Showing papers in "Symmetry Integrability and Geometry-methods and Applications in 2010"

Manifestly conformal descriptions and higher symmetries of bosonic singletons

Abstract: The usual ambient space approach to conformal fields is based on identifying the d-dimensional conformal space as the Dirac projective hypercone in a flat d+2-dimensional ambient space. In this work, we explicitly concentrate on singletons of any integer spin and propose an approach that allows one to have both locality and conformal symmetry manifest. This is achieved by using the ambient space representation in the fiber rather than in spacetime. This approach allows us to characterize a subalgebra of higher symmetries for any bosonic singleton, which is a candidate higher-spin algebra for mixed symmetry gauge fields on anti de Sitter spacetime. Furthermore, we argue that this algebra actually exhausts all higher symmetries.

Bidifferential calculus approach to AKNS hierarchies and their solutions

Abstract: We express AKNS hierarchies, admitting reductions to matrix NLS and matrix mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a universal result in this framework quickly generates an infinite family of exact solutions, including e.g. the matrix solitons in the focusing NLS case. Exploiting a general Miura transformation, we recover the generalized Heisenberg magnet hierarchy and establish a corresponding solution formula for it. Simply by exchanging the roles of the two derivations of the bidifferential graded algebra, we recover "negative flows", leading to an extension of the respective hierar- chy. In this way we also meet a matrix and vector version of the short pulse equation and also the sine-Gordon equation. For these equations corresponding solution formulas are also derived. In all these cases the solutions are parametrized in terms of matrix data that have to satisfy a certain Sylvester equation.

On Quadrirational Yang-Baxter Maps ?

Abstract: We use the classification of the quadrirational maps given by Adler, Bobenko and Suris to describe when such maps satisfy the Yang-Baxter relation. We show that the corresponding maps can be characterized by certain singularity invariance condition. This leads to some new families of Yang-Baxter maps corresponding to the geometric symmetries of pencils of quadrics.

κ-Minkowski Spacetimes and DSR Algebras: Fresh Look and Old Problems

Abstract: Some classes of Deformed Special Relativity (DSR) theories are reconsidered within the Hopf algebraic formulation. For this purpose we shall explore a minimal frame- work of deformed Weyl-Heisenberg algebras provided by a smash product construction of DSR algebra. It is proved that this DSR algebra, which uniquely unifies -Minkowski spacetime coordinates with Poincare generators, can be obtained by nonlinear change of generators from undeformed one. Its various realizations in terms of the standard (unde- formed) Weyl-Heisenberg algebra opens the way for quantum mechanical interpretation of DSR theories in terms of relativistic (Stuckelberg version) Quantum Mechanics. On this basis we review some recent results concerning twist realization of -Minkowski spacetime described as a quantum covariant algebra determining a deformation quantization of the corresponding linear Poisson structure. Formal and conceptual issues concerning quantum -Poincare and -Minkowski algebras as well as DSR theories are discussed. Particularly, the so-called "q-analog" version of DSR algebra is introduced. Is deformed special relativity quantization of doubly special relativity remains an open question. Finally, possible physi- cal applications of DSR algebra to description of some aspects of Planck scale physics are shortly recalled.

Exotic Galilean Symmetry and Non-Commutative Mechanics ?

Abstract: Some aspects of the "exotic" particle, associated with the two-parameter central extension of the planar Galilei group are reviewed. A fundamental property is that it has non-commuting position coordinates. Other and generalized non-commutative models are also discussed. Minimal as well as anomalous coupling to an external electromagnetic field is presented. Supersymmetric extension is also considered. Exotic Galilean symmetry is also found in Moyal field theory. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects.

Linearization of Second-Order Ordinary Differential Equations by Generalized Sundman Transformations

Abstract: The linearization problem of a second-order ordinary differential equation by the generalized Sundman transformation was considered earlier by Duarte, Moreira and Santos using the Laguerre form. The results obtained in the present paper demonstrate that their solution of the linearization problem for a second-order ordinary differential equation via the generalized Sundman transformation is not complete. We also give examples which show that the Laguerre form is not sufficient for the linearization problem via the generalized Sundman transformation.

The Multitrace Matrix Model of Scalar Field Theory on Fuzzy CP^n

Abstract: We perform a high-temperature expansion of scalar quantum field theory on fuzzy CP^n to third order in the inverse temperature. Using group theoretical methods, we rewrite the result as a multitrace matrix model. The partition function of this matrix model is evaluated via the saddle point method and the phase diagram is analyzed for various n. Our results confirm the findings of a previous numerical study of this phase diagram for CP^1.

Hopf Maps, Lowest Landau Level, and Fuzzy Spheres ?

Abstract: This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford al- gebra is exactly analogous to generalization of fuzzy two-spheres to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an interesting hierarchical structure made of "compounds" of lower dimensional spheres. We give a physical interpretation for such particular structure of fuzzy spheres by utilizing Landau models in generic even dimensions. With Grassmann algebra, we also introduce a graded version of the Hopf map, and discuss its relation to fuzzy supersphere in context of supersymmetric Landau model.

A View on Optimal Transport from Noncommutative Geometry

Abstract: We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry. Starting from a remark of Rieffel on compact manifolds, we first show that on any - i.e. non-necessary compact - complete Riemannian spin manifolds, the two distances coincide. Then, on convex manifolds in the sense of Nash embedding, we provide some natural upper and lower bounds to the distance between any two probability distributions. Specializing to the Euclidean space R-n, we explicitly compute the distance for a particular class of distributions generalizing Gaussian wave packet. Finally we explore the analogy between the spectral and the Wasserstein distances in the noncommutative case, focusing on the standard model and the Moyal plane. In particular we point out that in the two-sheet space of the standard model, an optimal-transport interpretation of the metric requires a cost function that does not vanish on the diagonal. The latest is similar to the cost function occurring in the relativistic heat equation.

Tools for verifying classical and quantum superintegrability.

Abstract: Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n 1 symmetries polynomial in the canonical momenta, so that they are in fact su- perintegrable. These newly discovered systems are all separable in some coordinate system and, typically, they depend on one or more parameters in such a way that the system is superintegrable exactly when some of the parameters are rational numbers. Most of the con- structions to date are for n = 2 but cases where n > 2 are multiplying rapidly. In this article we organize a large class of such systems, many new, and emphasize the underlying mecha- nisms which enable this phenomena to occur and to prove superintegrability. In addition to proofs of classical superintegrability we show that the 2D caged anisotropic oscillator and a Stackel transformed version on the 2-sheet hyperboloid are quantum superintegrable for all rational relative frequencies, and that a deformed 2D Kepler-Coulomb system is quantum superintegrable for all rational values of a parameter k in the potential.

Double Affine Hecke Algebras of Rank 1 and the Z 3 -Symmetric Askey-Wilson Relations

Abstract: We consider the double affine Hecke algebra H = H(k0,k1,k _ ,k _ ;q) associated with the root system (C _ 1 ,C1). We display three elements x, y, z in H that satisfy essentially the Z3-symmetric Askey-Wilson relations. We obtain the relations as follows. We work with an algebra ˆ H that is more general than H, called the universal double affine Hecke algebra of type (C _

On a Nonlocal Ostrovsky-Whitham Type Dynamical System, Its Riemann Type Inhomogeneous Regularizations and Their Integrability

Abstract: Short-wave perturbations in a relaxing medium, governed by a special reduction of the Ostrovsky evolution equation, and later derived by Whitham, are studied using the gradient-holonomic integrability algorithm. The bi-Hamiltonicity and complete integrability of the corresponding dynamical system is stated and an infinite hierarchy of commuting to each other conservation laws of dispersive type are found. The well defined regularization of the model is constructed and its Lax type integrability is discussed. A generalized hyd- rodynamical Riemann type system is considered, infinite hierarchies of conservation laws, related compatible Poisson structures and a Lax type representation for the special case N = 3 are constructed.

Modular Theory, Non-Commutative Geometry and Quantum Gravity ?

Abstract: This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spec- tral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.

Universal Bethe Ansatz and Scalar Products of Bethe Vectors

Abstract: An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_3)$ is given. This result is obtained in the framework of the universal Bethe ansatz, using presentation of the universal Bethe vectors in terms of the total currents of a "new" realization of the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_3)$.

Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions ⋆

Abstract: Theories defined in higher than four dimensions have been used in various frameworks and have a long and interesting history. Here we review certain attempts, developed over the last years, towards the construction of unified particle physics models in the context of higher-dimensional gauge theories with non-commutative extra dimensions. These ideas have been developed in two complementary ways, namely (i) starting with a higher-dimensional gauge theory and dimensionally reducing it to four dimensions over fuzzy internal spaces and (ii) starting with a four-dimensional, renormalizable gauge theory and dynamically generating fuzzy extra dimensions. We describe the above approaches and moreover we discuss the inclusion of fermions and the construction of realistic chiral theories in this context.

Quantum Spacetime: a Disambiguation

Abstract: We review an approach to non-commutative geometry, where models are constructed by quantisation of the coordinates. In particular we focus on the full DFR model and its irreducible components; the (arbitrary) restriction to a particular irreducible component is often referred to as the “canonical quantum spacetime”. The aim is to distinguish and compare the approaches under various points of view, including motivations, prescriptions for quantisation, the choice of mathematical objects and concepts, approaches to dynamics and to covariance. Some incorrect statements as “universality of Planck scale conflicts with Lorentz-Fitzgerald contraction and requires a modification of covariance”, or “stability of the geometric background requires an absolute

Multi-Component NLS Models on Symmetric Spaces: Spectral Properties versus Representations Theory

Abstract: We investigate scattering properties of a Moyal deformed version of the nonlinear Schrodinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general initial data has a unique globally defined solution, and also has soliton solutions if the interaction potential is suitably chosen. We demonstrate how to set up a scattering framework for equations of this type, including appropriate decay estimates of the free time evolution and the construction of wave operators defined for small scattering data in the general case and for arbitrary scattering data in the rotationally symmetric case.

Erlangen Program at Large-1: Geometry of Invariants

Abstract: This paper presents geometrical foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theories based on the representation theory of SL 2 (R) group. We describe here geometries of corresponding do- mains. The principal role is played by Clifford algebras of matching types. In this paper we also generalise the Fillmore-Springer-Cnops construction which describes cycles as points in the extended space. This allows to consider many algebraic and geometric invariants of cycles within the Erlangen program approach.

Gauge Theories on Deformed Spaces

Abstract: The aim of this review is to present an overview over available models and approaches to non-commutative gauge theory. Our main focus thereby is on gauge models formulated on flat Groenewold-Moyal spaces and renormalizability, but we will also review other deformations and try to point out common features. This review will by no means be complete and cover all approaches, it rather reflects a highly biased selection.

Field Theory on Curved Noncommutative Spacetimes

Abstract: We study classical scalar field theories on noncommutative curved spacetimes. Following the approach of Wess et al. (Classical Quantum Gravity 22 (2005), 3511 and Classical Quantum Gravity 23 (2006), 1883), we describe noncommutative spacetimes by using (Abelian) Drinfel'd twists and the associated ?-products and ?-differential geometry. In particular, we allow for position dependent noncommutativity and do not restrict our- selves to the Moyal-Weyl deformation. We construct action functionals for real scalar fields on noncommutative curved spacetimes, and derive the corresponding deformed wave equa- tions. We provide explicit examples of deformed Klein-Gordon operators for noncommuta- tive Minkowski, de Sitter, Schwarzschild and Randall-Sundrum spacetimes, which solve the noncommutative Einstein equations. We study the construction of deformed Green's func- tions and provide a diagrammatic approach for their perturbative calculation. The leading noncommutative corrections to the Green's functions for our examples are derived.

Global Eikonal Condition for Lorentzian Distance Function in Noncommutative Geometry

Abstract: Connes' noncommutative Riemannian distance formula is constructed in two steps, the first one being the construction of a path-independent geometrical functional using a global constraint on continuous functions. This paper generalizes this first step to Lorentzian geometry. We show that, in a globally hyperbolic spacetime, a single global timelike eikonal condition is sufficient to construct a path-independent Lorentzian distance function.

Spectral Distances: Results for Moyal Plane and Noncommutative Torus

Abstract: The spectral distance for noncommutative Moyal planes is considered in the framework of a non compact spectral triple recently proposed as a possible noncommutative analog of non compact Riemannian spin manifold. An explicit formula for the distance between any two elements of a particular class of pure states can be determined. The corresponding result is discussed. The existence of some pure states at infinite distance signals that the topology of the spectral distance on the space of states is not the weak topology. The case of the noncommutative torus is also considered and a formula for the spectral distance between some states is also obtained.

On a family of 2-variable orthogonal Krawtchouk polynomials.

Abstract: We give a hypergeometric proof involving a family of 2-variable Krawtchouk polynomials that were obtained earlier by Hoare and Rahman (SIGMA 4 (2008), 089, 18 pages) as a limit of the 9 j symbols of quantum angular momentum theory, and shown to be eigenfunctions of the transition probability kernel corresponding to a "poker dice" type probability model. The proof in this paper derives and makes use of the necessary and sufficient conditions of orthogonality in establishing orthogonality as well as indicating their geometrical significance. We also derive a 5-term recurrence relation satisfied by these polynomials.

Q-system Cluster Algebras, Paths and Total Positivity ?

Abstract: In the first part of this paper, we provide a concise review of our method of solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connection to the theory of planar networks introduced in the context of totally positive matrices by Fomin and Zelevinsky. As an illustration of the further generality of our method, we apply it to give a simple solution for the rank 2 affine cluster algebras studied by Caldero and Zelevinsky.

On the Origin of the Harmonic Term in Noncommutative Quantum Field Theory

Abstract: The harmonic term in the scalar field theory on the Moyal space removes the UV-IR mixing, so that the theory is renormalizable to all orders. In this paper, we review the three principal interpretations of this harmonic term: the Langmann-Szabo duality, the superalgebraic approach and the noncommutative scalar curvature interpretation. Then, we show some deep relationship between these interpretations.

A Family of Exactly Solvable Radial Quantum Systems on Space of Non-Constant Curvature with Accidental Degeneracy in the Spectrum

Abstract: Received October 05, 2010, in final form December 07, 2010; Published online December 15, 2010doi:10.3842/SIGMA.2010.097Abstract. A novel family of exactly solvable quantum systems on curved space is presented.The family is the quantum version of the classical Perlick family, which comprises all maxi-mally superintegrable 3-dimensional Hamiltonian systems with spherical symmetry. Thehigh number of symmetries (both geometrical and dynamical) exhibited by the classical sys-tems has a counterpartin the accidental degeneracyin the spectrum ofthe quantum systems.This family of quantum problem is completely solved with the techniques of the SUSYQM(supersymmetric quantum mechanics). We also analyze in detail the ordering problem ari-sing in the quantization of the kinetic term of the classical Hamiltonian, stressing the linkexisting between two physically meaningful quantizations: the geometrical quantization andthe position dependent mass quantization.

Finite Unification: Theory and Predictions ?

Abstract: All-loop Finite Unified Theories (FUTs) are very interesting N = 1 supersym- metric Grand Unified Theories (GUTs) which not only realise an old field theoretic dream but also have a remarkable predictive power due to the required reduction of couplings. The reduction of the dimensionless couplings in N = 1 GUTs is achieved by searching for renormalization group invariant (RGI) relations among them holding beyond the unification scale. Finiteness results from the fact that there exist RGI relations among dimensionless couplings that guarantee the vanishing of all beta-functions in certain N = 1 GUTs even to all orders. Furthermore developments in the soft supersymmetry breaking sector of N = 1 GUTs and FUTs lead to exact RGI relations, i.e. reduction of couplings, in this dimension- ful sector of the theory too. Based on the above theoretical framework phenomenologically consistent FUTS have been constructed. Here we present FUT models based on the SU(5) and SU(3) 3 gauge groups and their predictions. Of particular interest is the Higgs mass prediction of one of the models which is expected to be tested at the LHC.

Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors

Abstract: We review our construction of a bifundamental version of the fuzzy 2-sphere and its relation to fuzzy Killing spinors, first obtained in the context of the ABJM membrane model. This is shown to be completely equivalent to the usual (adjoint) fuzzy sphere. We discuss the mathematical details of the bifundamental fuzzy sphere and its field theory expansion in a model-independent way. We also examine how this new formulation affects the twisting of the fields, when comparing the field theory on the fuzzy sphere background with the compactification of the 'deconstructed' (higher dimensional) field theory.

Supersymmetrical Separation of Variables in Two-Dimensional Quantum Mechanics

Abstract: Two different approaches are formulated to analyze two-dimensional quantum models which are not amenable to standard separation of variables. Both methods are essen- tially based on supersymmetrical second order intertwining relations and shape invariance - two main ingredients of the supersymmetrical quantum mechanics. The first method ex- plores the opportunity to separate variables in the supercharge, and it allows to find a part of spectrum of the Schrodinger Hamiltonian. The second method works when the standard separation of variables procedure can be applied for one of the partner Hamiltonians. Then the spectrum and wave functions of the second partner can be found. Both methods are illustrated by the example of two-dimensional generalization of Morse potential for different values of parameters.

Quantum Fields on Noncommutative Spacetimes: Theory and Phenomenology

Abstract: In the present work we review the twisted field construction of quantum field theory on noncommutative spacetimes based on twisted Poincare invariance. We present the latest development in the field, in particular the notion of equivalence of such quan- tum field theories on a noncommutative spacetime, in this regard we work out explicitly the inequivalence between twisted quantum field theories on Moyal and Wick-Voros planes; the duality between deformations of the multiplication map on the algebra of functions on spacetime F(R 4 ) and coproduct deformations of the Poincare-Hopf algebra HP acting on F(R 4 ); the appearance of a nonassociative product on F(R 4 ) when gauge fields are also included in the picture. The last part of the manuscript is dedicated to the phenomenology of noncommutative quantum field theories in the particular approach adopted in this re- view. CPT violating processes, modification of two-point temperature correlation function in CMB spectrum analysis and Pauli-forbidden transition in Be 4 are all effects which show up in such a noncommutative setting. We review how they appear and in particular the constraint we can infer from comparison between theoretical computations and experimen- tal bounds on such effects. The best bound we can get, coming from Borexino experiment, is & 10 24 TeV for the energy scale of noncommutativity, which corresponds to a length scale . 10 43 m. This bound comes from a different model of spacetime deformation more adapted to applications in atomic physics. It is thus model dependent even though similar bounds are expected for the Moyal spacetime as well as argued elsewhere.