Compactified Minkowski Space Research Articles

Five-dimensional non-Lorentzian conformal field theories and their relation to six-dimensions

We study correlation functions in five-dimensional non-Lorentzian theories with an SU(1, 3) conformal symmetry. Examples of such theories have recently been obtained as Ω-deformed Yang-Mills Lagrangians arising from a null reduction of six-dimensional superconformal field theories on a conformally compactified Minkowski space. The correlators exhibit a rich structure with many novel properties compared to conventional correlators in Lorentzian conformal field theories. Moreover, identifying the instanton number with the Fourier mode number of the dimensional reduction offers a hope to formulate six-dimensional conformal field theories in terms of five-dimensional Lagrangian theories. To this end we show that the Fourier decompositions of six-dimensional correlation functions solve the Ward identities of the SU(1, 3) symmetry, although more general solutions are possible. Conversely we illustrate how one can reconstruct six-dimensional correlation functions from those of a five-dimensional theory, and do so explicitly at 2- and 3-points. We also show that, in a suitable decompactification limit Ω → 0, the correlation functions become those of the DLCQ description.

The dual conformal box integral in Minkowski space

The dual conformal box integral in Minkowski space is not fully determined by the conformal invariants z and z¯. Depending on the kinematic region its value is on a ‘branch’ of the Bloch-Wigner function which occurs in the Euclidean case. Dual special conformal transformations in Minkowski space can change the kinematic region in such a way that the value of the integral jumps to another branch of this function, encoding a subtle breaking of dual conformal invariance for the integral. We classify conformally equivalent configurations of four points in compactified Minkowski space. We show that starting with any configuration, one can reach up to four branches of the integral using dual special conformal transformations. We also show that most configurations with real z and z¯ can be conformally mapped to a configuration in the same kinematic region with two points at infinity, where the box integral can be calculated directly in Minkowski space using only the residue theorem.

START in a Five-Dimensional Conformal Domain

In this paper we give a brief review of the pseudo-Riemannian geometry of the five-dimensional homogeneous space for the conformal group O(4,2). Its topology is described and its relation to the conformally compactified Minkowski space is described. Its metric is calculated using a generalized half-space representation. Compactification via Lie-sphere geometry is outlined. Possible applications to Jaime Keller's START theory may follow by using its predecessor - the 5-optics of Yu. B. Rumer. The point of view of Rumer is given extensively in the last section of the paper. Keywords. Kaluza,Klein, Rumer, conformal symmetry, hyperbolic space, START, fifth dimension, action coordinate, 5-optics

Gravitation on a Homogeneous Domain

Among all plastic deformations of the gravitational Lorentz vacuum \cite{wr1} a particular role is being played by conformal deformations. These are conveniently described by using the homogeneous space for the conformal group SU(2,2)/S(U(2)x U(2)) and its Shilov boundary - the compactified Minkowski space \tilde{M} [1]. In this paper we review the geometrical structure involved in such a description. In particular we demonstrate that coherent states on the homogeneous Kae}hler domain give rise to Einstein-like plastic conformal deformations when extended to \tilde{M} [2].

Geometry and Shape of Minkowski's Space Conformal Infinity

We review and further analyze Penrose's 'light cone at infinity' - the conformal closure of Minkowski space. Examples of a potential confusion in the existing literature about it's geometry and shape are pointed out. It is argued that it is better to think about conformal infinity as of a needle horn supercyclide (or a limit horn torus) made of a family of circles, all intersecting at one and only one point, rather than that of a 'cone'. A parametrization using circular null geodesics is given. Compactified Minkowski space is represented in three ways: as a group manifold of the unitary group U(2) a projective quadric in six-dimensional real space of signature (4,2) and as the Grassmannian of maximal totally isotropic subspaces in complex four--dimensional twistor space. Explicit relations between these representations are given, using a concrete representation of antilinear action of the conformal Clifford algebra Cl(4,2) on twistors. Concepts of space-time geometry are explicitly linked to those of Lie sphere geometry. In particular conformal infinity is faithfully represented by planes in 3D real space plus the infinity point. Closed null geodesics trapped at infinity are represented by parallel plane fronts (plus infinity point). A version of the projective quadric in six-dimensional space where the quotient is taken by positive reals is shown to lead to a symmetric Dupin's type `needle horn cyclide' shape of conformal infinity.

Conformal Symmetry Patterns in Baryon Spectra

Attention is drawn to the fact that the spectra of the baryons of the lightest flavors, the nucleon and the Δ, carry quantum numbers characteristic for an unitary representation of the conformal group. We show that the above phenomenon is well explained for baryons whose internal structure is dominated by a quark-diquark configuration that resides in a conformally compactified Minkowski space time, R1 ⊗ S3, and is described by means of the conformal scale equation there. The R1 ⊗ S3 space-time represents the boundary of the conformally compactified AdS5, on which one expects to encounter a conformal theory in accord with the gauge-gravity duality. Within this context, our model is congruent with AdS5/CFT4.

On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the "cone at infinity" but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge * operator twistors (i.e. vectors of the pseudo-Hermitian space H_{2,2}) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4,2)/Z_2. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H_{p,q} are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.

Quantisation of Twistor Theory by Cocycle Twist

We present the main ingredients of twistor theory leading up to and including the Penrose-Ward transform in a coordinate algebra form which we can then `quantise' by means of a functorial cocycle twist. The quantum algebras for the conformal group, twistor space CP^3, compactified Minkowski space CMh and the twistor correspondence space are obtained along with their canonical quantum differential calculi, both in a local form and in a global *-algebra formulation which even in the classical commutative case provides a useful alternative to the formulation in terms of projective varieties. We outline how the Penrose-Ward transform then quantises. As an example, we show that the pull-back of the tautological bundle on CMh pulls back to the basic instanton on S^4\subset CMh and that this observation quantises to obtain the Connes-Landi instanton on \theta-deformed S^4 as the pull-back of the tautological bundle on our \theta-deformed CMh. We likewise quantise the fibration CP^3--> S^4 and use it to construct the bundle on \theta-deformed CP^3 that maps over under the transform to the \theta-deformed instanton.

Global anti-self-dual Yang-Mills fields in split signature and their scattering

This article concerns solutions to the anti-self-dual Yang Mills (ASDYM) equations in split signature that are global on the double cover of the appropriate conformally compactified Minkowski space = S2 × S2. Ward's ASDYM twistor construction is adapted to this geometry using a correspondence between points of and holomorphic discs in , twistor space, with boundary on the real slice ℝℙ3. Smooth global U(n) solutions to the ASDYM equations on are shown to be in 1 : 1 correspondence with pairs consisting of an arbitrary holomorphic vector bundle E over together with a positive definite Hermitian metric H on E|ℝℙ3. There are no topological or other restrictions on the bundle E. In ultrahyperbolic signature solutions are generically non-analytic or only finitely differentiable and such solutions arise from a corresponding choice of regularity for H. When E is trivial, the twistor data consists of the Hermitian matrix function H on ℝℙ3 up to constants and the correspondence provides a nonlinear generalisation of the X-ray transform. In general it provides a higher-dimensional analogue of the (inverse) scattering transform in which H plays the role of the reflection coefficient and E the algebraic data.

ELLIPTIC THERMAL CORRELATION FUNCTIONS AND MODULAR FORMS IN A GLOBALLY CONFORMAL INVARIANT QFT

Global conformal invariance (GCI) of quantum field theory (QFT) in two and higher space-time dimensions implies the Huygens' principle, and hence, rationality of correlation functions of observable fields [29]. The conformal Hamiltonian H has discrete spectrum assumed here to be finitely degenerate. We then prove that thermal expectation values of field products on compactified Minkowski space can be represented as finite linear combinations of basic (doubly periodic) elliptic functions in the conformal time variables (of periods 1 and τ) whose coefficients are, in general, formal power series in q½ = eiπτ involving spherical functions of the "space-like" fields' arguments. As a corollary, if the resulting expansions converge to meromorphic functions, then the finite temperature correlation functions are elliptic. Thermal 2-point functions of free fields are computed and shown to display these features. We also study modular transformation properties of Gibbs energy mean values with respect to the (complex) inverse temperature [Formula: see text]. The results are used to obtain the thermodynamic limit of thermal energy densities and correlation functions.

Partial wave expansion and Wightman positivity in conformal field theory

A new method for computing exact conformal partial wave expansions is developed and applied to approach the problem of Hilbert space (Wightman) positivity in a nonperturbative four-dimensional quantum field theory model. The model is based on the assumption of global conformal invariance on compactified Minkowski space (GCI). Bilocal fields arising in the harmonic decomposition of the operator product expansion (OPE) prove to be a powerful instrument in exploring the field content. In particular, in the theory of a field L of dimension 4 which has the properties of a (gauge invariant) Lagrangian, the scalar field contribution to the 6-point function of the twist 2 bilocal field is analyzed with the aim to separate the free field part from the nontrivial part.

Rationality of Conformally Invariant Local Correlation Functions on Compactified Minkowski Space

Rationality of the Wightman functions is proven to follow from energy positivity, locality and a natural condition of global conformal invariance (GCI) in any number D of space-time dimensions. The GCI condition allows to treat correlation functions as generalized sections of a vector bundle over the compactification of Minkowski space M and yields a strong form of locality valid for all non-isotropic intervals if assumed true for space-like separations.

Isotropic geometry and twistors in higher dimensions. II. Odd dimensions, reality conditions, and twistor superspaces

This work is a continuation of the study of higher-dimensional generalizations of the twistor correspondence in conformally compactified Minkowski space begun in Part I [J. Harnad and S. Shnider, J. Math. Phys. 33, 3197–3209 (1992)]. The case of odd-dimensional spaces is treated, with an equivariant diffeomorphism (Cartan map) defined between Grassmannians (or flag manifolds) of totally isotropic subspaces of a complex vector space with respect to a nondegenerate quadratic form, and minimal orbits in the Grassmannians (or flag manifolds) based upon spinor modules. Then the Cartan maps for the real forms of signature (p,q) are considered. The extensions to super-Grassmannians are also discussed, with emphasis upon the cases of dimensions 4, 6, and 10. It is found that in dimensions greater than 4, the natural choice for the superdouble flag space correspondence does not result in the expected supernull line foliation arising in the application to the supersymmetric Yang–Mills equations although in dimension 6 the model constructed is relevant to chiral supergravity.

On the structure of Yang–Mills fields in compactified Minkowski space

In this work the Yang–Mills functional with gauge group SU(2) in compactified Minkowski space is reduced to a mechanical Lagrangian which enables one to find new classes of exact dual and nonself-dual solutions. The technique used in the reduction allows a simple interpretation of meron-type solutions. The formalism is then used to produce a good simple analytic approximation to static solutions of Skyrme’s field equation, relevant to low energy hadronic interactions.

The Dirac operator on compactified Minkowski space

We analyze the discrete spectrum of the Dirac operator associated to the two non equivalent spin structures on the compactified Minkowski space S1 x S3, by the usual methods of harmonic analysis. We discuss in detail how the multiplicities of the eigenvalues depend on the radii of S1 and S3, by solving a Diophantine equation.

Riemannian twistors and Hermitian structures on low-dimensional space forms

A Riemannian Kerr Theorem is described, which associates to an integrable Hermitian structure (with singularities) on S4 a complex analytic surface in CP3. To such a Hermitian structure can be associated a harmonic morphism defined on each of the four-dimensional space forms with values in a Riemann surface. This unifies the classifications for the three-dimensional space forms of Baird and Wood and the recent classifications for R4 and S4 by Wood, into one simple equation. As a consequence we derive the classification for harmonic morphisms defined on four-dimensional hyperbolic space with values in a Riemann surface and a new implicit form for harmonic morphisms defined on three-dimensional hyperbolic space. Unification of the Kerr Theorems for S4 and for compactified Minkowski space M is achieved at the complex level after suitably embedding the spaces in compactified, complexified Minkowski space.

Isotropic geometry and twistors in higher dimensions. I. The generalized Klein correspondence and spinor flags in even dimensions

An equivariant diffeomorphism is given between flag varieties of totally isotropic subspaces of a complex inner product space and certain orbits under the complex spin group within flag varieties of the associated spinor modules. This allows a generalization of the Klein correspondence underlying twistor theory, applicable to arbitrary (even) dimensional conformally compactified Minkowski space and its isotropic subspaces.

Quantum deformation of compactified Minkowski space

The quantum GrassmanianG(2|0; ℂ 4|0 ) of “quantum 2-planes ℂ 2|0 in the quantum 4-plane ℂ 4|0 ”, which provides aq-deformation of compactified complexified Minkowski space, is proposed. A quantum analogue of classical Plucker embedding of the usual GrassmanianG(2; ℂ2) into a non-degenerate quadric in ℂℙ5 is found.

Topological gauge theory and twistors

By use of harmonic superfields, Witten's topological gauge theory is written as a field theory on a supertwistor space. Further, using the realisation of complex compactified Minkowski space as a quadric in CP 5, we show that the theory has a simple formulation in terms of analytic superfields on super-CP 5 × CP 3.

Group representations arising from Lorentz conformal geometry

It is shown that there exist conformally covariant differential operators D 2 l, k of all even orders 2 l, on differential forms of all orders k, in the double cover ▪ n of the n-dimensional compactified Minkowski space ▪ n. These act as intertwining differential operators for natural representations of O(2, n), the conformal group of ▪ n. For even n, the resulting decompositions of differential form representations of O ↑(2, n), the orthochronous conformal group, produce infinite families of unitary representations, the most interesting of which are carried by “positive mass-squared, positive frequency” quotients for 2l ⩾ ¦n − 2k¦. Physically, these generalize unitary representations of the conformal group associated with the modified wave operator D 2,0 = □ + ( (n − 2) 2 ) 2 , and the Maxwell operator on vector potentials D 2, (n − 2) 2 = δd . All the representation spaces produced, unitary and nonunitary, may be viewed as infinite systems of harmonic oscillators. As a by-product of the spectral resolution of the D 2 l, k , one gets some striking wave propagative properties for all of the equations D 2 l, k Φ = 0, including Huygens' principle in the curved spacetime ▪ n. The operators D 2 l, k have not been seen before except in the special cases k = 0 or n, and k = (n ± 2) 2 , l = 1 (the Maxwell operator). Thus much new information is obtained even in the physical case n = 4.