Conformally Covariant Research Articles

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.

Another Approach to Juhl's Conformally Covariant Differential Operators from S<sup>n</sup> to S<sup>n-1</sup>

A family $({\mathbf D}_\lambda)_{\lambda\in \mathbb C}$ of differential operators on the sphere $S^n$ is constructed. The operators are conformally covariant for the action of the subgroup of conformal transformations of $S^n$ which preserve the smaller sphere $S^{n-1}\subset S^n$. The family of conformally covariant differential operators from $S^n$ to $S^{n-1}$ introduced by A. Juhl is obtained by composing these operators on $S^n$ and taking restrictions to $S^{n-1}$.

Bergman and Calderón Projectors for Dirac Operators

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$ , we give a new construction of the Calderón projector on $\partial\bar{X}$ and of the associated Bergman projector on the space of L 2 harmonic spinors on $\bar{X}$ , and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of $D_{\bar{g}}$ and the scattering theory for the Dirac operator associated with the complete conformal metric $g=\bar{g}/\rho^{2}$ where ρ is a smooth function on $\bar{X}$ which equals the distance to the boundary near $\partial\bar{X}$ . We show that $\frac{1}{2}(\operatorname{Id}+\tilde{S}(0))$ is the orthogonal Calderón projector, where $\tilde{S}(\lambda)$ is the holomorphic family in {ℜ(λ)≥0} of normalized scattering operators constructed in Guillarmou et al. (Adv. Math., 225(5):2464–2516, 2010), which are classical pseudo-differential of order 2λ. Finally, we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.

On the Bergman Theory for Solenoidal and Irrotational Vector Fields. II. Conformal Covariance and Invariance of the Main Objects

This is a continuation of our work (Gonzalez-Cervantes et al. in On the Bergman theory for solenoidal and irrotational vector fields. I. General theory. Operator theory: advances and applications. Birkhauser, accepted) where for solenoidal and irrotational vector fields theory as well as for the Moisil–Theodoresco quaternionic analysis we introduced the notions of the Bergman space and the Bergman reproducing kernel and studied their main properties. In particular, we described the behavior of the Bergman theory for a given domain whenever the domain is transformed by a conformal map. The formulas obtained hint that the corresponding objects (spaces, operators, etc.) can be characterized as conformally covariant or invariant, and in the present paper we construct a series of categories and functors which allow us to give such characterizations in precise terms.

Conformal covariance of the Abelian sandpile height one field

We study the scaling limit for the height one field of the two-dimensional Abelian sandpile model. The scaling limit for the covariance having height one at two macroscopically distant sites, more generally the centred height one joint moment of a finite number of macroscopically distant sites, is identified and shown to be conformally covariant. The result is based on a representation of the height one joint intensities that is close to a block-determinantal structure.

Manifest superconformal covariance in six-dimensional (2,0) theory

A superconformal generalization of Dirac's formalism for manifest conformal covariance is presented and applied to the free (2,0) tensor multiplet field theory in six dimensions. A graded symmetric superfield, defined on a supercone in a higher-dimensional superspace is introduced. This superfield transforms linearly under the transformations of the supergroup OSp(8*|4), which is the superconformal group of the six-dimensional (2,0) theory. We find the relationship between the new superfield and the conventional (2,0) superfields in six dimensions and show that the implied superconformal transformation laws are correct. Finally, we present a manifestly conformally covariant constraint on the supercone, which reduces to the ordinary differential constraint for the superfields in the six-dimensional space-time.

Conformally Covariant Operators in Clifford Analysis

Following recent work of Peetre and Qian, the conformal covariance of convolution operators involving the fundamental solutions to iterates of the Dirac operator in Euclidean space is described using Vahlen matrices. This conformal covariance is applied to a number of problems, including Dirichlet problems over unbounded domains.

𝒲-gauge structures and their anomalies: An algebraic approach

Starting from flat two-dimensional gauge potentials we propose the notion of 𝒲-gauge structure in terms of a nilpotent Becchi–Rouet–Stora (BRS) differential algebra. The decomposition of the underlying Lie algebra with respect to an SL(2) subalgebra is crucial for the discussion of conformal covariance, in particular the appearance of a projective connection. Different SL(2) embeddings lead to different 𝒲-gauge structures. We present a general soldering procedure which allows one to express zero curvature conditions for the 𝒲-currents in terms of conformally covariant differential operators acting on the 𝒲-gauge fields and to obtain, at the same time, the complete nilpotent BRS differential algebra generated by 𝒲-currents, gauge fields, and the ghost fields corresponding to 𝒲-diffeomorphisms. As illustrations we treat the cases of SL(2) itself and the two different SL(2) embeddings in SL(3), viz., the 𝒲 (1)3- and 𝒲 (2)3-gauge structures, in some detail. In these cases we determine algebraically 𝒲-anomalies as solutions of the consistency conditions and discuss their Chern–Simons origin.

Phase transitions and criticality: Finite-size effects, conformal covariance and other current problems

Abstract This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ| 6 field theories in ( d =3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance ; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!

On the Conformally Covariant Energy-Momentum Tensor

A method is proposed with which one can easily obtain the conformally covariant energy-momentum tensors for various fields from the canonical ones.

Conformally covariant composite operators in quantum chromodynamics

Conformal covariance is shown to determine renormalization properties of composite operators in QCD and in the ϕ 6 3-model at the one-loop level. Its relevance to higher-order (renormalization group improved) perturbative calculations in the short-distance limit is also discussed. Light cone operator product expansions and spectral representations for wavefunctions in QCD are derived.

A manifestly conformally covariant CP 5 integral formula for massless fields

We formulate a new twistor-type contour integral expression for massless fields which displays manifest conformal covariance. Compact complex Minkowski space is treated as a quadric hypersurface in CP 5. When a point at infinity is chosen to break conformal invariance, our formula can be reduced to the standard twistor contour integral formula due to Penrose.

Symplectic structure and conserved quantities for some new conformally covariant systems

Let M be the n-dimensional Minkowski space, n ⩾ 3. One consequence of [1] is that the null space of the equation {(n − 2k + 2)d ∗d + (n − 2k − 2)dd ∗} Φ = 0 on differential k-forms Φ in M is conformally covariant. The same is true of a nonlinear equation obtained by adding to the above a term homogeneous of degree (n + 2) (n − 2) . This generalizes the well-known conformal covariance properties of the wave equation and the equations φ ± φ (n + 2) (n − 2) = 0 when k = 0, and of Maxwell's equations on a vector potential when k = (n ± 2) 2 (and n is even). We define a natural (conformally invariant) symplectic structure for the new equations, and use it to calculate the (n + 1)(n + 2) 2 conserved quantities corresponding to the standard conformal group generators.

Conformally covariant nonlinear equations on tensor-spinors

We prove that to each conformally covariant equation on tensor-spinors on a Riemannian or pseudo-Riemannian manifold with spin structure one can add a nonlinear term without losing the property of conformal covariance. It follows in particular that, on a manifold of dimension n, the nonlinear Dirac equation, Pψ + λ|ψ|1/(n−1)ψ = 0, where P is the Dirac operator and λ is a constant, is conformally covariant. This generalizes a result of Gursey [1]. Some results of Orsted [2], concerning a nonlinear equation associated with the Laplacian on function, and of Branson, concerning distinguished nonlinearities associated with his modified Laplacian on differential forms [3] are also derived as particular cases of this general result.

Conformally covariant massless spin-two field equations

An explicit proof is constructed to show that the field equations for a symmetric tensor fieldhab describing massless spin-2 particles in Minkowski space-time are not covariant under the 15-parameter groupSO4,2; this group is usually associated with conformal transformations on flat space, and here it will be considered as aglobal gauge group which acts upon matter fields defined on space-time. Notwithstanding the above noncovariance, the equations governing the rank-4 tensorSabcd constructed fromhab are shown to be covariant provided the contractionSab vanishes. Conformal covariance is proved by demonstrating the covariance of the equations for the equivalent 5-component complex field; in fact, covariance is proved for a general field equation applicable to massless particles of any spin >0. It is shown that the noncovariance of thehab equations may be ascribed to the fact that the transformation behaviour ofhab is not the same as that of a field consisting of a gauge only. Since this is in contradistinction to the situation for the electromagnetic-field equations, the vector form of the electromagnetic equations is cast into a form which can be duplicated for thehab-field. This procedure results in an alternative, covariant, field equation forhab.

On conformally covariant spinor field equations

A spinor field equation, covariant with respect to the general conformal group (including reflections), should consist in general of not less than eight linear equations and then, in Minkowski space, could be represented by not less than two massless Dirac equations. Their reduction through projectors to only one equation, while not spoiling conformal covariance implies unphysical consequences. It is shown instead that two Dirac equations may be brought unambiguously through a stereographic projection to a manifestly conformal covariant form inE 4,2 space. The physical implications are discussed and it is shown that if the fundamental elementary interactions are expressed in terms of conformal semispinors (which can never appear as free particles), then the corresponding physical Dirac spinors appear in the elementary interactions in terms of their chiral projections. This could indicate both the conformally invariant origin of weak interactions and their fundamental character. The possibility of constructing unified models from conformally invariant Lagrangians is envisaged.

The Transformation Behaviour of Fields in Conformally Covariant Quantum Field Theories

AbstractWe study the transformation behaviour of any covariant component ϕAB(x) obtained from a spinorial field operator ϕAB(x) by Fourier decomposition on the centre of the universal covering group of the conformal group for four dimensional Minkowski space.

On conformal covariance in Thirring model

We show that for the derivative coupling model and for the Thirring model with anomalous spin the theory is conformally covariant for a quantized set of coupling-constant values.

On the Scale Dimension of Conformally Covariant Spinor Fields

The spinor equation of Barut and Haugen is shown to admit no solutions having a definite scale dimension in Minkowski space.

Tensor representations of conformal algebra and conformally covariant operator product expansion

A conformally covariant formulation of operator product expansion is discussed as an expansion of the product of two representations into a direct sum of irreducible representations. The basic irreducible representations are analyzed and classified. The isomorphism between the conformal algebra and the O(4, 2) algebra is used to obtain a manifestly covariant formalism. The implications of the isomorphism in the derivation of the representations is discussed. The covariant O(4, 2) formalism directly relates dominant terms to nondominant terms in the light-cone limit. The essential coincidence of the problem of a conformal covariant operator product expansion to the problem of determining the form of the three-point function is stressed, together with the relevance of a selection rule for two-point functions following from exact (not spontaneously broken) conformal covariance. The role of Ward identities in a conformal covariant scheme is pointed out, and the mathematical implications on the n-point functions from causality are described.