Yang-Mills Detour Complexes and Conformal Geometry

Working over a pseudo-Riemannian manifold, for each vector bundle with connection we construct a sequence of three differential operators which is a complex (termed a Yang-Mills detour complex) if and only if the connection satisfies the full Yang-Mills equations. A special case is a complex controlling the deformation theory of Yang-Mills connections. In the case of Riemannian signature the complex is elliptic. If the connection respects a metric on the bundle then the complex is formally self-adjoint. In dimension 4 the complex is conformally invariant and generalises, to the full Yang-Mills setting, the composition of (two operator) Yang-Mills complexes for (anti-)self-dual Yang-Mills connections. Via a prolonged system and tractor connection a diagram of differential operators is constructed which, when commutative, generates differential complexes of natural operators from the Yang-Mills detour complex. In dimension 4 this construction is conformally invariant and is used to yield two new sequences of conformal operators which are complexes if and only if the Bach tensor vanishes everywhere. In Riemannian signature these complexes are elliptic. In one case the first operator is the twistor operator and in the other sequence it is the operator for Einstein scales. The sequences are detour sequences associated to certain Bernstein-Gelfand-Gelfand sequences.     

Hyperbolic Conformal Geometry with Clifford Algebra

In this paper, we study hyperbolic conformal geometry following a Clifford algebraic approach. Similar to embedding an affine space into a one-dimensional higher linear space, we embed the hyperboloid model of the hyperbolic n-space in into . The model is convenient for the study of hyperbolic conformal properties. Besides investigating various properties of the model, we also study conformal transformations using their versor representations.     

Conformal Geometry of the Supercotangent and Spinor Bundles

We study the actions of local conformal vector fields \({X \in {\rm conf}(M,g)}\) on the spinor bundle of (M, g) and on its classical counterpart: the supercotangent bundle \({\mathcal{M}}\) of (M, g). We first deal with the classical framework and determine the Hamiltonian lift of conf (M, g) to \({\mathcal{M}}\) . We then perform the geometric quantization of the supercotangent bundle of (M, g), which constructs the spinor bundle as the quantum representation space. The Kosmann Lie derivative of spinors is obtained by quantization of the comoment map.The quantum and classical actions of conf (M, g) turn, respectively, the space of differential operators acting on spinor densities and the space of their symbols into conf (M, g)-modules. They are filtered and admit a common associated graded module. In the conformally flat case, the latter helps us determine the conformal invariants of both conf (M, g)-modules, in particular the conformally odd powers of the Dirac operator.     

A Tolman-like Compact Model with Conformal Geometry

In this investigation, we study a model of a charged anisotropic compact star by assuming a relationship between the metric functions arising from a conformal symmetry. This mechanism leads to a first-order differential equation containing pressure anisotropy and the electric field. Particular forms of the electric field intensity, combined with the Tolman VII metric, are used to solve the Einstein–Maxwell field equations. New classes of exact solutions generated are expressed in terms of elementary functions. For specific parameter values based on the physical requirements, it is shown that the model satisfies the causality, stability and energy conditions. Numerical values generated for masses, radii, central densities, surface redshifts and compactness factors are consistent with compact objects such as PSR J1614-2230 and SMC X-1.     

Derivation of the Dirac Equation by Conformal Differential Geometry

A rigorous ab initio derivation of the (square of) Dirac’s equation for a particle with spin is presented. The Lagrangian of the classical relativistic spherical top is modified so to render it invariant with respect conformal changes of the metric of the top configuration space. The conformal invariance is achieved by replacing the particle mass in the Lagrangian with the conformal Weyl scalar curvature. The Hamilton-Jacobi equation for the particle is found to be linearized, exactly and in closed form, by an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” of the 4-spinor solution of Dirac’s equation. All quantum features arise from the subtle interplay between the conformal curvature acting on the particle as a potential and the particle motion which affects the geometric “pre-potential” associated to the conformal curvature itself. The theory, carried out here by assuming a Minkowski metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation.     

Quantization of the Space of Conformal Blocks

We consider the discrete Knizhnik–Zamolodchikov connection (qKZ) associated to gl(N), defined in terms of rational R-matrices. We prove that under certain resonance conditions, the qKZ connection has a non-trivial invariant subbundle which we call the subbundle of quantized conformal blocks. The subbundle is given explicitly by algebraic equations in terms of the Yangian Y(gl(N)) action. The subbundle is a deformation of the subbundle of conformal blocks in CFT. The proof is based on an identity in the algebra with two generators x,y and defining relation xy=yx+yy.     

Global Solutions of Einstein—Dirac Equation on the Conformal Space

The difference between the Riemann and Lorentz spinor manifolds of four dimensions is that the Dirac operator of the former is elliptic and that of the latter is hyperbolic.Moreover the spinor group of the former is a compact group and that of the latter is a noncompact group,which is isomorphic to SL(2,C).Hence the results and their interpretation coming from the two theories would be different.In this short note we study only the Lorentz spinor manifold and,especially,the solutions of Einstein-Dirac equations on the conformal space,which is closely related to the AdS/CFT correspondence.     

利于像差校正的共形整流罩内表面面形设计

共形光学系统中,椭球形等厚度整流罩使入射的平行光线经过整流罩后不再平行,变为发散的光线,进而使系统像差急剧增加,不利于后续像差的校正。通过对等厚度共形整流罩的内表面进行重新设计,打破了共形整流罩的等厚度条件,从而在使用较少校正光学透镜的基础上实现了系统像差的校正。通过分析不同级次非球面分别作为整流罩内表面面形时的像差校正效果,确定了将6次非球面作为共形整流罩内表面面形初始结构。通过对内表面进行优化设计,最终得到整流罩内表面面形。结果表明,该方法有效地减小了共形整流罩引入的像差。最后使用固定校正器对内表面变化后的整流罩进行了像差校正,设计结果表明,内表面的改变有效地减少了光学元件数量,消像差效果良好。     

Fredholm Modules on P.C.F. Self-Similar Fractals and Their Conformal Geometry

The aim of the present work is to show how, using the differential calculus associated to Dirichlet forms, it is possible to construct non-trivial Fredholm modules on post critically finite fractals by regular harmonic structures (D, r). The modules are (d _S , ∞)–summable, the summability exponent d _S coinciding with the spectral dimension of the generalized Laplacian operator associated with (D, r). The characteristic tools of the noncommutative infinitesimal calculus allow to define a d _S -energy functional which is shown to be a self-similar conformal invariant. Thiswork has been supported by the project “Teoria ellittica e forme di Dirichlet su spazi frattali” G.N.A.M.P.A. 2008 and by the G.R.E.F.I.-G.E.N.C.O. French-Italian Research Group.     

Space Geometry of Rotating Platforms: An Operational Approach

We study the space geometry of a rotating disk both from a theoretical and operational approach; in particular we give a precise definition of the space of the disk, which is not clearly defined in the literature. To this end we define an extended 3-space, which we call relative space: it is recognized as the only space having an actual physical meaning from an operational point of view, and it is identified as the physical space of the rotating platform. Then, the geometry of the space of the disk turns out to be non Euclidean, according to the early Einstein's intuition; in particular the Born metric is recovered, in a clear and self consistent context. Furthermore, the relativistic kinematics reveals to be self consistent, and able to solve the Ehrenfest's paradox without any need of dynamical considerations or ad hoc assumptions.     

Quantum Field Theory on a Discrete Space and Noncommutative Geometry

We analyze in detail the quantization of a simple noncommutative model of spontaneous symmetry breaking in zero dimensions taking into account the noncommutative setting seriously. The connection to the counting argument of Feynman diagrams of the corresponding theory in four dimensions is worked out explicitly. Special emphasis is put on the motivation as well as the presentation of some well-known basic notions of quantum field theory which in the zero-dimensional theory can be studied without being spoiled by technical complications due to the absence of divergencies.     

Logarithmic Intertwining Operators and the Space of Conformal Blocks Over the Projective Line

We show that the space of logarithmic intertwining operators among logarithmic modules for a vertex operator algebra is isomorphic to the space of 3-point conformal blocks over the projective line. This can be viewed as a generalization of Zhu??s result for ordinary intertwining operators among ordinary modules.     

The Geometry of a Parameter Space of Interacting Particle Systems

We study a four-parameter family of interacting particle systems containing the basic voter model and contact processes. Two processes in this family are related by duality or thinning if and only if their parameters belong to the same orbit of a certain one-dimensional group of linear mappings. This shows that many duals exist.     

Protecting the Conformal Symmetry via Bulk Renormalization on Anti deSitter Space

The problem of perturbative breakdown of conformal symmetry can be avoided, if a conformally covariant quantum field j{\varphi} on d-dimensional Minkowski spacetime is viewed as the boundary limit of a quantum field f{\phi} on d + 1-dimensional Anti-deSitter spacetime (AdS). We study the boundary limit in renormalized perturbation theory with polynomial interactions in AdS, and point out the differences as compared to renormalization directly on the boundary. In particular, provided the limit exists, there is no conformal anomaly. We compute explicitly the one-loop “fish diagram” on AdS₄ by differential renormalization, and calculate the anomalous dimension of the composite boundary field j²{\varphi^2} with bulk interaction kf⁴{\kappa \phi^4}.     

Lanczos-Lovelock and f(R) Gravity from Clifford Space Geometry

A rigorous construction of Clifford-space Gravity is presented which is compatible with the Clifford algebraic structure and permits the derivation of the generalized connections in Clifford spaces (C-space) in terms of derivatives of the C-space metric. We continue by arguing how Lanczos-Lovelock higher curvature gravity can be embedded into gravity in Clifford spaces and suggest how this might also occur for extended gravitational theories based on f(R),f(R _μν ),… actions, for polynomial-valued functions. Black-strings and black-brane metric solutions in higher dimensions D>4 play an important role in finding specific examples.     

Conformal Invariance and Noether Symmetry,Lie Symmetry of Birkhoffian Systems in Event Space

This paper focuses on studying a conformal invariance and a Noether symmetry, a Lie symmetry for a Birkhoffian system in event space. The definitions of the conformal invariance of the system are given. By investigation on the relations between the conformal invariance and the Noether symmetry, the conformal invariance and the Lie symmetry, the expressions of conformal factors of the system under these circumstances are obtained. The Noether conserved quantities and the Hojman conserved quantities directly derived from the conformal invariance are given. Two examples are given to illustrate the application of the results.     

The Map Between Conformal Hypercomplex/ Hyper-Kähler and Quaternionic(-Kähler) Geometry

We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between conformal hypercomplex manifolds (i.e. those that have a closed homothetic Killing vector) and quaternionic manifolds of one quaternionic dimension less. An important role is played by `ξ-transformations', relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds. In this map, the subclass of conformal hyper-Kähler manifolds is mapped to quaternionic-Kähler manifolds. We relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other.