Coupling solutions of BGG-equations in conformal spin geometry

BGG-equations are geometric overdetermined systems of partial differential equations (PDEs) on parabolic geometries. Normal solutions of BGG-equations are particularly interesting, and we give a simple formula for the necessary and sufficient additional integrability conditions on a solution. We then discuss a procedure for coupling known solutions of BGG-equations to produce new ones. Employing a suitable calculus for conformal spin structures, this yields explicit coupling formulas and conditions between almost Einstein scales, conformal Killing forms, and twistor spinors. Finally, we discuss a class of generic twistor spinors that provides an invariant decomposition of conformal Killing fields.     

Quantum geometry of a conformal field theory

We show that the central extension of the conformal algebra for the energy momentum tensor of quantized two-dimensional Weyl-Majorana fields coincides with Berry's curvature for adiabatic transport of quantum states on the space of diffeomorphisms. We present explicit expressions for Berry's curvature and connection.     

Outer curvature and conformal geometry of an imbedding

The differential geometry of an imbedded (e.g. string or membrane world sheet) surface in a higher-dimensional background is shown to be conveniently describable (except in the null limit case) in terms of what are designated as its first, second, and third fundamental tensors, which will have the respective symmetry properties η_μν = η_(μν) as a trivial algebraic identity, K_μν^ρ = K_(μν)^ρ as the “generalised Weingarten identity”, which is the (Frobenius type) integrability condition for the imbedding, and Ξ_λμν^ρ = Ξ_(λμν)^ρ as a “generalised Codazzi equation”, which depends on the background geometry being flat or of constant curvature, needing replacement by a more complicated expression for a generic value of the background curvature B_κλ^μ_ν. The “generalised Gauss equation” expressing the dependence on this background curvature of the internal curvature tensor R_κλ^μ_ν of the imbedded surface is converted into terms of the first and second fundamental tensors, and it is thereby demonstrated that the vanishing of the (conformally invariant) “conformation tensor”, i.e. the trace free part C_μν^ρ of the second fundamental tensor K_μv^ρ, is a sufficient condition for conformal flatness of the imbedded surface (and thus in particular for the vanishing of its (Weyl type) conformal curvature tensor C_κλ^μ_ν) provided the background is itself conformally flat. In a trio of which the first two members are the generalised Gauss and Codazzi equations, the “third” member is shown to give an expression in terms of C_μν^ρ for the (trace free, conformally invariant) “outer curvature” tensor Ω_κλ^μ_ν whose vanishing is the condition for feasibility of the natural generalisation of the Walker frame transportation ansatz. The vanishing of C_μν^ρ is shown to be sufficient in a conformally flat background for the vanishing also of Ω_κλ^μ_ν.     

Lobachevsky geometry of (super)conformal mechanics

We give a simple geometric explanation for the similarity transformation mapping one-dimensional conformal mechanics to free-particle system. Namely, we show that this transformation corresponds to the inversion of the Klein model of Lobachevsky space (non-compact complex projective plane) . We also extend this picture to the N=2k superconformal mechanics described in terms of Lobachevsky superspace .     

Spinor methods in conformal Killing transport

The equations of conformal Killing transport are discussed using tensor and spinor methods. It is shown that, in Minkowski space-time, the equations for a null conformal Killing vector ξ ^a are completely determined by the corresponding spinor ω ^A and its covariant derivative, which defines a spinor π _A′ . In conformally flat space-time, the covariant derivative of π _A′ is also involved. Some applications to twistor theory are briefly mentioned.     

Off-shell conformal methods for the superstring

Superconformal field theory is used to compute off-shell amplitudes for the superstring. Off-shell three- and four-point functions in Witten's superstring field theory are obtained. The problem of gauge invariance due to the associativity anomaly in the bosonic sector is resolved to order g².     

On the conformal geometry of transverse Riemann–Lorentz manifolds

Physical reasons suggested in [J.B. Hartle, S.W. Hawking, Wave function of the universe, Phys. Rev. D41 (1990) 1815–1834] for the Quantum Gravity Problem lead us to study type-changing metrics on a manifold. The most interesting cases are Transverse Riemann–Lorentz Manifolds. Here we study the conformal geometry of such manifolds.     

Notes on the hidden conformal symmetry in the near horizon geometry of the Kerr black hole

Toward the Kerr/CFT correspondence for the generic non-extremal Kerr black hole, the analysis of scattering amplitudes by near extremal Kerr provides a clue. This pursuit reveals a hidden conformal symmetry in the low frequency wave equation for a scalar field in a certain spacetime region referred to as the near region. For extremal case, the near region is expected to be the near horizon region in which the correspondence via the asymptotic symmetry is studied. We investigate the hidden conformal symmetry in the near horizon limit and consider the relation between the hidden conformal symmetry and the asymptotic symmetry in the near horizon limit. By using an appropriate definition of the quasi-local charge, we obtain the deviation of the entropy from the extremality.     

Generalised leptonic colour

It is conceivable that there is an SU?_(N)$\mathit{SU}{(N)}_{?}$ ‘colour’ gauge group for leptons, analogous to the gauged SUq₍₃₎$\mathit{SU}{(3)}_q$ colour group of the quarks. The standard model emerges as the low energy effective theory when the leptonic colour is spontaneously broken. The simplest such generalised leptonic colour models are constructed. We show that the see-saw mechanism for small neutrino masses, along with the theoretical constraint of electric charge quantisation, suggests that the models with N=3$N=3$, 5, 7 are the theoretically most promising cases. A striking feature of generalised leptonic colour is the physics associated with the extra leptonic degrees of freedom—the liptons. These particles can potentially be discovered at future colliders, such as the LHC, making the idea testable in the near future.     

Variational methods, multisymplectic geometry and continuum mechanics

This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order multisymplectic field theories with constraints, such as the incompressibility constraint. The results obtained in this paper set the stage for multisymplectic reduction and for the further development of Veselov-type multisymplectic discretizations and numerical algorithms. The latter will be the subject of a companion paper.     

Generalised Kerr-Schild space-times

If V is a space-time with metric tensor g_μν admitting a null, geodesic shear free vector field l_μ, then one may determine a function H so that the spacetime $\text{V}\text{?}$ with metric g_μν = g_μν + 2Hl_μl_ν satisfies the Einstein field equations for various material sources, and for no sources. When V is Minkowski space, $\text{V}\text{?}$ is a Kerr-Schild space-time. In case V is a vacuum space-time, one may choose H so that the source is a null fluid with no pressure. In case V is a Robertson-Walker universe H may be chosen so that the source has a stress-energy tensor with one timelike proper vector and three spacelike ones. There are two equal proper values associated with the latter vectors and one which differs from these. The stress-energy tensor describing this source may be interpreted as representing a perfect fluid with anisotropic pressures or as one describing the sum of a perfect fluid with isotropic pressures and a presureless null fluid. Vaidya's Kerr metric in a cosmological background [Pramana8 (1977) 512–517] is discussed as is the metric representing an accelerating point mass in an expanding universe.     

A quaternion quantum system

The (generalized) quantum mechanics of the two-dimensional quaternionic Hilbert space is constructed explicitly. In this simple example the unconventional properties of such a system as compared to complex quantum mechanics are discussed.     

Generalised Tsallis statistics in electron-positron collisions

The scaling of charged hadron fragmentation functions to the Tsallis distribution for the momentum fraction 0.01?x?0.2 is presented for various e⁺e⁻ collision energies. A possible microcanonical generalisation of the Tsallis distribution is proposed, which gives good agreement with measured data up to x≈1. The proposal is based on superstatistics and a Koba-Nielsen-Olesen (KNO) like scaling of multiplicity distributions in e⁺e⁻ experiments.     

Generalised splitting of spacetime

Normally, when a spacetime splitting is considered the ADM 3+1 split is brought to mind. In this paper, the idea of spacetime splitting is extended to include anm + n splitting of spacetime. The global spacetime has dimension (m + n) and the foliating spaces have dimensionm. There aren independent normals to each of these foliating spaces, thus givingn different extrinsic curvatures. The generalised Gauss-Weingarten and the generalised Gauss-Codazzi equations associated with this splitting are derived. These generalised equations reduce to the familar ADM equations when a 3+1 split is considered. The generalised equations are found to have a particularly elegant form when an orthogonal splitting of spacetime is examined.