Showing papers in "International Journal of Geometric Methods in Modern Physics in 2013"
TL;DR: In this paper, it was shown that the Chern-Simons action functional associated to an n-symplectic manifold is the action functional of the AKSZ σ-model whose target space is the given n -symmlectic manifold, and that this naturally generalizes from Lie algebras to dg-manifolds and to 3-bundles.
Abstract: Chern–Weil theory provides for each invariant polynomial on a Lie algebra 𝔤 a map from 𝔤-connections to differential cocycles whose volume holonomy is the corresponding Chern–Simons theory action functional. Kotov and Strobl have observed that this naturally generalizes from Lie algebras to dg-manifolds and to dg-bundles and that the Chern–Simons action functional associated this way to an n-symplectic manifold is the action functional of the AKSZ σ-model whose target space is the given n-symplectic manifold (examples of this are the Poisson σ-model or the Courant σ-model, including ordinary Chern–Simons theory, or higher-dimensional Abelian Chern–Simons theory). Here we show how, within the framework of the higher Chern–Weil theory in smooth ∞-groupoids, this result can be naturally recovered and enhanced to a morphism of higher stacks, the same way as ordinary Chern–Simons theory is enhanced to a morphism from the stack of principal G-bundles with connections to the 3-stack of line 3-bundles with connections.
50 citations
TL;DR: In this article, the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features, called the Lie-Hamilton systems, is discussed. But their results are illustrated by examples of physical and mathematical interest.
Abstract: This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie–Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearizability conditions, etc. Our results are illustrated by examples of physical and mathematical interest.
34 citations
TL;DR: In this paper, a Ricci soliton with the Reeb potential vector field or a transversal vector field is of constant sectional curvature 0.1 and a cosymplectic 3-manifold admits Ricci s solitons of constant curvature -1.
Abstract: A Kenmotsu 3-manifold M admitting a Ricci soliton (g, w) with a transversal potential vector field w (orthogonal to the Reeb vector field) is of constant sectional curvature -1. A cosymplectic 3-manifold admitting a Ricci soliton with the Reeb potential vector field or a transversal vector field is of constant sectional curvature 0.
25 citations
TL;DR: Magnot and Magnot as mentioned in this paper extended the notion of holonomy on diffeological bundles, reformulated the concept of regular Lie groups or Frolicher Lie groups, and proved the complete integrability of the Kadomtsev-Petviashvili equation.
Abstract: In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frolicher Lie groups, state an Ambrose–Singer theorem that enlarges the one stated in [J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math.128 (2004) 513–529], and conclude with a differential geometric treatment of KP hierarchy. The examples of Lie groups that are studied are principally those obtained by enlarging some graded Frolicher (Lie) algebras such as formal q-series of the quantum algebra of pseudo-differential operators. These deformations can be defined for classical pseudo-differential operators but they are used here on formal pseudo-differential operators in order to get a differential geometric framework to deal with the KP hierarchy that is known to be completely integrable with formal power series. Here, we get an integration of the Zakharov–Shabat connection form by means of smooth sections of a (differential geometric) bundle with structure group, some groups of q-deformed operators. The integration obtained by Mulase [Complete integrability of the Kadomtsev–Petviashvili equation Adv. Math.54 (1984) 57–66], and the key tools he developed, are totally recovered on the germs of the smooth maps of our construction. The tool coming from (classical) differential geometry used in this construction is the holonomy group, on which we have an Ambrose–Singer-like theorem: the Lie algebra is spanned by the curvature elements. This result is proved for any connection a diffeological principal bundle with structure group a regular Frolicher Lie group. The case of a (classical) Lie group modeled on a complete locally convex topological vector space is also recovered and the work developed in [J.-P. Magnot, Diffeologie du fibre d'Holonomie en dimension infinie, Math. Rep. Canadian Roy. Math. Soc.28(4) (2006); J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math. 128 (2004) 513–529] is completed.
24 citations
TL;DR: In this paper, an explicit correspondence between quantum mechanics and the classical theory of irreversible thermodynamics is presented, which maps irreversible Gaussian Markov processes into the semiclassical approximation of quantum mechanics.
Abstract: We present an explicit correspondence between quantum mechanics and the classical theory of irreversible thermodynamics as developed by Onsager, Prigogine et al. Our correspondence maps irreversible Gaussian Markov processes into the semiclassical approximation of quantum mechanics. Quantum-mechanical propagators are mapped into thermodynamical probability distributions. The Feynman path integral also arises naturally in this setup. The fact that quantum mechanics can be translated into thermodynamical language provides additional support for the conjecture that quantum mechanics is not a fundamental theory but rather an emergent phenomenon, i.e. an effective description of some underlying degrees of freedom.
24 citations
TL;DR: In this article, the authors define two classes of hypersurfaces in real space forms, golden-and product-shaped, by imposing the shape operator to be of golden or product type.
Abstract: We define two classes of hypersurfaces in real space forms, golden- and product-shaped, respectively, by imposing the shape operator to be of golden or product type. We obtain the whole families of above hypersurfaces, based on the classification of isoparametric hypersurfaces, as follows: in the golden case all are hyperspheres, a hyperbolic space and a generalized Clifford torus, while for the product case we obtain the unit hypersphere, the hyperplane, a hypersphere and its associated Clifford torus, respectively, according to the type of the ambient space form namely parabolic, hyperbolic or elliptic, respectively.
23 citations
TL;DR: In this article, it was shown that the conserved current associated with a generalized symmetry, assumed to be also a symmetry of the variation of the local inverse problem, is variationally equivalent to the variation for the strong Noether current for the corresponding local system of Lagrangians.
Abstract: We consider systems of local variational problems defining nonvanishing cohomology classes. Symmetry properties of the Euler–Lagrange expressions play a fundamental role since they introduce a cohomology class which adds up to Noether currents; they are related with invariance properties of the first variation, thus with the vanishing of a second variational derivative. In particular, we prove that the conserved current associated with a generalized symmetry, assumed to be also a symmetry of the variation of the corresponding local inverse problem, is variationally equivalent to the variation of the strong Noether current for the corresponding local system of Lagrangians. This current is conserved and a sufficient condition will be identified in order that such a current be global.
23 citations
TL;DR: In this article, a conformal Courant algebroid is introduced, which is a mild generalization of Courant bregman algebroids in which only conformal structure rather than a bilinear form is assumed, and is classified by pairs (L, H) with L a flat line bundle and H ∈ H3(M, L) a degree 3 class with coefficients in L.
Abstract: We introduce conformal Courant algebroids, a mild generalization of Courant algebroids in which only a conformal structure rather than a bilinear form is assumed. We introduce exact conformal Courant algebroids and show they are classified by pairs (L, H) with L a flat line bundle and H ∈ H3(M, L) a degree 3 class with coefficients in L. As a special case gerbes for the crossed module (U(1) → ℤ2) can be used to twist TM ⊕ T*M into a conformal Courant algebroid. In the exact case there is a twisted cohomology which is 4-periodic if L2 = 1. The structure of Conformal Courant algebroids on circle bundles leads us to construct a T-duality for orientifolds with free involution. This incarnation of T-duality yields an isomorphism of 4-periodic twisted cohomology. We conjecture that the isomorphism extends to an isomorphism in twisted KR-theory and give some calculations to support this claim.
22 citations
TL;DR: In this paper, wild embeddings like Alexanders horned ball and relate them to fractal spaces have been discussed and a C*-algebra corresponding to a wild embedding has been constructed.
Abstract: In this paper, we discuss wild embeddings like Alexanders horned ball and relate them to fractal spaces. We build a C*-algebra corresponding to a wild embedding. We argue that a wild embedding is the result of a quantization process applied to a tame embedding. Therefore, quantum states are directly the wild embeddings. Then we give an example of a wild embedding in the four-dimensional spacetime. We discuss the consequences for cosmology.
22 citations
TL;DR: In this article, the Ricci tensor is shown to be Weyl compatible, a concept enlarging the classical Derdzinski-Shen theorem about Codazzi tensors.
Abstract: In this paper, we introduce a new kind of tensor whose trace is the well-known Z tensor defined by the present authors. This is named Q tensor: the displayed properties of such tensor are investigated. A new kind of Riemannian manifold that embraces both pseudo-symmetric manifolds (PS)n and pseudo-concircular symmetric manifolds is defined. This is named pseudo-Q-symmetric and denoted with (PQS)n. Various properties of such an n-dimensional manifold are studied: the case in which the associated covector takes the concircular form is of particular importance resulting in a pseudo-symmetric manifold in the sense of Deszcz [On pseudo-symmetric spaces, Bull. Soc. Math. Belgian Ser. A44 (1992) 1–34]. It turns out that in this case the Ricci tensor is Weyl compatible, a concept enlarging the classical Derdzinski–Shen theorem about Codazzi tensors. Moreover, it is shown that a conformally flat (PQS)n manifold admits a proper concircular vector and the local form of the metric tensor is given. The last section is devoted to the study of (PQS)n space-time manifolds; in particular we take into consideration perfect fluid space-times and provide a state equation. The consequences of the Weyl compatibility on the electric and magnetic part of the Weyl tensor are pointed out. Finally a (PQS)n scalar field space-time is considered, and interesting properties are pointed out.
20 citations
TL;DR: In this paper, it was shown that every spherically symmetric Finsler metric of isotropic Berwald curvature is a Randers metric, which is a special case of the standard metric.
Abstract: A Finsler metric F is said to be spherically symmetric if the orthogonal group O(n) acts as isometries of F. In this paper, we show that every spherically symmetric Finsler metric of isotropic Berwald curvature is a Randers metric. We also construct explicitly a lot of new isotropic Berwald spherically symmetric Finsler metrics.
TL;DR: In this article, the Lagrangian and Hamiltonian formalism for mechanical systems using para/pseudo-Kahler manifolds is introduced, representing an interesting multidisciplinary field of research.
Abstract: The paper aims to introduce Lagrangian and Hamiltonian formalism for mechanical systems using para/pseudo-Kahler manifolds, representing an interesting multidisciplinary field of research. Moreover, the geometrical, relativistical, mechanical and physical results related to para/pseudo-Kahler mechanical systems are given, too.
TL;DR: In this paper, the (α, β, γ)-derivations of finite-dimensional Lie superalgebras over the field of complex numbers have been studied.
Abstract: This paper is primarily concerned with (α, β, γ)-derivations of finite-dimensional Lie superalgebras over the field of complex numbers. Some properties of (α, β, γ)-derivations of the Lie superalgebras are obtained. In particular, two examples for (α, β, γ)-derivations of low-dimensional non-simple Lie superalgebras are presented and the super-spaces of (α, β, γ)-derivations for simple Lie superalgebras are determined. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie superalgebras. A special case for the generalization of 1-cocycles with respect to the adjoint representation is exactly (α, β, γ)-derivations. Furthermore, two-dimensional twisted cocycles of the adjoint representation are investigated in detail.
TL;DR: In this article, the Ricci flow on Finsler manifolds with Berwald metrics cannot possibly be strictly parabolic, and a modified flow which is strictly parabolized is defined.
Abstract: This paper shows that the Ricci flow on Finsler manifolds with Berwald metrics cannot possibly be strictly parabolic. Then, we will define a modified flow which is strictly parabolic and by using it, we will prove the existence and uniqueness for solution of Ricci flow on Finsler manifolds.
TL;DR: In this paper, the authors considered a two-dimensional (2D) surface as a three-dimensional shell whose thickness is negligible in comparison with the dimension of the whole system.
Abstract: A two-dimensional (2D) surface can be considered as three-dimensional (3D) shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of 2D sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [xi, pj] = iℏ(δij - xixj/r2) rather than [xi, pj] = iℏδij that does not hold true anymore. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.
TL;DR: In this article, the homogeneous Kahler diffeomorphism (HDF) was introduced to express the Kahler two-form on the Siegel-Jacobi domain as the sum of the two forms on ℂ and the one on Siegel ball.
Abstract: We find the homogeneous Kahler diffeomorphism FC which expresses the Kahler two-form on the Siegel–Jacobi domain as the sum of the Kahler two-form on ℂ and the one on the Siegel ball . The classical motion and quantum evolution on determined by a linear Hamiltonian in the generators of the Jacobi group is described by a Riccati equation on and a linear first-order differential equation in z ∈ ℂ, where H1 denotes the three-dimensional Heisenberg group. When the transformation FC is applied, the first-order differential equation for the variable z ∈ ℂ decouples of the motion on the Siegel disk. Similar considerations are presented for the Siegel–Jacobi space , where denotes the Siegel upper half-plane.
TL;DR: The Veldkamp space of the smallest slim dense near hexagon is described in this article, which is isomorphic to PG(7,2) and its 28 - 1 = 255 Veldmap points fall into five distinct classes, each of which is uniquely characterized by the number of points/lines as well as a sequence of the cardinalities of points of given orders and/or that of (grid-)quads of given types.
Abstract: We give a detailed description of the Veldkamp space of the smallest slim dense near hexagon. This space is isomorphic to PG(7,2) and its 28 - 1 = 255 Veldkamp points (that is, geometric hyperplanes of the near hexagon) fall into five distinct classes, each of which is uniquely characterized by the number of points/lines as well as by a sequence of the cardinalities of points of given orders and/or that of (grid-)quads of given types. For each type we also give its weight, stabilizer group within the full automorphism group of the near hexagon and the total number of copies. The totality of (255 choose 2)/3 = 10,795 Veldkamp lines split into 41 different types. We give a complete classification of them in terms of the properties of their cores (i.e. subconfigurations of points and lines common to all the three hyperplanes comprising a given Veldkamp line) and the types of the hyperplanes they are composed of. These findings may lend themselves into important physical applications, especially in view of recent emergence of a variety of closely related finite geometrical concepts linking quantum information with black holes.
TL;DR: In this article, the authors investigated a variant of the classical mixmaster universe model of anisotropic cosmology, where the spatial sections are noncommutative 3-tori.
Abstract: In this paper we investigate a variant of the classical mixmaster universe model of anisotropic cosmology, where the spatial sections are noncommutative 3-tori. We consider ways in which the discrete dynamical system describing the mixmaster dynamics can be extended to act on the noncommutative torus moduli, and how the resulting dynamics differs from the classical one, for example, in the appearance of exotic smooth structures. We discuss properties of the spectral action, focussing on how the slow-roll inflation potential determined by the spectral action affects the mixmaster dynamics. We relate the model to other recent results on spectral action computation and we identify other physical contexts in which this model may be relevant.
TL;DR: In this paper, it was shown that the annihilation operators whose eigenstates are coherent states on the sphere take the expected form αx + iβp, where α and β are two operators that depend on the angular momentum and x and p are the position and geometric momentum, respectively.
Abstract: With a recently introduced geometric momentum that depends on the extrinsic curvature and offers a proper description of momentum on two-dimensional sphere, we show that the annihilation operators whose eigenstates are coherent states on the sphere take the expected form αx + iβp, where α and β are two operators that depend on the angular momentum and x and p are the position and the geometric momentum, respectively. Since the geometric momentum is manifestly a consequence of embedding the two-dimensional sphere in the three-dimensional flat space, the coherent states reflects some aspects beyond the intrinsic geometry of the surfaces.
TL;DR: In this paper, the gamma field is defined from the "diagonal tetrad" in a chart in which the metric has been put in a space-isotropic diagonal form, and the uniqueness problems are solved at once for all reference frames.
Abstract: Although the standard generally-covariant Dirac equation is unique in a topologically simple spacetime, it has been shown that it leads to non-uniqueness problems for the Hamiltonian and energy operators, including the non-uniqueness of the energy spectrum. These problems should be solved by restricting the choice of the Dirac gamma field in a consistent way. Recently, we proposed to impose the value of the rotation rate of the tetrad field. This is not necessarily easy to implement and works only in a given reference frame. Here, we propose that the gamma field should change only by constant gauge transformations. To get that situation, we are naturally led to assume that the metric can be put in a space-isotropic diagonal form. When this is the case, it distinguishes a preferred reference frame. We show that by defining the gamma field from the "diagonal tetrad" in a chart in which the metric has that form, the uniqueness problems are solved at once for all reference frames. We discuss the physical relevance of the metric considered and our restriction to first-quantized theory.
TL;DR: In this article, it was shown that a complete spacelike hypersurface is parabolic if the Riemannian universal covering of the fiber is so, and the corresponding Calabi-Bernstein problems were solved.
Abstract: We study non-compact complete spacelike hypersurfaces in generalized Robertson–Walker spacetimes of arbitrary dimension whose fiber is parabolic. Under boundedness assumptions on the warping function restricted on a spacelike hypersurface and on the hyperbolic angle of the hypersurface, we prove that a complete spacelike hypersurface is parabolic if the Riemannian universal covering of the fiber is so. As an application of this new technique, several uniqueness results on complete maximal spacelike hypersurfaces are obtained. Also, the corresponding Calabi–Bernstein problems are solved.
TL;DR: In this paper, the α-deformed Weyl-heisenberg algebra is used to obtain the su(2)- and su(1, 1)-algebras whenever α has specific values.
Abstract: At first, we introduce α-deformed algebra as a kind of generalization of the Weyl–Heisenberg algebra so that we get the su(2)- and su(1, 1)-algebras whenever α has specific values. After that, we construct coherent states of this algebra. Third, a realization of this algebra is given in the system of a harmonic oscillator confined at the center of a potential well. Then, we introduce two-boson realization of the α-deformed Weyl–Heisenberg algebra and use this representation to write α-deformed coherent states in terms of the two modes number states. Following these points, we consider mean number of excitations (we call them in general photons) and Mandel parameter as statistical properties of the α-deformed coherent states. Finally, the Fubini–Study metric is calculated for the α-coherent states manifold.
TL;DR: In this article, the authors present Weyl-Euler-Lagrange and Weyl Hamilton equations on a model of tangent manifolds of Constant W-Sectional Curvature.
Abstract: This paper aims to present Weyl–Euler–Lagrange and Weyl–Hamilton equations on which is a model of tangent manifolds of Constant W-Sectional Curvature. In this study some differential geometrical and physical results on the related Weyl-mechanical systems are given.
TL;DR: In this article, the integrability conditions for expanding hyperheavenly spaces with Λ in spinorial formalism are studied and it is shown that any conformal Killing vector reduces to homothetic or isometric Killing vector.
Abstract: Conformal Killing equations and their integrability conditions for expanding hyperheavenly spaces with Λ in spinorial formalism are studied. It is shown that any conformal Killing vector reduces to homothetic or isometric Killing vector. Reduction of respective Killing equation to one master equation is presented. Classification of homothetic and isometric Killing vectors is given. Type [D] ⊗ [any] is analyzed in detail and some expanding complex metrics of types [III, N] ⊗ [III, N] with Λ admitting isometric Killing vectors are found.
TL;DR: In this article, it was shown that linear differential operators with coefficients in the real superspace of weighted densities can be computed with coefficients vanishing on the Lie superalgebra of contact vector fields.
Abstract: Over the (1, n)-dimensional real superspace, we classify -invariant linear differential operators acting on the superspaces of weighted densities, where is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of with coefficients in the superspace of weighted densities, vanishing on the Lie superalgebra . We explicitly give 1-cocycles spanning these cohomology spaces.
TL;DR: In this article, a generalized current algebra is reconstructed by a graded manifold and a graded Poisson bracket, which can be used to obtain the structure of a Lie algebroid up to homotopy.
Abstract: Generalized current algebras introduced by Alekseev and Strobl in two dimensions are reconstructed by a graded manifold and a graded Poisson brackets. We generalize their current algebras to higher dimensions. QP-manifolds provide the unified structures of current algebras in any dimension. Current algebras give rise to structures of Leibniz/Loday algebroids, which are characterized by QP-structures. Especially, in three dimensions, a current algebra has a structure of a Lie algebroid up to homotopy introduced by Uchino and one of the authors, which has a bracket of a generalization of the Courant–Dorfman bracket. Anomaly cancellation conditions are reinterpreted as generalizations of the Dirac structure.
TL;DR: Dodonov and Dodonov as mentioned in this paper presented an analytic approach based on Mathematical Physics to the cavity Casimir effect in the presence of a two-level atom and showed that their method is simple and beautiful.
Abstract: In this paper, we treat the so-called dynamical Casimir effect in a cavity with a two-level atom and give an analytic approximate solution under the general setting. The aim of the paper is to show another approach based on Mathematical Physics to the paper [Approximate analytical results on the cavity Casimir effect in the presence of a two-level atom, Phys. Rev. A85 (2012) 063804, arXiv: 1112.0523 [quant-ph]] by A. V. Dodonov and V. V. Dodonov. We believe that our method is simple and beautiful.
TL;DR: In this paper, the Seiberg-Witten map is analyzed for non-commutative Yang-Mills theories with the related methods, developed in the literature, for its explicit construction, that hold for any gauge group.
Abstract: In this paper the Seiberg–Witten map is first analyzed for non-commutative Yang–Mills theories with the related methods, developed in the literature, for its explicit construction, that hold for any gauge group. These are exploited to write down the second-order Seiberg–Witten map for pure gravity with a constant non-commutativity tensor. In the analysis of pure gravity when the classical space–time solves the vacuum Einstein equations, we find for three distinct vacuum solutions that the corresponding non-commutative field equations do not have solution to first order in non-commutativity, when the Seiberg–Witten map is eventually inserted. In the attempt of understanding whether or not this is a peculiar property of gravity, in the second part of the paper, the Seiberg–Witten map is considered in the simpler case of Maxwell theory in vacuum in the absence of charges and currents. Once more, no obvious solution of the non-commutative field equations is found, unless the electromagnetic potential depends in a very special way on the wave vector.
TL;DR: In this paper, the Leray spectral sequence associated to a differential fibration is described by base and total differential graded algebras and the cohomology used is non-commutative differential sheaf cohomologies.
Abstract: This paper describes the Leray spectral sequence associated to a differential fibration. The differential fibration is described by base and total differential graded algebras. The cohomology used is noncommutative differential sheaf cohomology. For this purpose, a sheaf over an algebra is a left module with zero curvature covariant derivative. As a special case, we can recover the Serre spectral sequence for a noncommutative fibration.
TL;DR: In this paper, a Grassmann-grained variational bicomplex on graded manifolds is developed for degenerate Lagrangian non-relativistic systems, characterized by hierarchies of higher-order Noether identities and gauge symmetries.
Abstract: Graded Lagrangian formalism in terms of a Grassmann-graded variational bicomplex on graded manifolds is developed in a very general setting. This formalism provides the comprehensive description of reducible degenerate Lagrangian systems, characterized by hierarchies of non-trivial higher-order Noether identities and gauge symmetries. This is a general case of classical field theory and Lagrangian non-relativistic mechanics.