Symplectic Potential Research Articles

Dirac approach to constrained submanifolds in a double loop group: From Wess-Zumino-Novikov-Witten to Poisson-Lie σ-model

We study the restriction to a family of second class constrained submanifolds in the cotangent bundle of a double Lie group equipped with a 2-cocycle extended symplectic form to build the corresponding Dirac brackets. It is shown that, for 2-cocycle vanishing on each isotropic subspace of the associated Manin triple, the Dirac bracket contains no traces of the cocycle. We also investigate the restriction of the left translation action of the double Lie group on its cotangent bundle, where it fails to be a canonical transformation. However, the Hamiltonian symmetry is restored on some special submanifolds. The main application is to loop groups, showing that a WZNW-type model on the double Lie group with a quadratic Hamilton function in the momentum maps associated with the left translation action on the cotangent bundle with the canonical symplectic form, restricts to a collective system on some special submanifolds. There, the Lagrangian version coincides with the so-called Poisson-Lie σ -model.

Canonical connectivity and conservation laws in gravitation

The Riemann-canonical tetrad which defines, in the general case, six 2-directions of extreme values of the sectional Riemann curvature is introduced. It localizes the gravitational energy, defines the canonical 1-form of connectivity and canonical tetrad currents introduced by the author, which provide the conservation laws in the Einstein-Cartan theory.

A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

It is a classical fact that the cotangent bundle $$T^* {\mathcal {M}}$$ of a differentiable manifold $${\mathcal {M}}$$ enjoys a canonical symplectic form $$\Omega ^*$$ . If $$({\mathcal {M}},\mathrm{J} ,g,\omega )$$ is a pseudo-Kahler or para-Kahler $$2n$$ -dimensional manifold, we prove that the tangent bundle $$T{\mathcal {M}}$$ also enjoys a natural pseudo-Kahler or para-Kahler structure $$({\tilde{\hbox {J}}},\tilde{g},\Omega )$$ , where $$\Omega $$ is the pull-back by $$g$$ of $$\Omega ^*$$ and $$\tilde{g}$$ is a pseudo-Riemannian metric with neutral signature $$(2n,2n)$$ . We investigate the curvature properties of the pair $$({\tilde{\hbox {J}}},\tilde{g})$$ and prove that: $$\tilde{g}$$ is scalar-flat, is not Einstein unless $$g$$ is flat, has nonpositive (resp. nonnegative) Ricci curvature if and only if $$g$$ has nonpositive (resp. nonnegative) Ricci curvature as well, and is locally conformally flat if and only if $$n=1$$ and $$g$$ has constant curvature, or $$n>2$$ and $$g$$ is flat. We also check that (i) the holomorphic sectional curvature of $$({\tilde{\hbox {J}}},\tilde{g})$$ is not constant unless $$g$$ is flat, and (ii) in $$n=1$$ case, that $$\tilde{g}$$ is never anti-self-dual, unless conformally flat.

Stability and Convergence of Modified Du Fort–Frankel Schemes for Solving Time-Fractional Subdiffusion Equations

A class of modified Du Fort---Frankel-type schemes is investigated for fractional subdiffusion equations in the Jumarie's modified Riemann---Liouville form with constant, variable or distributed fr...

Chern–Simons line bundle on Teichmüller space

Let $X$ be a non-compact geometrically finite hyperbolic 3-manifold without cusps of rank 1. The deformation space $\mc{H}$ of $X$ can be identified with the Teichmuller space $\mc{T}$ of the conformal boundary of $X$ as the graph of a section in $T^*\mc{T}$. We construct a Hermitian holomorphic line bundle $\mc{L}$ on $\mc{T}$, with curvature equal to a multiple of the Weil-Petersson symplectic form. This bundle has a canonical holomorphic section defined by $e^{\frac{1}{\pi}{\rm Vol}_R(X)+2\pi i\CS(X)}$ where ${\rm Vol}_R(X)$ is the renormalized volume of $X$ and $\CS(X)$ is the Chern-Simons invariant of $X$. This section is parallel on $\mc{H}$ for the Hermitian connection modified by the $(1,0)$ component of the Liouville form on $T^*\mc{T}$. As applications, we deduce that $\mc{H}$ is Lagrangian in $T^*\mc{T}$, and that ${\rm Vol}_R(X)$ is a Kahler potential for the Weil-Petersson metric on $\mc{T}$ and on its quotient by a certain subgroup of the mapping class group. For the Schottky uniformisation, we use a formula of Zograf to construct an explicit isomorphism of holomorphic Hermitian line bundles between $\mc{L}^{-1}$ and the sixth power of the determinant line bundle.

Convergence of Kähler to real polarizations on flag manifolds via toric degenerations

In this paper we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr-Sommerfeld fibers. Our construction is based on a toric degeneration of flag varieties and a deformation of K\"ahler structure on toric varieties by symplectic potentials.

A finite difference construction of the spheroidal wave functions

A finite difference construction of the spheroidal wave functions

Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

We define the Global Centre Symmetry set (GCS) of a smooth closed m-dimensional submanifold M of R^n, $n \leq 2m$, which is an affinely invariant generalization of the centre of a k-sphere in R^{k+1}. The GCS includes both the centre symmetry set defined by Janeczko and the Wigner caustic defined by Berry. We develop a new method for studying generic singularities of the GCS which is suited to the case when M is lagrangian in R^{2m} with canonical symplectic form. The definition of the GCS, which slightly generalizes one by Giblin and Zakalyukin, is based on the notion of affine equidistants, so, we first study singularities of affine equidistants of Lagrangian submanifolds, classifying all the stable ones. Then, we classify the affine-Lagrangian stable singularities of the GCS of Lagrangian submanifolds and show that, already for smooth closed convex curves in R^2, many singularities of the GCS which are affine stable are not affine-Lagrangian stable.

Knot contact homology

The conormal lift of a link $K$ in $\R^3$ is a Legendrian submanifold $\Lambda_K$ in the unit cotangent bundle $U^* \R^3$ of $\R^3$ with contact structure equal to the kernel of the Liouville form. Knot contact homology, a topological link invariant of $K$, is defined as the Legendrian homology of $\Lambda_K$, the homology of a differential graded algebra generated by Reeb chords whose differential counts holomorphic disks in the symplectization $\R \times U^*\R^3$ with Lagrangian boundary condition $\R \times \Lambda_K$. We perform an explicit and complete computation of the Legendrian homology of $\Lambda_K$ for arbitrary links $K$ in terms of a braid presentation of $K$, confirming a conjecture that this invariant agrees with a previously-defined combinatorial version of knot contact homology. The computation uses a double degeneration: the braid degenerates toward a multiple cover of the unknot which in turn degenerates to a point. Under the first degeneration, holomorphic disks converge to gradient flow trees with quantum corrections. The combined degenerations give rise to a new generalization of flow trees called multiscale flow trees. The theory of multiscale flow trees is the key tool in our computation and is already proving to be useful for other computations as well.

Wigner functional of fermionic fields

The Wigner function, which provides a phase-space description of quantum systems, has various applications in quantum mechanics, quantum kinetic theory, quantum optics, radiation transport and others. The concept of Wigner function has been extended to quantum fields, scalar and electromagnetic. Then, one deals with the Wigner functional which gives a distribution of field and its conjugate momentum. We introduce here the Wigner functional of fermionic fields of the values in a Grassmann algebra. Properties of the functional are discussed and its equation of motion, which is of the Liouville's form, is derived.

\U0001d4ab\U0001d4af Symmetric Hamiltonian Model and Exactly Solvable Potentials

Searching for non-Hermitian, \U0001d4ab\U0001d4af-symmetric Hamiltonians [1] with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a \U0001d4ab\U0001d4af symmetric non-Hermitian Hamiltonian model which is given as where ω and α are real constants, and are first order differential operators. Because the Hamiltonian ℋ is pseudo-Hermitian, we have obtained the Hermitian equivalent of ℋ which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened and hyperbolic Rosen-Morse II potentials. Using convenient sinilarity transformations, we have obtained a physical Hamiltonian h for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics [2].

Measure of a 2-component link

A two-component link produces a torus as the product of the component knots in a two-point configuration space of a three-sphere. This space can be identified with a cotangent bundle and also with an indefinite Grassmannian. We show that the integration of the absolute value of the canonical symplectic form is equal to the area of the torus with respect to the pseudo-Riemannian structure, and that it attains the minimum only at the “best” Hopf links.

Poisson mechanics for perturbed MIC–Kepler problems at both positive and negative energies

The MIC–Kepler problem, an extension of the Kepler problem, is known to admit the symmetry group SU(2) × SU(2) or , according to whether the energy is negative or positive. In general, each of the co-adjoint orbits of a Lie group carries the canonical symplectic form called the KKS form, and a Hamiltonian dynamical system is defined on it if a suitable Hamiltonian is given. Perturbed MIC–Kepler problems can be treated in this setting if a perturbed Hamiltonian in normal form is determined according to whether the energy is negative or positive. Since the co-adjoint orbit in question can be viewed as a symplectic leaf of the associated Lie algebra or according to whether the energy is negative or positive, the perturbed MIC–Kepler problems in normal form can be described in the Poisson mechanics defined on respective symmetry Lie algebras. Thus, the equations of motion for perturbed systems can be described in the form of Poisson brackets for both cases of and on an equal footing. It will be shown further how two parameters assigning a co-adjoint orbit of SU(2) × SU(2) or are related to the parameters contained in the MIC–Kepler problem. The perturbation of the MIC–Kepler problem to be treated in this article is rather restricted to that caused by the presence of weak constant electric and magnetic fields orthogonal to each other. When regularized, the perturbed Hamiltonians at both positive and negative energies are put in the Birkhoff–Gustavson normal form and thereby the flows generated by the perturbed Hamiltonians are studied in Poisson mechanics in terms of variables associated with constants of motion for the MIC–Kepler problem.

Evolution equation for radiance and coherence

If the field of applications of radiometry is restricted to energy calculations of optical systems then the theorem about invariant of the phase radiance (or the reduced radiance) along the light ray takes the form of Liouville's equation. This differential equation is a good basis for the traditional radiometry of Lambertian sources, as well as radiometry of the quasi-homogeneous sources. It is known that the wave analogue of phase radiance is the Wigner definition function. This wave analogy helps to clarify the limits of applicability of the concepts and approximations, e.g. properties of optical media in which the radiation transfer can be described by Liouville's equation.

Globally solvable systems of complex vector fields

We consider a class of involutive systems of n smooth vector fields on the n+1 dimensional torus. We obtain a complete characterization for the global solvability of this class in terms of Liouville forms and of the connectedness of all sublevel and superlevel sets of the primitive of a certain 1-form in the minimal covering space.

Invariant Kählerian structures on the cotangent bundles of reductive spaces

We present a description of invariant Kahlerian structures on the cotangent bundle of a reductive homogeneous space (with a canonical symplectic form as a Kahler form) in terms of equivariant operator functions on a certain linear space. We have obtained an explicit expression (operator function) for the adapted complex structure on the cotangent bundle of an arbitrary compact reductive space.

Cubic polynomials on Lie groups: reduction of the Hamiltonian system

This paper analyzes the optimal control problem of cubic polynomials on compact Lie groups from a Hamiltonian point of view and its symmetries. The dynamics of the problem is described by a presymplectic formalism associated with the canonical symplectic form on the cotangent bundle of the semidirect product of the Lie group and its Lie algebra. Using these control geometric tools, the relation between the Hamiltonian approach developed here and the known variational one is analyzed. After making explicit the left trivialized system, we use the technique of Marsden–Weinstein reduction to remove the symmetries of the Hamiltonian system. In view of the reduced dynamics, we are able to guarantee, by means of the Lie–Cartan theorem, the existence of a considerable number of independent integrals of motion in involution.

General linear dynamics – quantum, classical or hybrid

We describe our recent proposal of a path integral formulation of classical Hamiltonian dynamics. Which leads us here to a new attempt at hybrid dynamics, which concerns the direct coupling of classical and quantum mechanical degrees of freedom. This is of practical as well as of foundational interest and no fully satisfactory solution of this problem has been established to date. Related aspects will be observed in a general linear ensemble theory, which comprises classical and quantum dynamics in the form of Liouville and von Neumann equations, respectively, as special cases. Considering the simplest object characterized by a two-dimensional state-space, we illustrate how quantum mechanics is special in several respects among possible linear generalizations.

Scalar-flat Kähler metrics on non-compact symplectic toric 4-manifolds

In a recent paper, Donaldson (J. Gökova Geom. Topol. 3:1–8, 2009) explains how to use an older construction of Joyce (Duke Math. J. 77:519–552, 1995) to obtain four dimensional local models for scalar-flat Kähler metrics with a 2-torus symmetry. In another paper (Geom. Funct. Anal. 19:83–136, 2009), using the same idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kähler metrics on \({\mathbb{R}^4}\) . In this article, we generalize Donaldson’s method and construct complete scalar-flat toric Kähler metrics on any symplectic toric 4-manifold with “strictly unbounded” moment polygon. These include the asymptotically locally Euclidean scalar-flat Kähler metrics previously constructed by Calderbank and Singer (Invent. Math. 156:405–443, 2004), as well as new examples of complete scalar-flat toric Kähler metrics which are asymptotic to Donaldson’s generalized Taub-NUT metrics. Our construction is in symplectic action-angle coordinates and determines all these metrics via their symplectic potentials. When the first Chern class is zero we obtain a new description of known Ricci-flat Kähler metrics.

Symplectic Geometry of Entanglement

We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using Kostant-Sternberg theorem, we show that separable states form a unique Kaehler orbit, whereas orbits of entanglement states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement and we show how to calculate it for various multiparticle systems providing also simple criteria of separability. The presented method is general and can be applied also under different additional symmetry conditions stemming, eg. from the indistinguishability of particles.