Perturbative Solutions of the Extended Constraint Equations in General Relativity

The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface in an asymptotically simple space-time satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-linear system of partial differential equations for the induced metric, the second fundamental form and two other tensorial quantities defined on , and are equivalent to the usual constraint equations that satisfies as a space-like hypersurface in a space-time satisfying Einstein’s vacuum equation. This article develops a method for finding perturbative, asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on Euclidean space. This method is fundamentally different from the ‘classical’ method of Lichnerowicz and York that is used to solve the usual constraint equations.     

Full-Wave Invisibility of Active Devices at All Frequencies

There has recently been considerable interest in the possibility, both theoretical and practical, of invisibility (or “cloaking”) from observation by electromagnetic (EM) waves. Here, we prove invisibility with respect to solutions of the Helmholtz and Maxwell’s equations, for several constructions of cloaking devices. The basic idea, as in the papers [GLU2, GLU3, Le, PSS1], is to use a singular transformation that pushes isotropic electromagnetic parameters forward into singular, anisotropic ones. We define the notion of finite energy solutions of the Helmholtz and Maxwell’s equations for such singular electromagnetic parameters, and study the behavior of the solutions on the entire domain, including the cloaked region and its boundary. We show that, neglecting dispersion, the construction of [GLU3, PSS1] cloaks passive objects, i.e., those without internal currents, at all frequencies k. Due to the singularity of the metric, one needs to work with weak solutions. Analyzing the behavior of such solutions inside the cloaked region, we show that, depending on the chosen construction, there appear new “hidden” boundary conditions at the surface separating the cloaked and uncloaked regions. We also consider the effect on invisibility of active devices inside the cloaked region, interpreted as collections of sources and sinks or internal currents. When these conditions are overdetermined, as happens for Maxwell’s equations, generic internal currents prevent the existence of finite energy solutions and invisibility is compromised. We give two basic constructions for cloaking a region D contained in a domain , from detection by measurements made at of Cauchy data of waves on Ω. These constructions, the single and double coatings, correspond to surrounding either just the outer boundary of the cloaked region, or both and , with metamaterials whose EM material parameters (index of refraction or electric permittivity and magnetic permeability) are conformal to a singular Riemannian metric on Ω. For the single coating construction, invisibility holds for the Helmholtz equation, but fails for Maxwell’s equations with generic internal currents. However, invisibility can be restored by modifying the single coating construction, by either inserting a physical surface at or using the double coating. When cloaking an infinite cylinder, invisibility results for Maxwell’s equations are valid if the coating material is lined on with a surface satisfying the soft and hard surface (SHS) boundary condition, but in general not without such a lining, even for passive objects.     

An Index for 4 Dimensional Super Conformal Theories

We present a trace formula for an index over the spectrum of four dimensional superconformal field theories on S ³ × time. Our index receives contributions from states invariant under at least one supercharge and captures all information – that may be obtained purely from group theory – about protected short representations in 4 dimensional superconformal field theories. In the case of the theory our index is a function of four continuous variables. We compute it at weak coupling using gauge theory and at strong coupling by summing over the spectrum of free massless particles in AdS ₅ × S ⁵ and find perfect agreement at large N and small charges. Our index does not reproduce the entropy of supersymmetric black holes in AdS ₅, but this is not a contradiction, as it differs qualitatively from the partition function over supersymmetric states of the theory. We note that entropy for some small supersymmetric AdS ₅ black holes may be reproduced via a D-brane counting involving giant gravitons. For big black holes we find a qualitative (but not exact) agreement with the naive counting of BPS states in the free Yang Mills theory. In this paper we also evaluate and study the partition function over the chiral ring in the Yang Mills theory.     

Jet flavor conversions in the quark-gluon plasma

A quark or gluon jet traversing through a quark-gluon plasma can be converted to a gluon or quark jet through scattering with the thermal quarks and gluons in the quark-gluon plasma. Their conversion rates due to two-body elastic and inelastic scattering have recently been evaluated in the lowest order in QCD. Including both energy loss and conversions of quark and gluon jets in the expanding quark-gluon plasma produced in relativistic heavy ion collisions, a net conversion of quark jets to gluon jets has been found. This reduces the difference between the nuclear modification factors for quark and gluon jets in heavy ion collisions and thus enhances the ratios of high transverse momentum protons and antiprotons to pions that are produced from the fragmentation of these jets. To account for the observed similar ratios in central Au + Au and p + p collisions at same energy requires, however, a much larger net quark to gluon jet conversion rate than that given by the lowest-order QCD, indicating the importance of higher-order processes and the strongly coupling properties of the quark-gluon plasma in describing the propagation of jets in the quark-gluon plasma. B.-W. Zhang: On leave from: Institute of Particle Physics, Huazhong Normal University, Wuhan 430079, China Correspondence: C.-M. Ko, Cyclotron Institute and Physics Department, Texas A&M University, College Station, TX 77843-3366, USA     

Wavepacket Preservation Under Nonlinear Evolution

We study nonlinear systems of hyperbolic PDE’s in , the hyperbolicity is understood in a wider sense, namely multiple roots of the characteristic equation are allowed and dispersive equations are permitted. They describe wave propagation in dispersive nonlinear media such as, for example, electromagnetic waves in nonlinear photonic crystals. The initial data is assumed to be a finite sum of wavepackets referred to as a multi-wavepacket. The wavepackets and the medium nonlinearity are characterized by two principal small parameters β and where: (i) is a factor describing spatial extension of involved wavepackets; (ii) is a factor describing the relative magnitude of the linear part of the evolution equation compared to its nonlinearity. A key element in our approach is a proper definition of a wavepacket. Remarkably, the introduced definition has a flexibility sufficient for a wavepacket to preserve its defining properties under a general nonlinear evolution for long times. In particular, the corresponding wave vectors and the band numbers of involved wavepackets are “conserved quantities”. We also prove that the evolution of a multi-wavepacket is described with high accuracy by a properly constructed system of envelope equations with a universal nonlinearity. The universal nonlinearity is obtained by a time averaging applied to the original nonlinearity, in simpler cases the averaged system turns into a system of Nonlinear Schrodinger equations.     

Approximating Multi-Dimensional Hamiltonian Flows by Billiards

The behavior of a point particle traveling with a constant speed in a region , undergoing elastic collisions at the regions’s boundary, is known as the billiard problem. Various billiard models serve as approximation to the classical and semi-classical motion in systems with steep potentials (e.g. for studying classical molecular dynamics, cold atom’s motion in dark optical traps and microwave dynamics). Here we develop methodologies for examining the validity and accuracy of this approximation. We consider families of smooth potentials , that, in the limit , become singular hard-wall potentials of multi-dimensional billiards. We define auxiliary billiard domains that asymptote, as to the original billiards, and provide, for regular trajectories, asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the C ^r norm and an iteration scheme for improving this approximation. Applying this theory to smooth potentials that limit to the multi-dimensional close to ellipsoidal billiards, we predict when the billiard’s separatrix splitting (which appears, for example, in the nearly flat and nearly oblate ellipsoids) persists for various types of potentials.     

The York map as a Shanmugadhasan canonical transformation in tetrad gravity and the role of non-inertial frames in the geometrical view of the gravitational field

A new parametrization of the 3-metric allows to find explicitly a York map by means of a partial Shanmugadhasan canonical transformation in canonical ADM tetrad gravity. This allows to identify the two pairs of physical tidal degrees of freedom (the Dirac observables of the gravitational field have to be built in term of them) and 14 gauge variables. These gauge quantities, whose role in describing generalized inertial effects is clarified, are all configurational except one, the York time, i.e. the trace of the extrinsic curvature of the instantaneous 3-spaces (corresponding to a clock synchronization convention) of a non-inertial frame centered on an arbitrary observer. In the Dirac Hamiltonian is the sum of the weak ADM energy (whose density is coordinate-dependent, containing the inertial potentials) and of the first-class constraints. The main results of the paper, deriving from a coherent use of constraint theory, are: (i) The explicit form of the Hamilton equations for the two tidal degrees of freedom of the gravitational field in an arbitrary gauge: a deterministic evolution can be defined only in a completely fixed gauge, i.e. in a non-inertial frame with its pattern of inertial forces. The simplest such gauge is the 3-orthogonal one, but other gauges are discussed and the Hamiltonian interpretation of the harmonic gauges is given. This frame-dependence derives from the geometrical view of the gravitational field and is lost when the theory is reduced to a linear spin 2 field on a background space-time. (ii) A general solution of the super-momentum constraints, which shows the existence of a generalized Gribov ambiguity associated to the 3-diffeomorphism gauge group. It influences: (a) the explicit form of the solution of the super-momentum constraint and then of the Dirac Hamiltonian; (b) the determination of the shift functions and then of the lapse one. (iii) The dependence of the Hamilton equations for the two pairs of dynamical gravitational degrees of freedom (the generalized tidal effects) and for the matter, written in a completely fixed 3-orthogonal Schwinger time gauge, upon the gauge variable , determining the convention of clock synchronization. The associated relativistic inertial effects, absent in Newtonian gravity and implying inertial forces changing from attractive to repulsive in regions with different sign of , are completely unexplored and may have astrophysical relevance in the interpretation of the dark side of the universe.     

Quark-gluon string-model description of baryon production in <Emphasis Type="Italic">K</Emphasis><Superscript>±</Superscript><Emphasis Type="Italic">N</Emphasis> interactions

The process of baryon production in Kp collisions at high energies is considered in the framework of the quark-gluon string model. The contribution of the string-junction mechanism to the strange-baryon production is analyzed. The results of numerical calculations are in reasonable agreement with the data on inclusive spectra of p, Λ, and on the /Λ asymmetry. The predictions for inclusive spectra and production asymmetry of Ξ and Ω baryons are presented. The text was submitted by the authors in English.     

The Spectral Shift Function and Spectral Flow

At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of both eta invariants and spectral flow. Using ideas of [24] Singer’s proposal was brought to an advanced level in [16] where a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry. This formula can be used for computing spectral flow in a general semifinite von Neumann algebra as described and reviewed in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae for spectral flow between a pair of unbounded self-adjoint operators D and D +  V with D having compact resolvent belonging to a general semifinite von Neumann algebra and the perturbation . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing a new idea from [3]. There it was observed that M. G. Krein’s spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein’s theory to the setting of semifinite spectral triples where D has compact resolvent belonging to and V is any bounded self-adjoint operator in . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory and the analytic theory of spectral flow. It is this interplay that enables us to take Singer’s idea much further to create a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism of [8–10]. One surprising conclusion that follows from our results is that the Krein spectral shift function is computed, in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2].     

Polyakov Soldering and Second-Order Frames: The Role of the Cartan Connection

The so-called ‘soldering’ procedure performed by Polyakov (Int J Math Phys A5, 833–842, 1990) for a -gauge theory is geometrically explained in terms of a Cartan connection on second-order frames of the projective space P¹. The relationship between a Cartan connection and the usual (Ehresmann) connection on a principal bundle allows to gain an appropriate insight into the derivation of the genuine ‘diffeomorphisms out of gauge transformations’ given by Polyakov himself. Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix-Marseille I, Aix-Marseille II et de l’Université du Sud Toulon-Var. Unité affiliée à la FRUMAM Fédération de Recherche 2291.     

Local Asymptotic Normality in Quantum Statistics

The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state ϕ _u of an algebra of canonical commutation relations. The convergence holds for all “local parameters” such that parametrizes a neighborhood of a fixed point . In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and investigate the relation between the two notions of convergence. For the reader’s convenience and completeness we review the relevant results of the classical as well as the quantum theory. Dedicated to Slava Belavkin on the occasion of his 60th anniversary     

Hidden Symmetries and Integrable Hierarchy of the $${\mathcal{N}}$$ = 4 Supersymmetric Yang-Mills Equations

We describe an infinite-dimensional algebra of hidden symmetries of supersymmetric Yang-Mills (SYM) theory. Our derivation is based on a generalization of the supertwistor correspondence. Using the latter, we construct an infinite sequence of flows on the solution space of the SYM equations. The dependence of the SYM fields on the parameters along the flows can be recovered by solving the equations of the hierarchy. We embed the SYM equations in the infinite system of the hierarchy equations and show that this SYM hierarchy is associated with an infinite set of graded symmetries recursively generated from supertranslations. Presumably, the existence of such nonlocal symmetries underlies the observed integrable structures in quantum SYM theory. On leave from Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia. Address after October 1st, 2006: Theoretical Physics Group, The Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, United Kingdom.     

Strominger–Yau–Zaslow Geometry,Affine Spheres and Painlevé III

We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitzéica equation as reductions of real forms of anti–self–dual Yang–Mills equations by two translations, or equivalently as a special case of the Hitchin equation. We use the Loftin–Yau–Zaslow construction to give an explicit expression for a six–real dimensional semi–flat Calabi–Yau metric in terms of a solution to the affine-sphere equation and show how a subclass of such metrics arises from 3^rd Painlevé transcendents.     

On a Unified Theory of Generalized Branes Coupled to Gauge Fields,Including the Gravitational and Kalb–Ramond Fields

We investigate a theory in which fundamental objects are branes described in terms of higher grade coordinates encoding both the motion of a brane as a whole, and its volume evolution. We thus formulate a dynamics which generalizes the dynamics of the usual branes. Geometrically, coordinates and associated coordinate frame fields {} extend the notion of geometry from spacetime to that of an enlarged space, called Clifford space or C-space. If we start from four-dimensional spacetime, then the dimension of C-space is 16. The fact that C-space has more than four dimensions suggests that it could serve as a realization of Kaluza-Klein idea. The “extra dimensions” are not just the ordinary extra dimensions, they are related to the volume degrees of freedom, therefore they are physical, and need not be compactified. Gauge fields are due to the metric of Clifford space. It turns out that amongst the latter gauge fields there also exist higher grade, antisymmetric fields of the Kalb–Ramond type, and their non-Abelian generalization. All those fields are naturally coupled to the generalized branes, whose dynamics is given by a generalized Howe–Tucker action in curved C-space.     

Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral <Emphasis Type="Italic">su</Emphasis>(2) WZNW Model

A zero modes’ Fock space is constructed for the extended chiral WZNW model. It gives room to a realization of the fusion ring of representations of the restricted quantum universal enveloping algebra at an even root of unity, and of its infinite dimensional extension by the Lusztig operators We provide a streamlined derivation of the characteristic equation for the Casimir invariant from the defining relations of A central result is the characterization of the Grothendieck ring of both and in Theorem 3.1. The properties of the fusion ring in are related to the braiding properties of correlation functions of primary fields of the conformal current algebra model.      

Hidden Grassmann Structure in the XXZ Model

For the critical XXZ model, we consider the space of operators which are products of local operators with a disorder operator. We introduce two anti-commutative families of operators which act on . These operators are constructed as traces over representations of the q-oscillator algebra, in close analogy with Baxter’s Q-operators. We show that the vacuum expectation values of operators in can be expressed in terms of an exponential of a quadratic form of . On leave of absence from Skobeltsyn Institute of Nuclear Physics, MSU, 119992, Moscow, Russia Membre du CNRS     

UA(1) anomaly and $ \eta^{{\prime}}_{}$ -mass from an infrared singular quark-gluon vertex

The U _A(1) problem of QCD is inevitably tied to the infrared behaviour of quarks and gluons with its most visible effect being the -mass. A dimensional argument of Kogut and Susskind showed that the mixing of the pseudoscalar flavour-singlet mesons with gluons can provide a screening of the Goldstone pole in this channel if the full quark-quark interaction is strongly infrared singular as ∼ 1/k ⁴ . We investigate this idea using previously obtained results for the Landau gauge ghost and gluon propagator, together with recent determinations for the singular behaviour of the quark-gluon vertex. We find that, even with an infrared vanishing gluon propagator, the singular structure of the quark-gluon vertex for certain kinematics is apposite for yielding a non-zero screening mass.     

Fragmentation function and hadronic production of the heavy supersymmetric hadrons

The light top-squark may be the lightest squark, and its lifetime may be ‘long enough’ in a kind of SUSY models that have not been ruled out yet experimentally, so colorless ‘supersymmetric hadrons (superhadrons)’ (q is a quark excluding the t-quark) may be formed as long as the light top-squark can be produced. The fragmentation function of into heavy ‘supersymmetric hadrons (superhadrons)’ (Q̄=c̄ or b̄) and hadronic production of the superhadrons are investigated quantitatively. The fragmentation function is calculated precisely. Due to the difference in spin of the SUSY component, the asymptotic behavior of the fragmentation function is different from those of the existing ones. The fragmentation function is also applied to compute the production of heavy superhadrons at the hadronic colliders Tevatron and LHC in the so-called fragmentation approach. The resultant cross-section for the heavy superhadrons is too small to observe at Tevatron, but large enough at LHC, when all the relevant parameters in the SUSY models are taken within the favored region for the heavy superhadrons. The production of ‘light superhadrons’ (q=u,d,s) is also roughly estimated with the same SUSY parameters. It is pointed out that the production cross-sections of the light superhadrons may be much greater than those of the heavy superhadrons, so that even at Tevatron the light superhadrons may be produced in great quantities. PACS 12.38.Bx; 13.87.Fh; 12.60.Jv; 14.80.Ly     

Dynamics of quark-gluon plasma from field correlators

It is argued that strong dynamics in the quark-gluon plasma and bound states of quarks and gluons is mostly due to nonperturbative effects described by field correlators. The emphasis in the paper is made on two explicit calculations of these effects from first principles—one analytic using gluelump Green’s functions and another using independent lattice data on correlators. The resulting hadron spectra are investigated in the range T _c ≤ T < 2T _c . The spectra of charmonia, bottomonia, light s mesons, glueballs, and quark-gluon states calculated numerically are in general agreement with lattice MEM data. The possible role of these bound states in the thermodynamics of quark—gluon plasma is discussed. The text was submitted by the authors in English.