Conformal Structure Research Articles

On higher-spin points and infinite distances in conformal manifolds

Distances in the conformal manifold, the space of CFTs related by marginal deformations, can be measured in terms of the Zamolodchikov metric. Part of the CFT Distance Conjecture posits that points in this manifold where part of the spectrum becomes free, called higher-spin points, can only be at infinite distance from the interior. There, an infinite tower of operators become conserved currents, and the conformal symmetry is enhanced to a higher-spin algebra. This proposal was initially motivated by the Swampland Distance Conjecture, one of pillars of the Swampland Program. In this work, we show that the conjecture can be tackled using only methods from the conformal toolkit, and without relying on the existence of a weakly-coupled gravity dual. Via conformal perturbation theory combined with properties of correlators and of the higher-spin algebra, we establish that higher-spin points are indeed at infinite distance in the conformal manifold. We make no assumptions besides the usual properties of local CFTs, such as unitarity and the existence of an energy-momentum tensor. In particular, we do not rely on a specific dimension of spacetime (although we assume d > 2), nor do we require the presence of supersymmetry.

Min-max harmonic maps and a new characterization of conformal eigenvalues

Given a surface M and a fixed conformal class c one defines \Lambda_{k}(M,c) to be the supremum of the k -th nontrivial Laplacian eigenvalue over all metrics g\in c of unit volume. It has been observed by Nadirashvili that the metrics achieving \Lambda_{k}(M,c) are closely related to harmonic maps to spheres. In the present paper, we identify \Lambda_{1}(M,c) and \Lambda_{2}(M,c) with min-max quantities associated to the energy functional for sphere-valued maps. As an application, we obtain several new eigenvalue bounds, including a sharp isoperimetric inequality for the first two Steklov eigenvalues. This characterization also yields an alternative proof of the existence of maximal metrics realizing \Lambda_{1}(M,c) , \Lambda_{2}(M,c) , and moreover allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.

19F-NMR studies of the impact of different detergents and nanodiscs on the A2A adenosine receptor.

For the A2A adenosine receptor (A2AAR), a class A G-protein-coupled receptor (GPCR), reconstituted in n-dodecyl-β-D-maltoside (DDM)/‌‌‌‌‌cholesteryl hemisuccinate (CHS) mixed micelles, previous 19F-NMR studies revealed the presence of multiple simultaneously populated conformational states. Here, we study the influence of a different detergent, lauryl maltose neopentyl glycol (LMNG) in mixed micelles with CHS, and of lipid bilayer nanodiscs on these conformational equilibria. The populations of locally different substates are pronouncedly different in DDM/‌‌‌‌‌CHS and LMNG/‌‌‌‌‌CHS micelles, whereas the A2AAR conformational manifold in LMNG/‌‌‌‌‌CHS micelles is closely similar to that in the lipid bilayer nanodiscs. Considering that nanodiscs represent a closer match of the natural lipid bilayer membrane, these observations support that LMNG/‌‌‌‌‌CHS micelles are a good choice for reconstitution trials of class A GPCRs for NMR studies in solution.

Geodesic Structure of Generalized Vaidya Spacetime through the K-Essence

This article investigates the radial and non-radial geodesic structures of the generalized K-essence Vaidya spacetime. Within the framework of K-essence geometry, it is important to note that the metric does not possess conformal equivalence to the conventional gravitational metric. This study employs a non-canonical action of the Dirac–Born–Infeld kind. In this work, we categorize the generalized K-essence Vaidya mass function into two distinct forms. Both the forms of the mass functions have been extensively utilized to analyze the radial and non-radial time-like or null geodesics in great detail inside the comoving plane. Indications of the existence of wormholes can be noted during the extreme phases of spacetime, particularly in relation to black holes and white holes, which resemble the Einstein–Rosen bridge. In addition, we have also detected a distinctive indication of the quantum tunneling phenomenon around the singularity (r→0). Furthermore, we have found that for certain types of solutions, there exist circular orbits through the event horizon as well as quasicircular orbits. Also, we have noted that there is no central singularity in our spacetime where both r and t tend towards zero. The existence of a central singularity is essential for any generalized Vaidya spacetime. This indicates that spacetime can be geodesically complete, which correlates with the findings of Kerr’s recent work (2023).

Gravitational observatories

We consider four-dimensional general relativity with vanishing cosmological constant defined on a manifold with a boundary. In Lorentzian signature, the timelike boundary is of the form σ × ℝ, with σ a spatial two-manifold that we take to be either flat or S2. In Euclidean signature we take the boundary to be S2 × S1. We consider conformal boundary conditions, whereby the conformal class of the induced metric and trace K of the extrinsic curvature are fixed at the timelike boundary. The problem of linearised gravity is analysed using the Kodama-Ishibashi formalism. It is shown that for a round metric on S2 with constant K, there are modes that grow exponentially in time. We discuss a method to control the growing modes by varying K. The growing modes are absent for a conformally flat induced metric on the timelike boundary. We provide evidence that the Dirichlet problem for a spherical boundary does not suffer from non-uniqueness issues at the linearised level. We consider the extension of black hole thermodynamics to the case of conformal boundary conditions, and show that the form of the Bekenstein-Hawking entropy is retained.

Automated design of dynamic programming schemes for RNA folding with pseudoknots

Although RNA secondary structure prediction is a textbook application of dynamic programming (DP) and routine task in RNA structure analysis, it remains challenging whenever pseudoknots come into play. Since the prediction of pseudoknotted structures by minimizing (realistically modelled) energy is NP-hard, specialized algorithms have been proposed for restricted conformation classes that capture the most frequently observed configurations. To achieve good performance, these methods rely on specific and carefully hand-crafted DP schemes. In contrast, we generalize and fully automatize the design of DP pseudoknot prediction algorithms. For this purpose, we formalize the problem of designing DP algorithms for an (infinite) class of conformations, modeled by (a finite number of) fatgraphs, and automatically build DP schemes minimizing their algorithmic complexity. We propose an algorithm for the problem, based on the tree-decomposition of a well-chosen representative structure, which we simplify and reinterpret as a DP scheme. The algorithm is fixed-parameter tractable for the treewidth tw of the fatgraph, and its output represents a O(ntw+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(n^{tw+1})$$\\end{document} algorithm (and even possibly O(ntw)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(n^{tw})$$\\end{document} in simple energy models) for predicting the MFE folding of an RNA of length n. We demonstrate, for the most common pseudoknot classes, that our automatically generated algorithms achieve the same complexities as reported in the literature for hand-crafted schemes. Our framework supports general energy models, partition function computations, recursive substructures and partial folding, and could pave the way for algebraic dynamic programming beyond the context-free case.

The use of CaO:Eu<sup>3+</sup> and Zn<sub>2</sub>SiO<sub>4</sub>:Mn<sup>2+</sup> phosphors to increase the color quality and illumination intensity of WLEDs

The lumen efficacy in remote phosphor structure displayed remarkable enhancements, which is notable for the development of white light-emitting diodes (WLEDs). Nevertheless, its quality of colour is deemed not as good as that of the conformal or in-cup phosphor structure. Therefore, the goal of this research is to achieve a better quality of colour and significant luminous flux value for remote phosphor configuration by using extra phosphor layers. In particular, the two-layer and three-layer structures with the implementation of green and red phosphors are proposed. Comparing these two structures can help pinpoint the best suited for the WLED production. The assessment of each structure’s effect on the WLEDs’ optical parameters was determined under various correlated temperatures of colour (5,600-8,500 K). The outcomes indicated that the three-layer structure enhanced the quality of colour with greater efficiency compared to the two-layer structure due to the increased color rendering index (CRI), color quality scale (CQS), and photoluminescence (PL), and reduced colour deviation. The scattering improvement of the three-layer structure is a key factor of these accomplishments, proven by the scattering theory of Mie. Therefore, the three-layer structure is potential for developing WLED production.

Flexible Conformal Multifunctional Time‐Varying Phase‐Modulated Metasurface with Polarization Control for Radar Feature Transformation

AbstractReconfigurable metasurfaces have shown unprecedented capabilities in controlling electromagnetic (EM) waves in recent years and play an essential role in information manipulation. However, most research on reconfigurable metasurfaces is limited to planar rigid structures, with only one control function and suitable for specific single polarization. Achieving simultaneous control of phase, polarization, and frequency in the conformal structure has significant practical implications in many applications, but remains obviously challenging. In this paper, a flexible multifunctional time‐varying phase‐modulated metasurface (MTPM) with orthogonal polarization control is proposed for conformal applications to manipulate the spectrum distribution. It enables time encoding capability in controlling the phase characteristics of x‐ and y‐polarized EM waves individually by switching the bias voltage of the PIN diode in two directions. Discrete harmonic spectrum and continuous Doppler frequency shifts are implemented separately on two orthogonal polarizations by applying different time coding sequences. As a practical application, a conformal MTPM for radar feature transformation is explored. Multiple discrete false targets and defocusing phenomena can be realized respectively for x‐ and y‐polarized waves. Both simulated and measured results verify the validity of the proposed method and prototype.

Torsion-free connections on G-structures

We prove that for a Lie group SOn(R)⊂G⊂GLn(R), any G-structure on a smooth manifold can be endowed with a torsion free connection which is locally the Levi-Civita connection of a Riemannian metric in a given conformal class. In this process, we classify the admissible groups.

Using in-abattoir 3-dimensional measurements from images of beef carcasses for the prediction of EUROP classification grade and carcass weight

Imaging technology can aid the automatic extraction of measurements from beef carcasses, which can be used for objective grading. Many abattoirs, however, rely on manual grading due to the required infrastructure and cost, making technology unfeasible. This study explores 3-dimensional (3D) imaging technology, requiring limited infrastructure, and its ability to predict carcass weight, conformation class and fat class for non-invasive, objective classification. Time-of-flight near-infrared cameras captured 3-dimensional point clouds of beef carcasses, on-line in one commercial abattoir in Scotland, over a 6-month period. Thirty-five 3D images were captured per carcass and processed using machine vison software. Seventy-four measurements were extracted from each point cloud. Removal of extreme outliers resulted in 285,109 datapoints for 17,250 carcasses. Coefficients of variation (CV) for each measurement on a per-animal basis were low and consistent, and measurements were averaged across images. Using a training and validation dataset (70:30), multiple linear regression models predicted EUROP conformation class, fat class, and carcass weight. Stepwise models included fixed effects (sex, breed type, kill date (and cold carcass weight for conformation and fat class)), and 3D image measurements. Including 3D measurements resulted in prediction accuracies of 70%, 50% and 23% for cold carcass weight, conformation, and fat class respectively. Mapping predictions on the traditional EUROP grid used in the UK showed that 99% of conformation classes and 93% of fat classes were classified within the correct or neighbouring grade. The results of this study indicate the potential for non-invasive, in-abattoir technology requiring limited infrastructure to predict carcass traits objectively.

Carcass characteristics and meat quality of thin-tailed Kivircik and fat-tailed Kangal Akkaraman lambs according to EUROP carcass fatness and conformation classes

A carcass classification method is expected to effectively categorize carcasses based on both carcass and meat quality criteria. Localisation of adipose tissue on the carcass differs between thin- and fat-tailed sheep breeds, which may influence the effectiveness of the EUROP classification system in distinguishing lamb carcasses. The aim of the study was to investigate the effects of EUROP carcass fatness and conformation classes on carcass characteristics and meat quality in thin-tailed Kivircik and fat-tailed Kangal Akkaraman lambs. Slaughtering, carcass, and meat quality data of 115 lambs (67 Kivircik and 48 Kangal Akkaraman) were examined. Hierarchical cluster analysis yielded 3 clusters in both breeds. Differences among lamb carcasses in different clusters were significant for nearly all carcass characteristics in both breeds. Fatness and conformation classes significantly influenced on most of the carcass characteristics in both breeds. However, the effect of fatness class on percentages of valuable joints, and the effect of conformation class on hind limb tissue composition were not significant in Kangal Akkaraman lambs. As carcass fatness and conformation classes increased in Kivircik lambs, meat colour became darker. In Kangal Akkaraman lambs, fatness class had no influence on meat colour. The effects of fatness and conformation classes on sensory characteristics were not significant in both breeds, except fatness effect on odour intensity in Kangal Akkaraman lambs. In conclusion, the EUROP classification system may not be as effective in distinguishing carcass and meat quality in Kangal Akkaraman lambs as in the Kivircik breed.

Semi-slant submanifolds in a locally conformal Kaehler space form

In this paper, we consider semi-slant submanifolds in a locally conformal Kaehler manifold and a locally conformal Kaehler space form. We give inequalities for the length of the second fundamental form and the mean curvature and, using Gauss and Codazzi equations, we get many useful results in locally conformal Kaehler space form.

Compactifications of indefinite 3-Sasaki structures and their quaternionic Kähler quotients

We show that 3-Sasaki structures admit a natural description in terms of projective differential geometry. First we establish that a 3-Sasaki structure may be understood as a projective structure whose tractor connection admits a holonomy reduction, satisfying a particular non-vanishing condition, to the (possibly indefinite) unitary quaternionic group Sp(p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Sp}}(p,q)$$\\end{document}. Moreover, we show that, if a holonomy reduction to Sp(p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Sp}}(p,q)$$\\end{document} of the tractor connection of a projective structure does not satisfy this condition, then it decomposes the underlying manifold into a disjoint union of strata including open manifolds with (indefinite) 3-Sasaki structures and a closed separating hypersurface at infinity with respect to the 3-Sasaki metrics. It is shown that the latter hypersurface inherits a Biquard–Fefferman conformal structure, which thus (locally) fibers over a quaternionic contact structure, and which in turn compactifies the natural quaternionic Kähler quotients of the 3-Sasaki structures on the open manifolds. As an application, we describe the projective compactification of (suitably) complete, non-compact (indefinite) 3-Sasaki manifolds and recover Biquard’s notion of asymptotically hyperbolic quaternionic Kähler metrics.

An embedding space approach to Carrollian CFT correlators for flat space holography

Carrollian conformal field theories (carrollian CFTs) are natural field theories on null infinity of an asymptotically flat spacetime or, more generally, geometries with conformal carrollian structure. Using a basis transformation, gravitational S-matrix elements can be brought into the form of correlators of a carrollian CFT. Therefore, it has been suggested that carrollian CFTs could provide a co-dimension one dual description to gravity in asymptotically flat spacetimes. In this work, we construct an embedding space formalism for three-dimensional carrollian CFTs and use it to determine two- and three-point correlators. These correlators are fixed by the global subgroup, ISO(1, 3), of the carrollian conformal symmetries, i.e., the Bondi-van der Burg-Metzner-Sachs symmetries (BMS). The correlators coincide with well-known two- and three-point scattering amplitudes in Minkowski space written with respect to a basis of asymptotic position states.

On slant Riemannian submersions from conformal Sasakian manifolds

In this paper, we study slant Riemannian submersions from conformal Sasakian manifolds onto Riemannian manifolds. We investigate the geometry of foliations associated with the submersion and give sufficient conditions for the submersion to be harmonic.

Petrov classification of analogue spacetimes

In an effort to invariantly characterize the conformal curvature structure of analogue spacetimes built from a nonrelativistic fluid background, we determine the Petrov type of a variety of laboratory geometries. Starting from the simplest examples, we increase the complexity of the background, and thereby determine how the laboratory fluid symmetry affects the corresponding Petrov type in the analogue spacetime realm of the sound waves. We find that for more complex flows isolated hypersurfaces develop, which are of a Petrov type differing from that of the surrounding fluid. Finally, we demonstrate that within the incompressible background approximation, as well as for all compressible quasi-one-dimensional flows, the only possible Petrov types are the algebraically general type I and the algebraically special types O and D.

The Lorentzian Lichnerowicz conjecture for real-analytic, three-dimensional manifolds

Abstract We prove that, for a compact, 3-dimensional, real-analytic, Lorentzian manifold, if the group of conformal transformations does not preserve any metric in the conformal class, then the metric is conformally flat.

Percolation on Triangulations: A Bijective Path to Liouville Quantum Gravity

We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by certain 2D lattice paths. Our bijection parallels in the discrete setting the mating-of-trees framework of LQG and Schramm-Loewner evolutions (SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two correspondences allows us to relate uniform site-percolated triangulations to 8 / 3 \sqrt {8/3} -LQG and SLE 6 _6 . In particular, we establish the convergence of several functionals of the percolation model to continuous random objects defined in terms of 8 / 3 \sqrt {8/3} -LQG and SLE 6 _6 . For instance, we show that the exploration tree of the percolation converges to a branching SLE 6 _6 , and that the collection of percolation cycles converges to the conformal loop ensemble CLE 6 _6 . We also prove convergence of counting measure on the pivotal points of the percolation. Our results play an essential role in several other works, including a program for showing convergence of the conformal structure of uniform triangulations and works which study the behavior of random walk on the uniform infinite planar triangulation.

Scattering of an inhomogeneous coupled Schrödinger system in the conformal space

Abstract This paper studies the inhomogeneous defocusing coupled Schrödinger system i ⁢ u ˙ j + Δ ⁢ u j = | x | - ρ ⁢ ( ∑ 1 ≤ k ≤ m a j ⁢ k ⁢ | u k | p ) ⁢ | u j | p - 2 ⁢ u j , ρ > 0 , j ∈ [ 1 , m ] . i\dot{u}_{j}+\Delta u_{j}=\lvert x\rvert^{-\rho}\bigg{(}\sum_{1\leq k\leq m}a_% {jk}\lvert u_{k}\rvert^{p}\biggr{)}\lvert u_{j}\rvert^{p-2}u_{j},\quad\rho>0,% \,j\in[1,m]. The goal of this work is to prove the scattering of energy global solutions in the conformal space made up of f ∈ H 1 ⁢ ( ℝ N ) {f\in H^{1}(\mathbb{R}^{N})} such that x ⁢ f ∈ L 2 ⁢ ( ℝ N ) {xf\in L^{2}(\mathbb{R}^{N})} . The present paper is a complement of the previous work by the first author and Ghanmi [T. Saanouni and R. Ghanmi, Inhomogeneous coupled non-linear Schrödinger systems, J. Math. Phys. 62 2021, 10, Paper No. 101508]. Indeed, the supplementary assumption x ⁢ u 0 ∈ L 2 {xu_{0}\in L^{2}} enables us to get the scattering in the mass-sub-critical regime p 0 < p ≤ 2 - ρ N + 1 {p_{0}<p\leq\frac{2-\rho}{N}+1} , where p 0 {p_{0}} is the Strauss exponent. The proof is based on the decay of global solutions coupled with some non-linear estimates of the source term in Strichartz norms and some standard conformal transformations. Precisely, one gets | t | α ⁢ ∥ u ⁢ ( t ) ∥ L r ⁢ ( ℝ N ) ≲ 1 \lvert t\rvert^{\alpha}\lVert u(t)\rVert_{L^{r}(\mathbb{R}^{N})}\lesssim 1 for some α > 0 {\alpha>0} and a range of Lebesgue norms. The decay rate in the mass super-critical regime is the same one as of e i ⋅ Δ ⁢ u 0 {e^{i\cdot\Delta}u_{0}} . This rate is different in the mass sub-critical regime, which requires some extra assumptions. The novelty here is the scattering of global solutions in the weighted conformal space for the class of source terms p 0 < p < 2 - ρ N - 2 + 1 {p_{0}<p<\frac{2-\rho}{N-2}+1} . This helps to better understand the asymptotic behavior of the energy solutions. Indeed, the source term has a negligible effect for large time and the above non-linear Schrödinger problem behaves like the associated linear one. In order to avoid a singular source term, one assumes that p ≥ 2 {p\geq 2} , which restricts the space dimensions to N ≤ 3 {N\leq 3} . In a paper in progress, the authors treat the same problem in the complementary case ρ < 0 {\rho<0} .

Generalized Willmore energies, Q-curvatures, extrinsic Paneitz operators, and extrinsic Laplacian powers

Over forty years ago, Paneitz, and independently Fradkin and Tseytlin, discovered a fourth-order conformally invariant differential operator, intrinsically defined on a conformal manifold, mapping scalars to scalars. This operator is a special case of the so-termed extrinsic Paneitz operator defined in the case when the conformal manifold is itself a conformally embedded hypersurface. In particular, this encodes the obstruction to smoothly solving the five-dimensional scalar Laplace equation, and suitable higher dimensional analogs, on conformally compact structures with constant scalar curvature. Moreover, the extrinsic Paneitz operator can act on tensors of general type by dint of being defined on tractor bundles. Motivated by a host of applications, we explicitly compute the extrinsic Paneitz operator. We apply this formula to obtain: an extrinsically-coupled [Formula: see text]-curvature for embedded four-manifolds, the anomaly in renormalized volumes for conformally compact five-manifolds with negative constant scalar curvature, Willmore energies for embedded four-manifolds, the local obstruction to smoothly solving the five-dimensional singular Yamabe problem, and new extrinsically-coupled fourth- and sixth-order operators for embedded surfaces and four-manifolds, respectively.