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This review is a contribution to a book dedicated to the memory of Michael E. Fisher. The first example of a quantum many body system not expected to have any quasiparticle excitations was the Wilson-Fisher conformal field theory. The absence of quasiparticles can be established in the compressible, metallic state of the Sachdev-Ye-Kitaev model of fermions with random interactions. The solvability of the latter model has enabled numerous computations of the non-quasiparticle dynamics of chaotic many-body states, such as those expected to describe quantum black holes. We review thermodynamic properties of the SYK model, and describe how they have led to an understanding of the universal structure of the low energy density of states of charged black holes without low energy supersymmetry. [ABSTRACT FROM AUTHOR]

We consider an infinite planar Delaunay graph Gϵ which is obtained by locally deforming the coordinate embedding of a general isoradial graph Gcr, with respect to a real deformation parameter ϵ. Using Kenyon's exact and asymptotic results for the critical Green's function on an isoradial graph, we calculate the leading asymptotics of the first- and second-order terms in the perturbative expansion of the log-determinant of the Laplace-Beltrami operator Δ(ϵ), the David-Eynard Kähler operator D(ϵ), and the conformal Laplacian Δ ​(ϵ) on the deformed Delaunay graph Gϵ. We show that the scaling limits of the second-order bi-local term for both the Laplace-Beltrami and David-Eynard Kähler operators exist and coincide, with a shared value independent of the choice of the initial isoradial graph Gcr. Our results allow us to define a discrete analog of the stress-energy tensor for each of the three operators. Furthermore, we can identify a central charge (c=-2) in the case of both the Laplace-Beltrami and David-Eynard Kähler operators. While the scaling limit is consistent with the stress-energy tensor and the value of the central charge for the Gaussian free field (GFF), the discrete central charge value of c=-2 for the David-Eynard Kähler operator is, however, at odds with the value of c=-26 expected by Polyakov's theory of 2D quantum gravity; moreover, there are problems with convergence of the scaling limit of the discrete stress-energy tensor for the David-Eynard Kähler operator. The second-order bi-local term for the conformal Laplacian involves anomalous terms corresponding to the creation of discrete curvature dipoles in the deformed Delaunay graph Gϵ; we examine the difficulties in defining a convergent scaling limit in this case. Connections with some discrete statistical models at criticality are explored. [ABSTRACT FROM AUTHOR]

The aim of this review paper is to survey the fractal morphology of scale-invariant patterns. We are particularly focusing on the scale and conformal invariance, as well as on the fractal non-uniformity (multifractality), inhomogeneity (lacunarity), and anisotropy (succolarity). We argue that these features can be properly quantified by the following six adimensional numbers: the fractal (e.g., similarity, box-counting, or Assouad) dimension, conformal dimension, degree of multifractal non-uniformity, coefficient of multifractal asymmetry, index of lacunarity, and index of fractal anisotropy. The difference between morphological properties of mathematical and real-world fractals is especially outlined in this review paper. [ABSTRACT FROM AUTHOR]

The possibility of using spin connection components as basic quantization variables of a conformal version of general relativity is studied. The considered model contains gravitational degrees of freedom and a scalar dilaton field. The standard tetrad formalism is applied. Properties of spin connections in this model are analyzed. Secondary quantization of the chosen variables is performed. The gravitational part of the model action turns out to be quadratic with respect to the spin connections. So at the quantum level, the model looks trivial, i.e., without quantum self-interactions. Meanwhile the correspondence to general relativity is preserved at the classical level. [ABSTRACT FROM AUTHOR]

The basic idea of the 'partially nonclassical method', developed in the present paper, is to apply the invariance requirement of the Lie group method using not all differential consequences of the invariant surface condition but only part of them. It differs from the 'classical' method, in which the invariant surface condition is not used, and from the 'nonclassical' method, in which all the differential consequences are used. It provides additional possibilities for the symmetry analysis of partial differential equations (PDEs), as compared with the 'classical' and 'nonclassical' methods, in the so-named no-go case when the group generator, associated with one of the independent variables, is identically zero. The method is applied to the flat steady-state boundary layer problem, reduced to an equation for the stream function, and it is found that applying the partially nonclassical method in the no-go case yields new symmetry reductions and new exact solutions of the boundary layer equations. A computationally convenient unified framework for the classical, nonclassical and partially nonclassical methods (λ -formulation) is developed. The issue of conformal invariance in the context of the Lie group method is considered, stemming from the observation that the classical Lie method procedure yields transformations not leaving the differential polynomial of the PDE invariant but modifying it by a conformal factor. The physical contexts, in which that observation could be important, are discussed using the derivation of the Lorentz transformations of special relativity as an example. [ABSTRACT FROM AUTHOR]

The action of an ideal fluid in Euler variables with a variable number of particles is used for the phenomenological description of the processes of particle creation in strong external fields. It has been demonstrated that the conformal invariance of the creation law imposes quite strict restrictions on the possible types of sources. It is shown that combinations with the particle number density in the creation law can be interpreted as dark matter within the framework of this model. [ABSTRACT FROM AUTHOR]

This paper presents new classes of exact radial solutions to the nonlinear ordinary differential equation that arises as a saddle-point condition for a Euclidean scalar field theory in D-dimensional spacetime. These solutions are found by exploiting the dimensional consistency of the radial differential equation for a single massless scalar field, which allows it to transform into an autonomous equation. For massive theories, the radial equation is not exactly solvable, but the massless solutions provide useful approximations to the results for the massive case. The solutions presented here depend on the power of the interaction and on the spatial dimension, both of which may be noninteger. Scalar equations arising in the study of conformal invariance fit into this framework, and classes of new solutions are found. These solutions exhibit two distinct behaviors as D → 2 from above. [ABSTRACT FROM AUTHOR]

In this article, we prove two global existence and full convergence theorems for flow lines of the Möbius-invariant Willmore flow, and we use the latter result in order to prove that fully and smoothly convergent flow lines of the Möbius-invariant Willmore flow are stable with respect to small perturbations of their initial immersions in any C 4 , γ -norm, provided they converge to a umbilic-free C 4 -local minimizer of the Willmore functional among C 4 -immersions of a smooth compact torus into either R 3 or S 3 . The proofs of our two main theorems rely on the author’s recent achievements about the Möbius-invariant Willmore flow, on Weiner’s investigation of the stability of the Clifford torus with respect to the Willmore functional, and on Escher’s, Mayer’s, and Simonett’s work from the 1990 s on invariant center manifolds for uniformly parabolic quasilinear evolution equations and their special applications to the Willmore- and surface diffusion flow near round 2-spheres in R 3 . [ABSTRACT FROM AUTHOR]

The aim of this review paper is to survey the fractal morphology of scale-invariant patterns. We are particularly focusing on the scale and conformal invariance, as well as on the fractal non-uniformity (multifractality), inhomogeneity (lacunarity), and anisotropy (succolarity). We argue that these features can be properly quantified by the following six adimensional numbers: the fractal (e.g., similarity, box-counting, or Assouad) dimension, conformal dimension, degree of multifractal non-uniformity, coefficient of multifractal asymmetry, index of lacunarity, and index of fractal anisotropy. The difference between morphological properties of mathematical and real-world fractals is especially outlined in this review paper.

The basic idea of the ‘partially nonclassical method’, developed in the present paper, is to apply the invariance requirement of the Lie group method using not all differential consequences of the invariant surface condition but only part of them. It differs from the ‘classical’ method, in which the invariant surface condition is not used, and from the ‘nonclassical’ method, in which all the differential consequences are used. It provides additional possibilities for the symmetry analysis of partial differential equations (PDEs), as compared with the ‘classical’ and ‘nonclassical’ methods, in the so-named no-go case when the group generator, associated with one of the independent variables, is identically zero. The method is applied to the flat steady-state boundary layer problem, reduced to an equation for the stream function, and it is found that applying the partially nonclassical method in the no-go case yields new symmetry reductions and new exact solutions of the boundary layer equations. A computationally convenient unified framework for the classical, nonclassical and partially nonclassical methods (λ-formulation) is developed. The issue of conformal invariance in the context of the Lie group method is considered, stemming from the observation that the classical Lie method procedure yields transformations not leaving the differential polynomial of the PDE invariant but modifying it by a conformal factor. The physical contexts, in which that observation could be important, are discussed using the derivation of the Lorentz transformations of special relativity as an example.

The possibility of using spin connection components as basic quantization variables of a conformal version of general relativity is studied. The considered model contains gravitational degrees of freedom and a scalar dilaton field. The standard tetrad formalism is applied. Properties of spin connections in this model are analyzed. Secondary quantization of the chosen variables is performed. The gravitational part of the model action turns out to be quadratic with respect to the spin connections. So at the quantum level, the model looks trivial, i.e., without quantum self-interactions. Meanwhile the correspondence to general relativity is preserved at the classical level.

We use the concept of gauge transformations in the proof of the invariance of the statistics of zero-vorticity lines in the case of the inverse energy cascade in wave optical turbulence; we study it in the framework of the hydrodynamic approximation of the two-dimensional nonlinear Schrödinger equation for the weight velocity field . The multipoint probability distribution density functions of the vortex field satisfy an infinite chain of Lundgren–Monin–Novikov equations (statistical form of the Euler equations). The equations are considered in the case of the external action in the form of white Gaussian noise and large-scale friction, which makes the probability distribution density function statistically stationary. The main result is that the transformations are local and conformally transform the -point statistics of zero-vorticity lines or the probability that a random curve passes through points for , , where , is invariant under conformal transformations. [ABSTRACT FROM AUTHOR]

A promising method for understanding the geometric properties of a spacetime in the vicinity of the horizon of a Kerr-like black hole can be developed by applying the antipodal boundary condition on the two opposite regions in the extended Penrose diagram. By considering a conformally invariant Lagrangian on a Randall–Sundrum warped five-dimensional spacetime, an exact vacuum solution is found, which can be interpreted as an instanton solution on the Riemannian counterpart spacetime, R + 2 × R 1 × S 1 , where R + 2 is conformally flat. The antipodal identification, which comes with a CPT inversion, is par excellence, suitable when quantum mechanical effects, such as the evaporation of a black hole by Hawking radiation, are studied. Moreover, the black hole paradoxes could be solved. By applying the non-orientable Klein surface, embedded in R 4 , there is no need for instantaneous transport of information. Further, the gravitons become "hard" in the bulk, which means that the gravitational backreaction on the brane can be treated without the need for a firewall. By splitting the metric in a product ω 2 g ˜ μ ν , where ω represents a dilaton field and g ˜ μ ν the conformally flat "un-physical" spacetime, one can better construct an effective Lagrangian in a quantum mechanical setting when one approaches the small-scale area. When a scalar field is included in the Lagrangian, a numerical solution is presented, where the interaction between ω and Φ is manifest. An estimate of the extra dimension could be obtained by measuring the elapsed traversal time of the Hawking particles on the Klein surface in the extra dimension. Close to the Planck scale, both ω and Φ can be treated as ordinary quantum fields. From the dilaton field equation, we obtain a mass term for the potential term in the Lagrangian, dependent on the size of the extra dimension. [ABSTRACT FROM AUTHOR]

In this paper, we study existence of solutions to a conformally invariant integral equation involving Poisson-type kernels. Such integral equation has a stronger non-local feature and is not the dual of any PDE. We obtain the existence of solutions in the antipodal symmetry class. [ABSTRACT FROM AUTHOR]

Optical turbulence is described in terms of multipoint probability density distribution functions (PDF) fn using the Lundgren–Monin–Novikov (LMN) equation (statistical form of the Euler equation) for the field of vortex w = ∇ × u in a 2D flow (u is the weight velocity field). The evolution of Lagrangian particles occurs along the characteristics of the fn equation from the LMN hierarchy. The vorticity is preserved along the characteristics in the absence of an external random force. It is shown that the G group of conformal transformations invariantly transforms the characteristics of the equation with zero vorticity and the family of fn equations for PDF along these lines, or the statistics of zero-vorticity lines. Along other level lines w = const ≠ 0, the statistics is not conformally invariant. In addition, the action of G conserves the PDF class. [ABSTRACT FROM AUTHOR]

The action of an ideal fluid in Euler variables with a variable number of particles is used for the phenomenological description of the processes of particle creation in strong external fields. It has been demonstrated that the conformal invariance of the creation law imposes quite strict restrictions on the possible types of sources. It is shown that combinations with the particle number density in the creation law can be interpreted as dark matter within the framework of this model.

In any medium there are fluctuations due to temperature or due to the quantum nature of its constituents. If a material body is immersed into such a medium, its shape and the properties of its constituents modify the properties of the surrounding medium and its fluctuations. If in the same medium there is a second body then — in addition to all direct interactions between them — the modifications due to the first body influence the modifications due to the second body. This mutual influence results in a force between these bodies. If the excitations of the medium, which mediate the effective interaction between the bodies, are massless, this force is long-ranged and nowadays known as a Casimir force. If the fluctuating medium consists of the confined electromagnetic field in vacuum, one speaks of the quantum mechanical Casimir effect. In the case that the order parameter of material fields fluctuates – such as differences of number densities or concentrations – and that the corresponding fluctuations of the order parameter are long-ranged, one speaks of the critical Casimir effect. This holds, e.g., in the case of systems which undergo a second-order phase transition and which are thermodynamically located near the corresponding critical point, or for systems with a broken continuous symmetry exhibiting Goldstone mode excitations. Here we review the currently available exact results concerning the critical Casimir effect in systems encompassing the one-dimensional Ising, XY, and Heisenberg models, the two-dimensional Ising model, the Gaussian and the spherical models, as well as the mean field results for the Ising and the XY model. Special attention is paid to the influence of the boundary conditions on the behavior of the Casimir force. We present results both for the case of classical critical fluctuations if the system possesses a critical point at a non-zero temperature, as well as the case of quantum systems undergoing a continuous phase transition at zero temperature as function of certain parameters. As confinements we consider the film, the sphere–plane, and the sphere–sphere geometries. We discuss systems governed by short-ranged, by subleading long-ranged (i.e., of the van der Waals type), and by leading long-ranged interactions. In order to put the critical Casimir effect into the proper context and in order to make the review as self-contained as possible, basic facts about the theory of phase transitions, the theory of critical phenomena in classical and quantum systems, and finite-size scaling theory are recalled. Whenever possible, a discussion of the relevance of the exact results towards an understanding of available experiments is presented. The eventual applicability of the present results for certain devices is pointed out, too. [ABSTRACT FROM AUTHOR]

Polyakov [Nuclear Phys. B, 396, 1993] proposed a program for expanding the group of symmetries of hydrodynamic models to the conformal invariance of statistics in inverse cascades, where the conformal group is infinite-dimensional. This study presents group G of transformations of the equation for an -point probability density function fn (PDF) from an infinite chain of the Lundgren–Monin–Novikov equations (the statistical form of the Euler equations) for the field of a vortex of two-dimensional flow. The main result obtained is that the group G transforms conformally the characteristics of the zero-vorticity equation and, invariantly, the family of the fn equations for the PDF along these lines. Along with other characteristics, the equation is not invariant. The action of G retains the PDF class. The results obtained can be used to study the invariance of the statistical properties in optical turbulence. [ABSTRACT FROM AUTHOR]

Let Ω be the multiply connected domain in the extended complex plane ℂ ¯ obtained by removing m non-overlapping rectilinear segments from the infinite strip S = { z : Im z < π / 2 } . In this paper, we present an iterative method for numerical computation of a conformally equivalent bounded multiply connected domain G in the interior of the unit disk D and the exterior of m non-overlapping smooth Jordan curves. We demonstrate the utility of the proposed method through two applications. First, we estimate the capacity of condensers of the form (S,E) where E ⊂ S is a union of disjoint segments. Second, we determine the streamlines associated with uniform incompressible, inviscid and irrotational flow past disjoint segments in the strip S. [ABSTRACT FROM AUTHOR]

In this work, we study a new class of attractor models which we shall call generalized R p -attractor models. This class of models is based on a generalization of the Einstein frame potential of R p f (R) gravity models in the Jordan frame. We present the attractor properties of the corresponding nonminimally coupled Jordan frame theory, and we calculate the observational indices of inflation in the Einstein frame. As we show, there is a large class of nonminimally coupled scalar theories, with an arbitrary nonminimal coupling which satisfies certain conditions, that yield the same Einstein frame potential, this is why these models are characterized attractors. As we demonstrate, the generalized R p -attractor models are viable and well fitted within the Planck constraints. This includes the subclass of the generalized R p -attractor models, namely the Einstein frame potential of R p inflation in the Jordan frame, a feature also known in the literature. We also highlight an important issue related to the R p inflation in the Jordan frame, which is known to be nonviable. By conformal invariance, the R p inflation model should also be viable in the Jordan frame, which is not. We pinpoint the source of the problem using two different approaches in the f (R) gravity Jordan frame, and as we demonstrate, the problem arises in the literature due to some standard simplifications made for the sake of analyticity. We demonstrate the correct way to analyze R p inflation in the Jordan frame, using solely the slow-roll conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exact time-dependent black hole solution on a warped five-dimensional Randall–Sundrum space–time in conformal dilaton gravity. The zeroes of the model are described by a meromorphic quintic polynomial, which has no essential singularities. The quintic equation can be reduced to the Brioschi form by means of the Weierstrass elliptic curve over ℚ. The model fits the antipodal boundary condition, i.e. antipodal points in the projected space are identified using the embedding of a Klein surface in ℂ 2 , using the ℤ 2 symmetry on the two sides of the brane. If one writes (5) g μ ν = ω 4 ∕ 3 (5) g ̃ μ ν , (5) g ̃ μ ν = (4) g ̃ μ ν + n μ n ν , (4) g ̃ μ ν = ω ̄ 2 (4) ḡ μ ν , with n μ the normal to the brane and ω the dilaton field, then (4) ḡ μ ν is conformally flat. It is the contribution from the bulk which determines the real pole on the effective four-dimensional space–time. There is no objection applying 't Hooft's back reaction method in constructing the unitary S-matrix for the Hawking radiation. Again, there is no "inside" of the black hole. The zeroes can also be analyzed by the icosahedron equation and by the Hopf-fibration of the Klein surface. [ABSTRACT FROM AUTHOR]

In this work, we find a harmonic Green function for a doubly connected domain using the theory of conformal mappings and then we define a Poisson kernel. Using these functions an integral representation formula for the solution of the Dirichlet problem for the Poisson equation is derived.

A promising method for understanding the geometric properties of a spacetime in the vicinity of the horizon of a Kerr-like black hole can be developed by applying the antipodal boundary condition on the two opposite regions in the extended Penrose diagram. By considering a conformally invariant Lagrangian on a Randall–Sundrum warped five-dimensional spacetime, an exact vacuum solution is found, which can be interpreted as an instanton solution on the Riemannian counterpart spacetime, R+2×R1×S1, where R+2 is conformally flat. The antipodal identification, which comes with a CPT inversion, is par excellence, suitable when quantum mechanical effects, such as the evaporation of a black hole by Hawking radiation, are studied. Moreover, the black hole paradoxes could be solved. By applying the non-orientable Klein surface, embedded in R4, there is no need for instantaneous transport of information. Further, the gravitons become “hard” in the bulk, which means that the gravitational backreaction on the brane can be treated without the need for a firewall. By splitting the metric in a product ω2g˜μν, where ω represents a dilaton field and g˜μν the conformally flat “un-physical” spacetime, one can better construct an effective Lagrangian in a quantum mechanical setting when one approaches the small-scale area. When a scalar field is included in the Lagrangian, a numerical solution is presented, where the interaction between ω and Φ is manifest. An estimate of the extra dimension could be obtained by measuring the elapsed traversal time of the Hawking particles on the Klein surface in the extra dimension. Close to the Planck scale, both ω and Φ can be treated as ordinary quantum fields. From the dilaton field equation, we obtain a mass term for the potential term in the Lagrangian, dependent on the size of the extra dimension.

We investigate the trajectories of point charges in the background of finite-action vacuum solutions of Maxwell’s equations known as knot solutions. More specifically, we work with a basis of electromagnetic knots generated by the so-called ‘de Sitter method’. We find a variety of behaviors depending on the field configuration and the parameter set used. This includes an acceleration of particles by the electromagnetic field from rest to ultrarelativistic speeds, a quick convergence of their trajectories into a few narrow cones asymptotically for sufficiently high value of the coupling, and a pronounced twisting and turning of trajectories in a coherent fashion. This work is part of an effort to improve the understanding of knotted electromagnetic fields and the trajectories of charged particles they generate, and may be relevant for experimental applications. [ABSTRACT FROM AUTHOR]

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup ⁎ inf type Harnack inequality of Schoen for integral equations. [ABSTRACT FROM AUTHOR]

The loop O(n) model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin O(n) model. It has been predicted by Nienhuis that for 0≤n≤2 the loop O(n) model exhibits a phase transition at a critical parameter xc(n) For 0

We study the spin transport of light in weakly inhomogeneous axion field in a flat Robertson-Walker universe and derive the spin Hall effect for circularly polarized rays. Regarding primordial quantum fluctuations of the axion field in the de Sitter phase as the origin of the inhomogeneity, we show that the conformal invariance of the correlator determines the root-mean-square (r.m.s) fluctuation of the path of circularly polarized cosmic rays. We explain how the r.m.s fluctuation can be experimentally determined.

We establish convergence to SLE6 of the Exploration Processes for the correlated bond-triangular type models studied by two of us in an earlier work (Chayes and Lei in Rev. Math. Phys. 19:511–565, 2007). This (rigorously) establishes that these models are in the same universality class as the standard site percolation model on the triangular lattice. In the context of these models, the result is proven for all domains with boundary Minkowski dimension less than two. Moreover, the proof of convergence applies in the context of general critical 2D percolation models and for general domains, under the stipulation that Cardy’s Formula can be established for domains in this generality.

We show how to extract Cardy’s Formula for a general class of domains given the requisite interior analyticity statement. This is accomplished by a careful study of the interplay between discretization schemes and extraction of limiting boundary values. Of particular importance to the companion work (Binder et al. in J. Stat. Phys., 2010) we establish these results for slit domains and for the critical percolation models introduced in Chayes and Lei (Rev. Math. Phys. 19:511–565, 2007).

We consider conformally invariant energies W on the group GL + (2) of 2 × 2 -matrices with positive determinant, i.e., W : GL + (2) → R such that W (A F B) = W (F) for all A , B ∈ { a R ∈ GL + (2) | a ∈ (0 , ∞) , R ∈ SO (2) } , where SO (2) denotes the special orthogonal group and provides an explicit formula for the (notoriously difficult to compute) quasiconvex envelope of these functions. Our results, which are based on the representation W (F) = h ( λ 1 λ 2 ) of W in terms of the singular values λ 1 , λ 2 of F, are applied to a number of example energies in order to demonstrate the convenience of the singular-value-based expression compared to the more common representation in terms of the distortion K : = 1 2 ‖ F ‖ 2 det F . Applying our results, we answer a conjecture by Adamowicz (in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni, vol 18(2), pp 163, 2007) and discuss a connection between polyconvexity and the Grötzsch free boundary value problem. Special cases of our results can also be obtained from earlier works by Astala et al. (Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, Princeton, 2008) and Yan (Trans Am Math Soc 355(12):4755–4765, 2003). Since the restricted domain of the energy functions in question poses additional difficulties with respect to the notion of quasiconvexity compared to the case of globally defined real-valued functions, we also discuss more general properties related to the W 1 , p -quasiconvex envelope on the domain GL + (n) which, in particular, ensure that a stricter version of Dacorogna's formula is applicable to conformally invariant energies on GL + (2) . [ABSTRACT FROM AUTHOR]

It is well-known that the conformal structure of a relativistic spacetime is of profound physical and conceptual interest. In this note, we consider the analogous structure for Newtonian theories. We show that the Newtonian Weyl tensor is an invariant of this structure. [ABSTRACT FROM AUTHOR]

We show how the theory of the critical behaviour of d-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain in a bridge configuration. We further define multi-bridge networks, where several vertices are in local bridge configurations. We consider all cases of ordinary, mixed and special surface transitions, and polymer chains made of self-avoiding walks, or of mutually-avoiding walks, or at the tricritical Θ -point. In the Θ -point case, generalising the good-solvent case, we relate the critical exponent for simple bridges, γ b Θ , to that of terminally-attached arches, γ 11 Θ , and to the correlation length exponent ν Θ. We find γ b Θ = γ 11 Θ + ν Θ . In the case of the special transition, we find γ b Θ (sp) = 1 2 [ γ 11 Θ (sp) + γ 11 Θ ] + ν Θ . For general networks, the explicit expression of configurational exponents then naturally involves bulk and surface exponents for multiple random paths. In two-dimensions, we describe their Euclidean exponents from a unified perspective, using Schramm–Loewner Evolution (SLE) in Liouville quantum gravity (LQG), and the so-called KPZ relation between Euclidean and LQG scaling dimensions. This is done in the cases of ordinary, mixed and special surface transitions, and of the Θ -point. We provide compelling numerical evidence for some of these results both in two- and three-dimensions. [ABSTRACT FROM AUTHOR]

New images of black holes in recent times have created a strong validity of the Friedmann universe. Since the Friedmann universe possess singularity, synonymously black holes, its quantum aspect becomes more important for study. We have explored the Friedmann universe with reference to higher order gravity theory, a natural generalization of Einstein theory of gravity. Particle creation in vacuum space-time in the presence of strong gravitation field has been explained and thus emphasized upon gravitons creation as well as non-conformal invariance of the higher order theory in this context. [ABSTRACT FROM AUTHOR]

We investigate the space–time of a spinning cosmic string in conformal invariant gravity, where the interior consists of a gauged scalar field. We find exact solutions of the exterior of a stationary spinning cosmic string, where we write the metric as g μ ν = ω 2 g ̃ μ ν , with ω a dilaton field which contains all the scale dependences. The "unphysical" metric g ̃ μ ν is related to the (2 + 1) -dimensional Kerr space–time. The equation for the angular momentum J decouples, for the vacuum situation as well as for global strings, from the other field equations and delivers a kind of spin-mass relation. For the most realistic solution, J falls off as ∼ 1 r and ∂ r J → 0 close to the core. The space–time is Ricci flat. The formation of closed timelike curves can be pushed to space infinity for suitable values of the parameters and the violation of the weak energy condition can be avoided. For the interior, a numerical solution is found. This solution can easily be matched at the boundary on the exterior exact solution by special choice of the parameters of the string. This example shows the power of conformal invariance to bridge the gap between general relativity and quantum field theory. [ABSTRACT FROM AUTHOR]

We review the Baňados–Teitelboim–Zanelli (BTZ) black hole solution in connection with the spinning string solution. We find a new exact solution, which can be related to the (2 + 1)-dimensional spinning point particle solution. There is no need for a cosmological constant, and so the solution can be up-lifted to (3 + 1) dimensions. The exact solution in a conformally invariant gravity model, where the space–time is written as g μ ν = ω 2 g ~ μ ν , is horizon free and has an ergo-circle, while g ~ μ ν is the BTZ solution. The dilaton ω determines the scale of the model. It is conjectured that the conformally invariant non-vacuum BTZ solution will solve the boundary and causality problems which one encounters in spinning cosmic string solutions. [ABSTRACT FROM AUTHOR]

Recently it was shown that self-organized criticality is an important component of the dynamics of cumulus clouds (Najafi et al., 2021). Here we introduce a new algorithm to simulate cumulus clouds in two-dimensional square lattices, based on two important facts: the cohesive energy of wet air parcels and a sand-pile-type diffusion of cloud segments. The latter is realized by considering the evaporation/condensation of air parcels in various regions of the cloud, which enables them to diffuse to the neighboring regions. The results stemming from this model are in excellent agreement with observations reported in the above-cited paper, in which the exponents were determined for two-dimensional earth-to-sky RGB cloud images. The exponents obtained at the lowest condensation level in our model are consistent with the exponents observed in nature. We find that the cloud fields obtained from our model are fractal, with the outer perimeter having a fractal dimension of D f = 1. 25 ± 0. 01. Furthermore, the distributions of the radius of gyration and the loop length follow a power-law function with exponents τ r = 2. 3 ± 0. 1 and τ l = 2. 1 ± 0. 1 , respectively. The loop Green function is found to be logarithmic with the radius of gyration of the loops following the observational results. The winding angle statistic of the external perimeter of the cloud field is also analyzed, showing an exponent in agreement with the fractal dimension, which may serve as the conformal invariance of the system. • Self-organized criticality is at the core of Cumulus cloud formation. • Cumulus clouds emerge from Metropolis Monte Carlo simulations in 2D. • SOC-Ising model and Loop Erased Random walk share the same universality class. • 2D Cumulus cloud projections have conformal invariance. [ABSTRACT FROM AUTHOR]

In this note, we prove that the kernel of the linearized equation around a positive energy solution in ℝn${\mathbb{R}^{n}}$, n≥3${n\geq 3}$, to the problem -Δ⁢W-γ⁢|x|-2⁢V=|x|-s⁢W2⋆⁢(s)-1$-\Delta W-\gamma|x|^{-2}V=|x|^{-s}W^{2^{\star}(s)-1}$ is one-dimensional when s+γ>0$s+\gamma>0$. Here, s∈[0,2)${s\in[0,2)}$, 0≤γ