring two different metrics. One metric describes the causal structure for matter, while the other characterizes the causal structure for gravitational interactions, which is related to variations in fundamental constants such as the gravitational constant and the speed of light. For this reason, we focused on the gravitational metric to calculate the conformal transformation and compose Penrose diagrams for the singularities arising from the varying fundamental constants. We have shown that, in one case, the parameter such as the scale factor, the density and the pressure resemble those of the finite scale factor singularity (FSF). Despite singularity appears in constant conformal time in our case and in the case of FSF the Misner-Sharp horizon looks different. Our another case is similar to sudden future singularity (SFS), but there are differences in the conformal structures. We have also shown that in our cases the behavior of Misner-Sharp horizon strongly depends on initial conditions. The last analytical solution which we introduced is identical to conformal structure of the standard exotic singularity for the matter.

MOF-Derived Ni/NiO-C Nanocomposites as Bifunctional Electrocatalysts Capable of Driving Both ORR and OER.

Bifunctional electrocatalysts, capable of efficiently driving both the oxygen reduction reaction (ORR) and the oxygen evolution reaction (OER), are crucial for advancing electrochemical processes. While noble-metal-based catalysts are widely recognized for their role in oxygen processes, current state-of-the-art designs are limited to either ORR or OER activity, presenting a notable research gap. In addressing this challenge, we have developed a novel Ni/NiO-C nanocomposite catalyst derived from a nickel-based metal-organic framework (Ni-SKU-5). For the ORR, the Ni/NiO-C catalyst exhibits an onset potential of 0.95 V vs RHE in a 1.0 M KOH solution, coupled with a Tafel slope of -99 mV dec-1 at 1600 rpm. Moreover, the catalyst displays excellent stability, maintaining a performance of over 90% after 10 h of continuous reaction. Furthermore, the catalyst proves effective in the OER, boasting an overpotential of 370 mV (at 10 mA cm-2) and a Tafel slope of 114 mV dec-1, highlighting its bifunctionality. The bifunctional overpotential of the Ni/NiO-C composite is measured at 820 mV, surpassing that of the 20% Pt/C electrocatalyst (860 mV), highlighting its potential for practical applications. Comparative experiments establish the origin of the robust bifunctional reactivity toward the conformal hybrid structure, porous framework, and the synergistic effect operating among the constituents of the nanocomposite.

Deformational rigidity of integrable metrics on the torus

Abstract It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: we consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations, the deformed metric is again Liouville. The principal idea of the argument is that the preservation of rational invariant tori in the foliation of the phase space forces a linear combination on the Fourier coefficients of the deformation to vanish. Showing that the resulting linear system is non-degenerate will then yield the claim. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. To put our results in perspective, we review existing results about integrable metrics on the torus.