Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor

The emphasis of this review is both the geometric realization of the 2-point velocity correlation tensor field B_ij (x,x′,t) and isometries of the correlation space K³ equipped with a (pseudo-) Riemannian metrics ds²(t) generated by the tensor field. The special form of this tensor field for homogeneous isotropic turbulence specifies ds²(t) as the semi-reducible pseudo-Riemannian metric. This construction presents the template for the application of methods of Riemannian geometry in turbulence to observe, in particular, the deformation of length scales of turbulent motion localized within a singled out fluid volume of the flow in time. This also allows to use common concepts and technics of Lagrangian mechanics for a Lagrangian system (M^t, ds²(t)), M^t ? K³. Here the metric ds²(t), whose components are the correlation functions, evolves due to the von Kármán-Howarth equation. We review the explicit geometric realization of ds²(t) in K³ and present symmetries (or isometric motions in K³) of the metric ds²(t) which coincide with the sliding deformation of a surface arising under the geometric realization of ds²(t). We expose the fine structure of a Lie algebra associated with this symmetry transformation and construct the basis of differential invariants. Minimal generating set of differential invariants is derived. We demonstrate that the well-known Taylor microscale λ_g is a second-order differential invariant and show how λ_g can be obtained by the minimal generating set of differential invariants and the operators of invariant differentiation. Finally, we establish that there exists a nontrivial central extension of the infinite-dimensional Lie algebra constructed wherein the central charge is defined by the same bilinear skew-symmetric form c as for the Witt algebra which measures the number of internal degrees of freedom of the system. For turbulence, we give the asymptotic expansion of the transversal correlation function for the geometry generated by a quadratic form.     

Classical and quantum integrability for a class of potentials in two dimensions

A method for the construction of the second constant of motion in fourth order is carried out. Correspondingly the fourth order potential equation is obtained whose solutions directly provide the classical integrable systems. Second constant of motion is obtained for a large class of potentials. Quantum invariants are also obtained with second order quantum corrections of the order O(?²) to the corresponding classical invariants. The phase space diagrams for these cases are drawn using a mathematical computer software package in two dimensions.     

Integrable Substructure in a Korteweg Capillarity Model. A Karman-Tsien Type Constitutive Relation

A classical Korteweg capillarity system with a Karman-Tsien type (κ, ρ) constitutive relation is shown, via a Madelung transformation and use of invariants of motion, to admit integrable Hamiltonian subsystems.     

Graph invariants of Vassiliev type and application to 4D quantum gravity

We consider graph invariants of Vassiliev type extended by the quantum group link invariants. When they are expanded byx whereq=e ^x , the expansion coefficients are known as the Vassiliev invariants of finite type. In the present paper, we define tangle operators of graphs given by a functor from a category of colored and oriented graphs embedded into a 3-space to a category of representations of the quasi-triangular ribbon Hopf algebra extended byU _q (sl(2),C)), which are subject to a quantum group analog of the spinor identity. In terms of them, we obtain the graph invariants of Vassiliev type expressed to be identified with Chern Simons vacuum expectation values of Wilson loops including intersection points. We also consider the 4d canonical quantum gravity of Ashtekar. It is verified that the graph invariants of Vassiliev type satisfy constraints of the quantum gravity in the loop space representation of Rovelli and Smolin.This is not the author's present address.     

Topological Invariants in Braid Theory

Many invariants of knots and links have their counterparts in braid theory. Often, these invariants are most easily calculated using braids. A braid is a set of n strings stretching between two parallel planes. This review demonstrates how integrals over the braid path can yield topological invariants. The simplest such invariant is the winding number – the net number of times two strings in a braid wrap about each other. But other, higher-order invariants exist. The mathematical literature on these invariants usually employs techniques from algebraic topology that may be unfamiliar to physicists and mathematicians in other disciplines. The primary goal of this paper is to introduce higher-order invariants using only elementary differential geometry.Some of the higher-order quantities can be found directly by searching for closed one-forms. However, the Kontsevich integral provides a more general route. This integral gives a formal sum of all finite order topological invariants. We describe the Kontsevich integral, and prove that it is invariant to deformations of the braid.Some of the higher-order invariants can be used to generate Hamiltonian dynamics of n particles in the plane. The invariants are expressed as complex numbers; but only the real part gives interesting topological information. Rather than ignoring the imaginary part, we can use it as a Hamiltonian. For n = 2, this will be the Hamiltonian for point vortex motion in the plane. The Hamiltonian for n = 3 generates more complicated motions.     

The interpretation of the C metric. The charged case whene 2 ≤m 2

The charged C metric involves three parametersm, e andA representing mass, charge and acceleration respectively. Using a method developed in a previous paper, we show that whene ² m ² the metric may be interpreted in terms of two Reissner-Nordström particles, each of massm and with charges +e and –e, in accelerated motion and connected by a spring. The method depends on the fact that for certain regions of the coordinate space the charged C metric may be transformed into the Weyl form for a static axisymmetric system. In this form the horizons of the C metric become line sources. One of the regions leads to a Weyl metric with two line sources, one of finite length which corresponds to the outer horizon of a Reissner-Nordström particle and the other semi-infinite corresponding to a horizon associated with uniform accelerated motion. A further coordinate transformation leads to a metric valid for a larger region of space-time in which there are two charged particles in accelerated motion. WhenAm is small, the electromagnetic invariants approximate to those for the Born field for two accelerated charges in special relativity.     

Gauge Theoretical Equivariant Gromov–Witten Invariants and the Full Seiberg–Witten Invariants¶of Ruled Surfaces

Let F be a differentiable manifold endowed with an almost K?hler structure (J,ω), α a J-holomorphic action of a compact Lie group on F, and K a closed normal subgroup of which leaves ω invariant. The purpose of this article is to introduce gauge theoretical invariants for such triples (F,α,K). The invariants are associated with moduli spaces of solutions of a certain vortex type equation on a Riemann surface Σ. Our main results concern the special case of the triple

Link invariants of finite type and perturbation theory

The Vassiliev-Gusarov link invariants of finite type are known to be closely related to perturbation theory for Chern-Simons theory. In order to clarify the perturbative nature of such link invariants, we introduce an algebra V _x containing elements g _i satisfying the usual braid group relations and elements a _i satisfying g _i–g _infi ^sup-1 =a _i, where is a formal variable that may be regarded as measuring the failure of g _infi ^sup2 to equal 1. Topologically, the elements a _i signify intersections. We show that a large class of link invariants of finite type are in one-to-one correspondence with homogeneous Markov traces on V _x. We sketch a possible application of link invariants of finite type to a manifestly diffeomorphisminvariant perturbation theory for quantum gravity in the loop representation.     

On Equilibrium Configurations in Continuum Mechanics of Structural Defects

We consider the determination of the theory by a second order tensor field g_ik and affinity Γ^f_ik. By variational principle for Einstein-Hilbert Lagrangian solid state equilibrium positions of the ideal and real crystal will be described. On account of external Galilei-invariance this theory affords an invariant three dimensional geometry at most being able to produce a stable static equilibrium of defects. The motion of defects is related to the theory of invariants of the internal group of field equations produced by this theory in strong analogy to Maxwell's electrodynamics. The elastic ether concept for the theory of light affords the idea of a gauge field approximation of continuum mechanics fitting linearized Einstein-Hilbert Lagrangian approach. The stress and strain space duality has to be understood on this background.     

Subfactor Realisation of Modular Invariants

 We study the problem of realising modular invariants by braided subfactors and the related problem of classifying nimreps. We develop the fusion rule structure of these modular invariants. This structure is a useful tool in the analysis of modular data from quantum double subfactors, particularly those of the double of cyclic groups, the symmetric group on 3 letters and the double of the subfactors with principal graph the extended Dynkin diagram D ₅ ⁽¹⁾. In particular for the double of S ₃, 14 of the 48 modular modular invariants are nimless, and only 28 of the remaining 34 nimble invariants can be realised by subfactors. Received: 14 February 2003 / Accepted: 3 April 2003 Published online: 19 May 2003 Communicated by H. Araki, D. Buchholz and K. Fredenhagen     

Third and fourth order invariants for one-dimensional time-dependent classical systems

The construction of invariants up to fourth order in velocities has been carried out for one-dimensional, time-dependent classical dynamical systems. While the exact results are recovered for the first and second order integrable systems, the results for the third and fourth order invariants are expressed in terms of nonlinearpotential equations. Noticing the separability of the potential in space and time variables these nonlinear equations are reduced to a tractable form. A possible solution for the third order case suggests a new integrable systemV(q, t) ∼t ^−4/3 q ^1/2. Alexander von Humboldt-Stiftung Fellow, on leave from the Department of Physics, Ramjas College (University of Delhi), Delhi 110 007, India.     

Hydrogen bonding in homochiral dimers of hydroxyesters studied by Raman optical activity spectroscopy

Spectroscopic analysis of homochiral dimerization is important for the understanding of the homochirality of life and enantioselective catalysis. In this paper, (S)‐methyl lactate and related molecules were studied to provide detailed structural information on hydrogen bonding in homochiral dimers of chiral α‐hydroxyesters through the experimental and theoretical study of Raman optical activity. Different homochiral dimers can be distinguished by comparing their simulated Raman optical activity spectra with the experimental results. Hydrogen bonding motions are decoded with the aid of vibrational motion analysis, which are apparently involved in vibrational motions below 800 cm^–1. A common feature related to the chain‐bending mode also indicates the absolute configuration of methyl lactate and related molecules. The differing behavior of electric dipole–electric quadrupole invariants (β(A)²) compared with the electric dipole–magnetic dipole invariant (β(G′)²), suggests that the intermolecular hydrogen bonding motion behaves differently from the intramolecular one in the asymmetric molecular electric and magnetic fields. These results may help understand hydrogen‐bonded self‐recognition and other dynamical features in chiral recognition. Copyright © 2011 John Wiley & Sons, Ltd.     

On invariant and covariant Schwartz distributions in the case of a compact linear group

A proof is given for the representations of invariant and covariant (Schwartz) distributions onR ⁿ , which are often used in theoretical physics. We express invariant distributions as distributions of standard polynomial invariants and decompose covariant distributions in standard polynomial covariants. Our consideration is restricted to compact groups acting linearly onR ⁿ . The representation for invariant distributions is obtained provided the standard invariants form an algebraically independent generating set in the ring of invariant polynomials. As for the standard covariants we assume that in the class of covariant polynomials they provide a unique decomposition into a sum of the standard covariants multiplied with invariant polynomials.     

Differential Invariants of Immersions of Manifolds with Metric Fields

The problem of finding all r^th order differential invariants of immersions of manifolds with metric fields, with values in a left (G¹_m×G¹_n)-manifold is formulated. For obtaining the basis of higher order differential invariants the orbit reduction method is used. As a new result it appears that r^th order differential invariants depending on an immersion f:M N of smooth manifolds M and N and metric fields on them can be factorized through metrics, curvature tensors and their covariant differentials up to the order (r–2), and covariant differentials of the tangent mapping Tf up to the order r. The concept of a covariant differential of Tf is also introduced in this paper. The obtained results are geometrically interpreted as well.This research is supported by grants GAR 201/03/0512 and MSM 143100006.     

Global invariants and equilibrium states in lattice gases

It is now well known that, in addition to the physical conserved quantities, lattice gases also have other unphysical ones related to the discretization of their phase space. From an abstract point of view a lattice gas can be considered like a full discrete Markov processL and these spurious conserved quantities yield the existence of a nonspatially homogeneous equilibrium state forL ^k. We show that a particular set of these conserved quantities is of special interest: Its elements will be called regular. These regular invariants are simply built from the local ones and their projection on each node is always a locally conserved quantity. Moreover, for most models they are one-to-one related to the Gibbs states ofL ^k which remain factorized. It turns out that all the classical known spurious invariants are regular and one can exhibit simple conditions to build models with only regular invariants. For the latter it is then justified to determine the transport coefficients of the locally conserved densities with the Green-Kubo procedure.     

Chern-Simons theory,coloured-oriented braids and link invariants

A method to obtain explicit and complete topological solution of SU(2) Chern-Simons theory onS ³ is developed. To this effect the necessary aspects of the theory of coloured-oriented braids and duality properties of conformal blocks for the correlators ofSU(2) _k Wess-Zumino conformal field theory are presented. A large class of representations of the generators of the groupoid of coloured-oriented braids are obtained. These provide a whole lot of new link invariants of which Jones polynomials are the simplest examples. These new invariants are explicity calculated as illustrations for knots up to eight crossings and twocomponent multicoloured links up to seven crossings.     

Collective hyperspherical coordinates for polyatomic molecules and clusters

For n-body dynamics an analysis is made of the properties of configuration space within a symmetric hyperspherical framework. Coordinates are conveniently broken up into spatial (or external) rotations, kinematic invariants (related to the inertia moments) and kinematic (or internal) rotations. Their usefulness is demonstrated for the study of constrained intramolecular motions and of concerted reactions and for collective modes of polyatomic molecules and clusters. For a fixed hyperradius, which is a measure of total inertia, the space of kinematic invariants is the surface of a right spherical triangle that leads to the tetrahedral (for n = 4) or octahedral (for n ≥ 5) tessellation of the sphere. Alternative parametrizations are discussed, including the proper one to deal with the umbrella inversion motion of ammonia.     

On the Quantum Invariants for the Spherical Seifert Manifolds

We study the Witten–Reshetikhin–Turaev SU(2) invariant for the Seifert manifolds S ³/Gamma where Γ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of modular forms with half-integral weight, and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the Eichler integral. We further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to Γ.     

The ultra-violet question in maximally supersymmetric field theories

We discuss various approaches to the problem of determining which supersymmetric invariants are permitted as counterterms in maximally supersymmetric super Yang–Mills and supergravity theories in various dimensions. We review the superspace non-renormalisation theorems based on conventional, light-cone, harmonic and certain non-Lorentz covariant superspaces, and we write down explicitly the relevant invariants. While the first two types of superspace admit the possibility of one-half BPS counterterms, of the form F ⁴ and R ⁴ respectively, the last two do not. This suggests that UV divergences begin with one-quarter BPS counterterms, i.e. d ² F ⁴ and d ⁴ R ⁴, and this is supported by an entirely different approach based on algebraic renormalisation. The algebraic formalism is discussed for non-renormalisable theories and it is shown how the allowable supersymmetric counterterms can be determined via cohomological methods. These results are in agreement with all the explicit computations that have been carried out to date. In particular, they suggest that maximal supergravity is likely to diverge at four loops in D = 5 and at five loops in D = 4, unless other infinity suppression mechanisms not involving supersymmetry or gauge invariance are at work.     

Further examples of integrable systems in two dimensions

The construction of the second constant of motion of second order for two-dimensional classical systems is carried out in terms ofz=q ₁ +iq ₂ andq=q ₁ −iq ₂. As a result a class of Toda-type potentials admitting second order invariants is explored.