Cotangent Bundle Research Articles

Maslov Index on Symplectic Manifolds. With Supplement by A. T. Fomenko “Constructing the Generalized Maslov Class for the Total Space $$W=\mathbb{T}^*(M)$$ of the Cotangent Bundle”

Maslov Index on Symplectic Manifolds. With Supplement by A. T. Fomenko “Constructing the Generalized Maslov Class for the Total Space $$W=\mathbb{T}^*(M)$$ of the Cotangent Bundle”

Junctions of mass-deformed nonlinear sigma models on SO(2N)/U(N) and Sp(N)/U(N)

We construct on-shell $\mathcal{N}=2$ nonlinear sigma models on $SO(2N)/U(N)$ and $Sp(N)/U(N)$ by holomorphically embedding the models in the hyper-K\"ahler nonlinear sigma model on the cotangent bundle of the Grassmann manifold ${T}^{*}{G}_{2N,N}$ in the $\mathcal{N}=1$ superspace formalism. We apply the moduli matrix formalism to the mass-deformed nonlinear sigma models on the quadrics to study three-pronged junctions by using a recently proposed diagram method.

On a 2-form Derived by Riemannian Metric in the Tangent Bundle

In a recent paper [Salimov, A., Asl, M.B., Kazimova, S.: Problems of lifts in symplectic geometry. Chin Ann. Math. Ser. B. 40(3), (2019), 321-330] the authors have investigated the curious fact that the canonıcal symplectic structure dp = dpi ∧ dxi on cotangent bundle may be given by the introduction of symplectic isomorphism between tangent and cotangent bundles. Our analysis began with the observation that the complete lift of the symplectic structure from the base manifold to its tangent bundle is being a closed 2-form and consequently we proved that its image by the simplectic isomorphism is the natural 2-form dp. We apply this construction in the case where the basic manifold of bundles is a Riemannian manifold with metric g and consider a new 1-form ω = gijyjdxi and its exterior differential on the tangent bundle, from which the symplectic structure is derived.

Three Examples in the Dynamical Systems Theory

We present three explicit curious simple examples in the theory of dynamical systems. The first one is an example of two analytic diffeomorphisms $R$, $S$ of a closed two-dimensional annulus that possess the intersection property but their composition $RS$ does not ($R$ being just the rotation by $\pi/2$). The second example is that of a non-Lagrangian $n$-torus $L_0$ in the cotangent bundle $T^\ast{\mathbb T}^n$ of ${\mathbb T}^n$ ($n\geq 2$) such that $L_0$ intersects neither its images under almost all the rotations of $T^\ast{\mathbb T}^n$ nor the zero section of $T^\ast{\mathbb T}^n$. The third example is that of two one-parameter families of analytic reversible autonomous ordinary differential equations of the form $\dot{x}=f(x,y)$, $\dot{y}=\mu g(x,y)$ in the closed upper half-plane $\{y\geq 0\}$ such that for each family, the corresponding phase portraits for $01$ are topologically non-equivalent. The first two examples are expounded within the general context of symplectic topology.

A geometric approach to contact Hamiltonians and contact Hamilton–Jacobi theory

We propose a novel approach to contact Hamiltonian mechanics which, in contrast to the one dominating in the literature, serves also for non-trivial contact structures. In this approach Hamiltonians are no longer functions on the contact manifold M itself but sections of a line bundle over M or, equivalently, 1-homogeneous functions on a certain -principal bundle , which is equipped with a homogeneous symplectic form ω. In other words, our understanding of contact geometry is that it is not an ‘odd-dimensional cousin’ of symplectic geometry but rather a part of the latter, namely ‘homogeneous symplectic geometry’. This understanding of contact structures is much simpler than the traditional one and very effective in applications, reducing the contact Hamiltonian formalism to the standard symplectic picture. We develop in this language contact Hamiltonian mechanics in the autonomous, as well as the time-dependent case, and the corresponding Hamilton–Jacobi theory. Fundamental examples are based on canonical contact structures on the first jet bundles of sections of line bundles L, which play in contact geometry a fundamental rôle, similar to that played by cotangent bundles in symplectic geometry.

A Bruhat Atlas for the Mehta–van der Kallen Stratification of T∗GLn/B

Mehta and van der Kallen put a Frobenius splitting on the type A cotangent bundle T∗GLn/B, thereby defining a stratification by compatibly split subvarieties, and they determined a few of the elements of this stratification. We embed T∗GLn/B as a stratum in a larger stratified (and Frobenius split) space GLn/B × Matn whose stratification we determine, thereby giving a full description of the one of Mehta–van der Kallen. The main technique is to endow GLn/B × Matn with a Bruhat atlas, covering it with open sets that are stratified-isomorphic to Bruhat cells (in GL2n/B2n). Among the consequences are that each stratum closure is normal and Cohen–Macaulay.

Calabi–Yau structure and Bargmann type transformation on the Cayley projective plane

Our purpose is to show the existence of a Calabi–Yau structure on the punctured cotangent bundle $T^{*}_{0}(P^2\mathbb{O})$ of the Cayley projective plane $P^{2}\mathbb{O}$ and to construct a Bargmann type transformation from a space of holomorphic functions on $T^{*}_{0}(P^2\mathbb{O})$ to $L_{2}$-space on $P^{2}\mathbb{O}$. The space of holomorphic functions corresponds to the Fock space in the case of the original Bargmann transformation. A Kähler structure on $T^{*}_{0}(P^{2}\mathbb{O})$ was given by identifying it with a quadric in the complex space $\mathbb{C}^{27} \backslash \{0\}$ and the natural symplectic form of the cotangent bundle $T^{*}_{0}(P^2\mathbb{O})$ is expressed as a Kähler form. Our construction of the transformation is the pairing of polarizations, one is the natural Lagrangian foliation given by the projection map $\boldsymbol{q}:T^{*}_{0}(P^2\mathbb{O}) \longrightarrow P^{2}\mathbb{O}$ and the other is the polarization given by the Kähler structure. The transformation gives a quantization of the geodesic flow in terms of one parameter group of elliptic Fourier integral operators whose canonical relations are defined by the graph of the geodesic flow action at each time. It turns out that for the Cayley projective plane the results are not same with other cases of the original Bargmann transformation for Euclidean space, spheres and other projective spaces.

On Complete, Horizontal and Vertical Lifts From a Manifold With fλ(6,4) Structure to Its Cotangent Bundle

Manifolds with fλ(6,4) structure was defined and studied in the past. Later the geometry of tangent and cotangent bundles in a differentiable manifold with fλ(6,4) structure was studied. The aim of the present paper is to study complete, horizontal and vertical lifts from a manifold with fλ(6,4)- structure to its cotangent bundle.

On supersymmetric interface defects, brane parallel transport, order-disorder transition and homological mirror symmetry

We concentrate on a treatment of a Higgs-Coulomb duality as an absence of manifest phase transition between ordered and disordered phases of 2d mathcal{N} = (2, 2) theories. We consider these examples of QFTs in the Schrödinger picture and identify Hilbert spaces of BPS states with morphisms in triangulated categories of D-brane boundary conditions. As a result of Higgs-Coulomb duality D-brane categories on IR vacuum moduli spaces are equivalent, this resembles an analog of homological mirror symmetry. Following construction ideas behind the Gaiotto-Moore-Witten algebra of the infrared one is able to introduce interface defects in these theories and associate them to D-brane parallel transport functors. We concentrate on surveying simple examples, analytic when possible calculations, numerical estimates and simple physical picture behind curtains of geometric objects. Categorification of hypergeometric series analytic continuation is derived as an Atiyah flop of the conifold. Finally we arrive to an interpretation of the braid group action on the derived category of coherent sheaves on cotangent bundles to flag varieties as a categorification of Berry connection on the Fayet-Illiopolous parameter space of a sigma-model with a quiver variety target space.

Cotangent bundle reduction and Routh reduction for polysymplectic manifolds

We discuss Lagrangian and Hamiltonian field theories that are invariant under a symmetry group. We apply the polysymplectic reduction theorem for both types of field equations and we investigate aspects of the corresponding reconstruction process. We identify the polysymplectic structures that lie at the basis of cotangent bundle reduction and Routh reduction in this setting and we relate them by means of the Routhian function and its associated Legendre transformation. Throughout the paper we provide examples that illustrate various aspects of the results.

Dimensional reduction of Courant sigma models and Lie theory of Poisson groupoids

We show that the 2d Poisson Sigma Model on a Poisson groupoid arises as an effective theory of the 3d Courant Sigma Model associated with the double of the underlying Lie bialgebroid. This field-theoretic result follows from a Lie-theoretic one involving a coisotropic reduction of the odd cotangent bundle by a generalized space of algebroid paths. We also provide several examples, including the case of symplectic groupoids in which we relate the symplectic realization construction of Crainic–Marcut to a particular gauge fixing of the 3d theory.

3d mirror symmetry of the cotangent bundle of the full flag variety

Aganagic and Okounkov proved that the elliptic stable envelope provides the pole cancellation matrix for the enumerative invariants of quiver varieties known as vertex functions. This transforms a basis of a system of q-difference equations holomorphic in variables $${\varvec{z}}$$ with poles in variables $${\varvec{a}}$$ to a basis of solutions holomorphic in $${\varvec{a}}$$ with poles in $${\varvec{z}}$$ . The resulting functions are expected to be the vertex functions of the 3d mirror dual variety. In this paper, we prove that the functions obtained by applying the elliptic stable envelope to the vertex functions of the cotangent bundle of the full flag variety are precisely the vertex functions for the same variety under an exchange of the parameters $$\text{\AA} \leftrightarrow {\varvec{z}}$$ . As a corollary of this, we deduce the expected 3d mirror relationship for the elliptic stable envelope.

On the locus of higher order jets of entire curves in complex projective varieties

For a given complex projective variety, the existence of entire curves is strongly constrained by the positivity properties of the cotangent bundle. The Green–Griffiths–Lang conjecture stipulates that entire curves drawn on a variety of general type should all be contained in a proper algebraic subvariety. We present here new results on the existence of differential equations that strongly restrain the locus of entire curves in the general context of foliated or directed varieties, under appropriate positivity conditions.

Ramified cover of varieties with nef cotangent bundle

We construct examples to show that having nef cotangent bundle is not preserved under finite ramified covers. Our examples also show that a projective manifold with Stein universal cover may not have nef cotangent bundle, disproving a conjecture of Liu–Maxim–Wang [7].

On the isomorphism problem for the rings of differential operators on smooth affine varieties

We show that given two smooth affine varieties over C such that their rings of differential operators are Morita equivalent, then corresponding cotangent bundles are isomorphic as symplectic varieties.

Vertical rescaled Cheeger-Gromoll metric and harmonicity on the cotangent bundle

In this paper, we introduce the harmonicity on the cotangent bundle equipped with vertical rescaled Cheeger-Gromoll metric which rescales the vertical part by a non-zero differentiable function f. We establish a necessary and sufficient condition under which a covector field is harmonic with respect to this metric. We also construct some examples of harmonic vector fields. Finally, we study the harmonicity of a map between a Riemannian manifold and a cotangent bundle of another Riemannian manifold and vice versa.

Symplectic embeddings into disk cotangent bundles

In this paper, we compute the embedded contact homology (ECH) capacities of the disk cotangent bundles $D^*S^2$ and $D^*\mathbb{R} P^2$. We also find sharp symplectic embeddings into these domains. In particular, we compute their Gromov widths. In order to do that, we explicitly calculate the ECH chain complexes of $S^*S^2$ and $S^* \mathbb{R} P^2$ using a direct limit argument on the action inspired by Bourgeois's Morse-Bott approach and ideas from Nelson-Weiler's work on the ECH of prequantization bundles. Moreover, we use integrable systems techniques to find explicit symplectic embeddings. In particular, we prove that the disk cotangent bundles of a hemisphere and of a punctured sphere are symplectomorphic to an open ball and a symplectic bidisk, respectively.

Higher elastica: geodesics in jet space

Carnot groups are subRiemannian manifolds. As such, they admit geodesic flows, which are left-invariant Hamiltonian flows on their cotangent bundles. Some of these flows are integrable; some are not. The k-jets space of for real-valued functions on the real line forms a Carnot group of dimension k+2. In this study, it is shown that its geodesic flow is integrable and that its geodesics generalize Euler’s elastica, with the case k=2 corresponding to the elastica.

A mixed method for 3D nonlinear elasticity using finite element exterior calculus

AbstractIn this study, a mixed finite element model for 3D nonlinear elasticity using a Hu–Washizu (HW) type variational principle is presented. This mixed variational principle takes the deformed configuration and sections from its cotangent bundle as the input arguments. The critical points of the proposed HW functional enforce compatibility of these sections with the configuration, in addition to mechanical equilibrium and constitutive relations. Using this variational principle we construct a mixed FE approximation that distinguishes a vector from a 1‐form, a feature not commonly found in FE approximations for nonlinear elasticity. This distinction plays a pivotal role in identifying suitable FE spaces for approximating the 1‐forms appearing in the variational principle. These discrete approximations are constructed using ideas borrowed from finite element exterior calculus, which are in turn used to construct a discrete approximation to our HW functional. The discrete equations describing mechanical equilibrium, compatibility, and constitutive rule, are obtained by seeking extremum of the discrete functional with respect to the respective degrees of freedom. The discrete extremum problem is then solved numerically; we use Newton's method for this purpose. This mixed FE technique is then applied to a few benchmark problems wherein conventional displacement based approximations encounter locking and checker boarding. These studies help establish that our mixed FE approximation, which requires no artificial stabilizing terms, is free of these numerical bottlenecks.

Global Propagator for the Massless Dirac Operator and Spectral Asymptotics

We construct the propagator of the massless Dirac operator W on a closed Riemannian 3-manifold as the sum of two invariantly defined oscillatory integrals, global in space and in time, with distinguished complex-valued phase functions. The two oscillatory integrals—the positive and the negative propagators—correspond to positive and negative eigenvalues of W, respectively. This enables us to provide a global invariant definition of the full symbols of the propagators (scalar matrix-functions on the cotangent bundle), a closed formula for the principal symbols and an algorithm for the explicit calculation of all their homogeneous components. Furthermore, we obtain small time expansions for principal and subprincipal symbols of the propagators in terms of geometric invariants. Lastly, we use our results to compute the third local Weyl coefficients in the asymptotic expansion of the eigenvalue counting functions of W.