Cotangent Bundle Research Articles

Quantization in fibering polarizations, Mabuchi rays and geometric Peter–Weyl theorem

In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter–Weyl theorem. We present a novel approach to half-form corrected geometric quantization in a specific type of non-Kähler polarizations and study one important class of examples, namely cotangent bundles of compact connected Lie groups K. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of K×K-invariant Kähler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the Kähler case. An important role is played by invariance of the limit polarization under a torus action.Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for K with the Borel–Weil description of irreducible representations of K.

Classical mechanics in noncommutative spaces: confinement and more

We consider a semi-classical approximation to the dynamics of a point particle in a noncommutative space. In this approximation, the noncommutativity of space coordinates is described by a Poisson bracket. For linear Poisson brackets, the corresponding phase space is given by the cotangent bundle of a Lie group, with the Lie group playing the role of a curved momentum space. We show that the curvature of the momentum space may lead to rather unexpected physical phenomena such as an upper bound on the velocity of a free nonrelativistic particle, bounded motion for repulsive central force, and no-fall-into-the-centre for attractive Coulomb potential. We also consider a superintegrable Hamiltonian for the Kepler problem in 3-space with su(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {su}(2)$$\\end{document} noncommutativity. The leading correction to the equations of motion due to noncommutativity is shown to be described by an effective monopole potential.

N=(2,2) superfields and geometry revisited

Abstract We take a fresh look at the relation between generalised Kähler geometry and N = (2, 2) supersymmetric sigma models in two dimensions formulated in terms of (2, 2) superfields. Dual formulations in terms of different kinds of superfield are combined to give a formulation with a doubled target space and both the original superfield and the dual superfield. For Kähler geometry, we show that this doubled geometry is Donaldson’s deformation of the holomorphic cotangent bundle of the original Kähler manifold. This doubled formulation gives an elegant geometric reformulation of the equations of motion. We interpret the equations of motion as the intersection of two Lagrangian submanifolds (or of a Lagrangian submanifold with an isotropic one) in the infinite dimensional symplectic supermanifold which is the analogue of phase space. We then consider further extensions of this formalism, including one in which the geometry is quadrupled, and discuss their geometry.

Almost-Poisson Brackets for Nonholonomic Systems with Gyroscopic Terms and Hamiltonisation

We extend known constructions of almost-Poisson brackets and their gauge transformations to nonholonomic systems whose Lagrangian is not mechanical but possesses a gyroscopic term linear in the velocities. The new feature introduced by such a term is that the Legendre transformation is an affine, instead of linear, bundle isomorphism between the tangent and cotangent bundles of the configuration space and some care is needed in the development of the geometric formalism. At the end of the day, the affine nature of the Legendre transform is reflected in the affine dependence of the brackets that we construct on the momentum variables. Our study is motivated by a wide class of nonholonomic systems involving rigid bodies with internal rotors which are of interest in control. Our construction provides a natural geometric framework for the (known) Hamiltonisations of the gyrostatic generalisations of the Suslov and Chaplygin sphere problems.

Asymptotic safe nonassociative quantum gravity with star R-flux products, Goroff–Sagnotti counter-terms, and geometric flows

Nonassociative modifications of general relativity, GR, defined by star products with R-flux deformations in string gravity consist an important subclass of modified gravity theories, MGTs. A longstanding criticism for elaborating quantum gravity, QG, argue that the asymptotic safety does not survive once certain perturbative terms (in general, nonassociative and noncommutative) are included in the projection space. The goal of this work is to prove that a generalized asymptotic safety scenario allows us to formulate physically viable nonassociative QG theories using effective models defined by generic off-diagonal solutions and nonlinear symmetries in nonassociative geometric flow and gravity theories. We elaborate on a new nonholonomic functional renormalization techniques with parametric renormalization group, RG, flow equations for effective actions supplemented by certain canonical two-loop counter-terms. The geometric constructions and quantum deformations are performed for nonassociative phase spaces modelled as R-flux deformed cotangent Lorentz bundles. Our results prove that theories involving nonassociative modifications of GR can be well defined both as classical nonassociative MGTs and QG models. Such theories are characterized by generalized G. Perelman thermodynamic variables which are computed for certain examples of nonassociative geometric and RG flows.

Stability of the Generalized Lagrangian Mean Curvature Flow in Cotangent Bundle

Stability of the Generalized Lagrangian Mean Curvature Flow in Cotangent Bundle

The characteristic cycle of a non-confluent ℓ-adic GKZ hypergeometric sheaf

An ℓ-adic GKZ hypergeometric sheaf is defined analogously to a GKZ hypergeometric D-module. We introduce an algorithm of computing the characteristic cycle of an ℓ-adic GKZ hypergeometric sheaf of certain type. Our strategy is to apply a formula of the characteristic cycle of the direct image of an ℓ-adic sheaf. We verify the requirements for the formula to hold by calculating the dimension of the direct image of a certain closed conical subset of cotangent bundle. We also define an ℓ-adic GKZ-type sheaf whose specialization tensored with a constant sheaf is isomorphic to an ℓ-adic non-confluent GKZ hypergeometric sheaf. On the other hand, the topological model of an ℓ-adic GKZ-type sheaf is isomorphic to the image by the de Rham functor of a non-confluent GKZ hypergeometric D-module whose characteristic cycle has been calculated. This gives an easier way to determine the characteristic cycle of an ℓ-adic non-confluent GKZ hypergeometric sheaf of certain type.

Structures Associated with the Borromean Rings’ Complement in the Poincaré Ball

Guided by physical needs, we deal with the rotationally isotropic Poincar´e ball, when considering the complement of Borromean rings embedded in it. We consistently describe the geometry of the complement and realize the fundamental group as isometry subgroup in three dimensions. Applying this realization, we reveal normal stochastization and multifractal behavior within the examined model of directed random walks on the rooted Cayley tree, whose sixbranch graphs are associated with dendritic polymers. According to Penner, we construct the Teichm¨uller space of the decorated ideal octahedral surface related to the quotient space of the fundamental group action. Using the conformality of decoration, we define six moduli and the mapping class group generated by cyclic permutations of the ideal vertices. Intending to quantize the geometric area, we state the connection between the induced geometry and the sine-Gordon model. Due to such a correspondence we obtain the differential two-form in the cotangent bundle of the moduli space.

Mini-Workshop: Poisson and Poisson-type algebras

The first historical encounter with Poisson-type algebras is with Hamiltonian mechanics. With the abstraction of many notions in Physics, Hamiltonian systems were geometrized into manifolds that model the set of all possible configurations of the system, and the cotangent bundle of this manifold describes its phase space, which is endowed with a Poisson structure. Poisson brackets led to other algebraic structures, and the notion of Poisson-type algebra arose, including transposed Poisson algebras, Novikov–Poisson algebras, or commutative pre-Lie algebras, for example. These types of algebras have long gained popularity in the scientific world and are not only of their own interest to study, but are also an important tool for researching other mathematical and physical objects.

Riemannian distance and symplectic embeddings in cotangent bundle

Given an open neighborhood [Formula: see text] of the zero section in the cotangent bundle of [Formula: see text] we define a distance-like function [Formula: see text] on [Formula: see text] using certain symplectic embeddings from the standard ball [Formula: see text] to [Formula: see text]. We show that when [Formula: see text] is the unit-disk cotangent bundle of a Riemannian metric on [Formula: see text], [Formula: see text] recovers the metric. As an intermediate step, we give a new construction of a symplectic embedding of the ball of capacity 4 to the product of Lagrangian disks [Formula: see text], and we give a new proof of the strong Viterbo conjecture about normalized capacities for [Formula: see text]. We also give bounds of the symplectic packing number of two balls in a unit-disk cotangent bundle relative to the zero section [Formula: see text].

Bordism classes of loops and Floer’s equation in cotangent bundles

Bordism classes of loops and Floer’s equation in cotangent bundles

Notes on the geometry of cotangent bundle and unit cotangent sphere bundle

Let $(\mathtt{N},\mathfrak{g})$ be a Riemannian manifold, by using the musical isomorphisms ♪ and $\natural$ induced by $\mathfrak{g}$, we built a bridge between the geometry of the tangent bundle $\mathtt{TN}$ (resp. the unit tangent sphere bundle $\mathtt{T}_{1}\mathtt{N}$) equipped with the Sasaki metric $\mathfrak{g}_{S}$ (resp. the induced Sasaki metric $\bar{\mathfrak{g}}_{S}$) and that of the cotangent bundle $\mathtt{T}^{\ast}\mathtt{N}$ (resp. the unit cotangent sphere bundle $\mathtt{T}_{1}^{\ast}\mathtt{N}$) endowed with the Sasaki metric $\mathfrak{g}_{\widetilde{S}}$ (resp. the induced Sasaki metric $\tilde{\mathfrak{g}}_{\widetilde{S}}$). Moreover, we prove that $\mathtt{T}_{1}^{\ast}\mathtt{N}$ carries a contact metric structure and study some of its proprieties.

Novel theorems for the cotangent bundle endowed with metallic structures on a differentiable manifold

The present paper aims to study the complete, horizontal and diagonal lifts of metallic structures in the cotangent bundle. Furthermore, the Nijenhuis tensor of a metallic structure is calculated and its integrability conditions by means of partial differential equations are established.

Free scalar field theory on a Sobolev space over a bounded interval

This paper discusses several functional analytic issues relevant for field theories in the context of the Hamiltonian formulation for a free, massless, scalar field defined on a closed interval of the real line. The fields that we use belong to a Sobolev space with a scalar product. As we show this choice is useful because it leads to an explicit representation of the points in the fibers of the phase space (the cotangent bundle of the configuration space). The dynamical role of the boundary of the spatial manifold where the fields are defined is analyzed.

A Nonvanishing Conjecture for Cotangent Bundles

A Nonvanishing Conjecture for Cotangent Bundles

On the cotangent bundle with vertical modified riemannian extensions

Let $M$ be an n-dimensional differentiable manifold with a torsion-free linear connection $\nabla $ which induces on its cotangent bundle ${T^*}M$. The main purpose of the present paper is to study some properties of the vertical modified Riemannian extension on ${T^*}M$ which is given as a new metric in [17]. At first, we investigate a metric connection with torsion on ${T^*}M$. And then, we present the holomorphy properties with respect to a compatible almost complex structure. urthermore, we study locally decomposable Golden pseudo-Riemannian structures on the cotangent bundle endowed with vertical modified Riemannian extension.

The unbounded Lagrangian spectral norm and wrapped Floer cohomology

We investigate the question of whether the spectral metric on the orbit space of a fiber in the disk cotangent bundle of a closed manifold, under the action of the compactly supported Hamiltonian diffeomorphism group, is bounded. We utilize wrapped Floer cohomology to define the spectral invariant of an admissible Lagrangian submanifold within a Weinstein domain. We show that the pseudo-metric derived from this spectral invariant is a valid Ham-invariant metric. Furthermore, we establish that the spectral metric on the orbit space of an admissible Lagrangian is bounded if and only if the wrapped Floer cohomology vanishes. Consequently, we prove that the Lagrangian Hofer diameter of the orbit space for any fiber in the disk cotangent bundle of a closed manifold is infinite.

Poisson–Lie analogues of spin Sutherland models revisited

Some generalizations of spin Sutherland models descend from ‘master integrable systems’ living on Heisenberg doubles of compact semisimple Lie groups. The master systems represent Poisson–Lie counterparts of the systems of free motion modeled on the respective cotangent bundles and their reduction relies on taking quotient with respect to a suitable conjugation action of the compact Lie group. We present an enhanced exposition of the reductions and prove rigorously for the first time that the reduced systems possess the property of degenerate integrability on the dense open subset of the Poisson quotient space corresponding to the principal orbit type for the pertinent group action. After restriction to a smaller dense open subset, degenerate integrability on the generic symplectic leaves is demonstrated as well. The paper also contains a novel description of the reduced Poisson structure and a careful elaboration of the scaling limit whereby our reduced systems turn into the spin Sutherland models.

Proof of the Ginzburg-Kazhdan conjecture

We prove that the affine closure of the cotangent bundle of the basic affine space of a reductive group has conical symplectic singularities, which confirms a conjecture of Ginzburg and Kazhdan. We also show that this variety is Q-factorial and has terminal singularities.

Classical and Quantised Resolvent Algebras for the Cylinder

AbstractBuchholz and Grundling (Commun Math Phys 272:699–750, 2007) introduced a $$\hbox {C}^*$$ C ∗ -algebra called the resolvent algebra as a canonical quantisation of a symplectic vector space and demonstrated that this algebra has several desirable features. We define an analogue of their resolvent algebra on the cotangent bundle $$T^*\mathbb {T}^n$$ T ∗ T n of an n-torus by first generalising the classical analogue of the resolvent algebra defined by the first author of this paper in earlier work (van Nuland in J Funct Anal 277:2815–2838, 2019) and subsequently applying Weyl quantisation. We prove that this quantisation is almost strict in the sense of Rieffel and show that our resolvent algebra shares many features with the original resolvent algebra. We demonstrate that both our classical and quantised algebras are closed under the time evolutions corresponding to large classes of potentials. Finally, we discuss their relevance to lattice gauge theory.