Conformal Covariance Research Articles

Cosmological Einstein-λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-Perfect-Fluid Solutions with Asymptotic Dust or Asymptotic Radiation Equations of State

This article introduces the notions of asymptotic dust and asymptotic radiation equations of state. With these non-linear generalizations of the well known dust or (incoherent) radiation equations of state the perfect-fluid equations lose any conformal covariance or privilege. We analyse the conformal field equations induced with these equations of state. It is shown that the Einstein-λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-perfect-fluid equations with an asymptotic radiation equation of state allow for large sets of Cauchy data that develop into solutions which admit smooth conformal boundaries in the future and smooth extensions beyond. In the case of asymptotic dust equations of state sharp results on the future asymptotic behaviour are not available yet.

Conformal covariance of connection probabilities and fields in 2D critical percolation

AbstractFitting percolation into the conformal field theory framework requires showing that connection probabilities have a conformally invariant scaling limit. For critical site percolation on the triangular lattice, we prove that the probability that n vertices belong to the same open cluster has a well‐defined scaling limit for every . Moreover, the limiting functions transform covariantly under Möbius transformations of the plane as well as under local conformal maps, that is, they behave like correlation functions of primary operators in conformal field theory. In particular, they are invariant under translations, rotations and inversions, and for any . This implies that and , for some constants C2 and C3.We also define a site‐diluted spin model whose n‐point correlation functions Cn can be expressed in terms of percolation connection probabilities and, as a consequence, have a well‐defined scaling limit with the same properties as the functions . In particular, . We prove that the magnetization field associated with this spin model has a well‐defined scaling limit in an appropriate space of distributions. The limiting field transforms covariantly under Möbius transformations with exponent (scaling dimension) 5/48. A heuristic analysis of the four‐point function of the magnetization field suggests the presence of an additional conformal field of scaling dimension 5/4, which counts the number of percolation four‐arm events and can be identified with the so‐called “four‐leg operator” of conformal field theory.

Quantum operations on conformal nets

On a conformal net [Formula: see text], one can consider collections of unital completely positive maps on each local algebra [Formula: see text], subject to natural compatibility, vacuum preserving and conformal covariance conditions. We call quantum operations on [Formula: see text] the subset of extreme such maps. The usual automorphisms of [Formula: see text] (the vacuum preserving invertible unital *-algebra morphisms) are examples of quantum operations, and we show that the fixed point subnet of [Formula: see text] under all quantum operations is the Virasoro net generated by the stress-energy tensor of [Formula: see text]. Furthermore, we show that every irreducible conformal subnet [Formula: see text] is the fixed points under a subset of quantum operations. When [Formula: see text] is discrete (or with finite Jones index), we show that the set of quantum operations on [Formula: see text] that leave [Formula: see text] elementwise fixed has naturally the structure of a compact (or finite) hypergroup, thus extending some results of [M. Bischoff, Generalized orbifold construction for conformal nets, Rev. Math. Phys. 29 (2017) 1750002]. Under the same assumptions, we provide a Galois correspondence between intermediate conformal nets and closed subhypergroups. In particular, we show that intermediate conformal nets are in one-to-one correspondence with intermediate subfactors, extending a result of Longo in the finite index/completely rational conformal net setting [R. Longo, Conformal subnets and intermediate subfactors, Comm. Math. Phys. 237 (2003) 7–30].

Boundary Green’s functions and Minkowski content measure of multi-force-point SLE[formula omitted

We consider a transient chordal SLEκ(ρ1,…,ρm) curve η in H from w to ∞ with force points v1>⋯>vm in (−∞,w−], which intersects and is not boundary-filling on (−∞,vm). The main result is that there is an atomless locally finite Borel measure μη on η∩(−∞,vm] such that for any v<vm, the d-dimensional Minkowski content of η∩[v,vm] exists and equals μη[v,vm], where d=(∑ρj+4)(κ−4−2∑ρj)2κ is the Hausdorff dimension of η∩[v,vm]. In the case that all ρj=0, this measure agrees with the covariant measure derived in Alberts and Sheffield (2011) for chordal SLEκ up to a multiplicative constant. We call such measure a Minkowski content measure, extend it to a class of subsets of Rn, and prove that they satisfy conformal covariance. To construct the Minkowski content measure on η∩[v,vm], we follow the standard approach to derive the existence and estimates of the one- and two-point boundary Green’s functions of η on (−∞,vm), which are the limits of the rescaled probability that η passes through small disks or open real intervals centered at points on (−∞,vm).

Lorentzian 2D CFT from the pAQFT Perspective

We provide a detailed construction of the quantum theory of the massless scalar field on two-dimensional, globally hyperbolic (in particular, Lorentzian) manifolds using the framework of perturbative algebraic quantum field theory. From this we obtain subalgebras of observables isomorphic to the Heisenberg and Virasoro algebras on the Einstein cylinder. We also show how the conformal version of general covariance, as first introduced by Pinamonti as an extension of the construction due to Brunetti, Fredenhagen and Verch, may be applied to natural Lagrangians, which allow one to specify a theory consistently across multiple spacetimes, in order to obtain a simple condition for the conformal covariance of classical dynamics, which is then shown to quantise in the case of a quadratic Lagrangian. We then compare the covariance condition for the stress-energy tensor in the classical and quantum theory in order to obtain a transformation law involving the Schwarzian derivative of the new coordinate, in accordance with a well-known result in the Euclidean literature.

Double Copy for Celestial Amplitudes.

Celestial amplitudes which use conformal primary wave functions rather than plane waves as external states offer a novel opportunity to study properties of amplitudes with manifest conformal covariance and give insight into a potential holographic celestial conformal field theory at the null boundary of asymptotically flat space. Since translation invariance is obscured in the conformal basis, features of amplitudes that heavily rely on it appear to be lost. Among these are the remarkable relations between gauge theory and gravity amplitudes known as the double copy. Nevertheless, properties of amplitudes reflecting fundamental aspects of the perturbative regime of quantum field theory are expected to survive a change of basis. Here we show that there exists a well-defined procedure for a celestial double copy. This requires a generalization of the usual squaring of numerators which entails first promoting them to generalized differential operators acting on external wave functions and then squaring them. We demonstrate this procedure for three- and four-point celestial amplitudes and give an argument for its validity to all multiplicities.

Ground state Dirac bubbles and Killing spinors

We prove a classification result for ground state solutions of the critical Dirac equation on \(\mathbb {R}^n\), \(n\geqslant 2\). By exploiting its conformal covariance, the equation can be posed on the round sphere \(\mathbb {S}^n\) and the non-zero solutions at the ground level are given by Killing spinors, up to conformal diffeomorphisms. Moreover, such ground state solutions of the critical Dirac equation are also related to the Yamabe equation for the sphere, for which we crucially exploit some known classification results.

On the Conformal Mappings and the Global Operator G

Some important global properties of the slice regular functions have been obtained from the global operator $$\begin{aligned}G:= \Vert \mathbf {x}\Vert ^2 \partial _0 + \mathbf {x} \sum _{i=1}^3 x_i \partial _i , \end{aligned}$$ such as a global characterization, a global Cauchy integral theorem and a global Borel–Pompeiu formula, see Colombo et al. (Trans Am Math Soc 365:303–318, 2013), Gonzalez Cervantes (Complex Anal Oper Theory 13:2527–2539, 2019) and Gonzalez Cervantes and Gonzalez-Campos (Complex Var Elliptic Equ 65:1–10, 2020, https://doi.org/10.1080/17476933.2020.1738410) [4, 13, 14], respectively. The aim of this work is to show: some relationships between G and the composition operator with the conformal mappings, a conformal covariance property of G along with its interpretations in terms of a covariant functor, all consequences of these facts for the slice regular functions, a Leibnitz rule associated to the operator G and a characterization of the real Components of slice regular functions in terms of a Non-constant Coefficient second order differential equation.

SLE loop measures

We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of $$\hbox {SLE}_\kappa $$ loop measures for $$\kappa \in (0,8)$$ . First, we construct rooted $$\hbox {SLE}_\kappa $$ loop measures in the Riemann sphere $$\widehat{\mathbb {C}}$$ , which satisfy Mobius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parametrized by its $$(1+\frac{\kappa }{8})$$ -dimensional Minkowski content. Second, by integrating rooted $$\hbox {SLE}_\kappa $$ loop measures, we construct the unrooted $$\hbox {SLE}_\kappa $$ loop measure in $$\widehat{\mathbb {C}}$$ , which satisfies Mobius invariance and reversibility. Third, we use Brownian loop measures to extend the rooted and unrooted $$\hbox {SLE}_\kappa $$ loop measures from $$\widehat{\mathbb {C}}$$ to subdomains of $$\widehat{\mathbb {C}}$$ , which respectively satisfy conformal covariance and conformal invariance, and then further use the conformal invariance to extend unrooted $$\hbox {SLE}_\kappa $$ loop measures to some Riemann surfaces. Finally, using a similar approach, we construct $$\hbox {SLE}_\kappa $$ bubble measures in simply/multiply connected domains rooted at a boundary point. The space-time homogeneity of rooted $$\hbox {SLE}_\kappa $$ loop measures in $$\widehat{\mathbb {C}}$$ confirms a conjecture by Greg Lawler on the existence of such measures.

Conformal two-point correlation functions from the operator product expansion

We compute the most general embedding space two-point function in arbitrary Lorentz representations in the context of the recently introduced formalism in [1, 2]. This work provides a first explicit application of this approach and furnishes a number of checks of the formalism. We project the general embedding space two-point function to position space and find a form consistent with conformal covariance. Several concrete examples are worked out in detail. We also derive constraints on the OPE coefficient matrices appearing in the two-point function, which allow us to impose unitarity conditions on the two-point function coefficients for operators in any Lorentz representations.

The Frank–Lieb approach to sharp Sobolev inequalities

Frank and Lieb gave a new, rearrangement-free, proof of the sharp Hardy–Littlewood–Sobolev inequalities by exploiting their conformal covariance. Using this they gave new proofs of sharp Sobolev inequalities for the embeddings [Formula: see text]. We show that their argument gives a direct proof of the latter inequalities without passing through Hardy–Littlewood–Sobolev inequalities, and, moreover, a new proof of a sharp fully nonlinear Sobolev inequality involving the [Formula: see text]-curvature. Our argument relies on nice commutator identities deduced using the Fefferman–Graham ambient metric.

Split Property for Free Massless Finite Helicity Fields

We prove the split property for any finite helicity free quantum fields. Finite helicity Poincar\'e representations extend to the conformal group and the conformal covariance plays an essential role in the argument. The split property is ensured by the trace class condition: Tr (exp(-s L_0)) is finite for all s>0 where L_0 is the conformal Hamiltonian of the M\"obius covariant restriction of the net on the time axis. We extend the argument for the scalar case presented in [7]. We provide the direct sum decomposition into irreducible representations of the conformal extension of any helicity-h representation to the subgroup of transformations fixing the time axis. Our analysis provides new relations among finite helicity representations and suggests a new construction for representations and free quantum fields with non-zero helicity.

Ising Model: Local Spin Correlations and Conformal Invariance

We study the 2-dimensional Ising model at critical temperature on a simply connected subset {Omega_{delta}} of the square grid {deltamathbb{Z}^{2}}. The scaling limit of the critical Ising model is conjectured to be described by Conformal Field Theory; in particular, there is expected to be a precise correspondence between local lattice fields of the Ising model and the local fields of Conformal Field Theory. Towards the proof of this correspondence, we analyze arbitrary spin pattern probabilities (probabilities of finite spin configurations occurring at the origin), explicitly obtain their infinite-volume limits, and prove their conformal covariance at the first (non-trivial) order. We formulate these probabilities in terms of discrete fermionic observables, enabling the study of their scaling limits. This generalizes results of Hongler (Conformal invariance of Ising model correlations. Ph.D. thesis, [Hon10]), Hongler and Smirnov (Acta Math 211(2):191–225, [HoSm13]), Chelkak, Hongler, and Izyurov (Ann. Math. 181(3), 1087–1138, [CHI15]) to one-point functions of any local spin correlations. We introduce a collection of tools which allow one to exactly and explicitly translate any spin pattern probability (and hence any lattice local field correlation) in terms of discrete complex analysis quantities. The proof requires working with multipoint lattice spinors with monodromy (including construction of explicit formulae in the full plane), and refined analysis near their source points to prove convergence to the appropriate continuous conformally covariant functions.

The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane

We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the $\mathsf{MST}$, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting $\mathsf{MST}$. The topology of convergence is the space of spanning trees introduced by Aizenman et al. [Random Structures Algorithms 15 (1999) 319–365], and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works.

The scaling limits of near-critical and dynamical percolation

We prove that near-critical percolation and dynamical percolation on the triangular lattice \eta \mathbb T have a scaling limit as the mesh \eta tends to 0, in the “quad-crossing” space \mathcal H of percolation configurations introduced by Schramm and Smirnov. The proof essentially proceeds by “perturbing” the scaling limit of the critical model, using the pivotal measures studied in our earlier paper. Markovianity and conformal covariance of these new limiting objects are also established.

Gluon amplitudes as 2d conformal correlators

Recently, spin-one wavefunctions in four dimensions that are conformal primaries of the Lorentz group SL(2,C) were constructed. We compute low-point, tree-level gluon scattering amplitudes in the space of these conformal primary wavefunctions. The answers have the same conformal covariance as correlators of spin-one primaries in a 2d CFT. The BCFW recursion relation between three- and four-point gluon amplitudes is recast into this conformal basis.

Conformal Covariance and the Split Property

We show that for a conformal local net of observables on the circle, the split property is automatic. Both full conformal covariance (i.e. diffeomorphism covariance) and the circle-setting play essential roles in this fact, while by previously constructed examples it was already known that even on the circle, M\"obius covariance does not imply the split property. On the other hand, here we also provide an example of a local conformal net living on the two-dimensional Minkowski space, which - although being diffeomorphism covariant - does not have the split property.

Asymptotic expansions and conformal covariance of the mass of conformal differential operators

We give an explicit description of the full asymptotic expansion of the Schwartz kernel of the complex powers of m-Laplace type operators L on compact Riemannian manifolds in terms of Riesz distributions. The constant term in this asymptotic expansion turns out to be given by the local zeta function of L. In particular, the constant term in the asymptotic expansion of the Green’s function L^{-1} is often called the mass of L, which (in case that L is the Yamabe operator) is an important invariant, namely a positive multiple of the ADM mass of a certain asymptotically flat manifold constructed out of the given data. We show that for general conformally invariant m-Laplace operators L (including the GJMS operators), this mass is a conformal invariant in the case that the dimension of M is odd and that ker L = 0, and we give a precise description of the failure of the conformal invariance in the case that these conditions are not satisfied.

Prescribing integral curvature equation

In this paper we define new curvature functions on $\mathbb{S}^n$ via integral operators. For certain even orders, these curvature functions are equivalent to the classic curvature functions defined via differential operators, but not for all even orders. Existence result for antipodally symmetric prescribed curvature functions on $\mathbb{S}^n$ is obtained. As a corollary, the existence of a conformal metric for an antipodally symmetric prescribed $Q-$curvature functions on $\mathbb{S}^3$ is proved. Curvature functions on general compact manifolds as well as the conformal covariance property for the corresponding integral operator are also addressed.

Liouville Quantum Gravity on the Riemann Sphere

In this paper, we rigorously construct Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov. We establish some of its fundamental properties like conformal covariance under PSL $${_2(\mathbb{C})}$$ -action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly formula. We also make precise conjectures about the relationship of the theory to scaling limits of random planar maps conformally embedded onto the sphere.