Narrow Results

A dynamical system is canonically associated to every Drinfeld double of any affine Kac–Moody group. In particular, the choice of the affine Lu–Weinstein double gives a smooth one-parameter deformation of the standard WZW model. The deformed WZW model is exactly solvable and it admits the chiral decomposition. Its classical action is not invariant with respect to the left and right action of the loop group, however, it satisfies the weaker condition of the Poisson–Lie symmetry. The structure of the deformed WZW theory is characterized by several ordinary and dynamical r-matrices with spectral parameter. They describe the q-deformed current algebras, appear in the definition of q-primary fields and characterize the quasitriangular exchange (braiding) relations. The symplectic structure of the deformed chiral WZW theory is cocharacterized by the same elliptic dynamical r-matrix that appears in the Bernard generalization of the Knizhnik–Zamolodchikov equation, with q entering the modular parameter of the Jacobi theta functions. This reveals a remarkable connection between the classical q-deformed WZW model and the quantum standard WZW theory on elliptic curves.

We construct a family of global Fourier Integral Operators, defined for arbitrary large times, representing a global parametrix for the Schrödinger propagator when the potential is quadratic at infinity. This construction is based on the geometric approach to the corresponding Hamilton–Jacobi equation and thus sidesteps the problem of the caustics generated by the classical flow. Moreover, a detailed study of the real phase function allows us to recover a WKB semiclassical approximation which necessarily involves the multivaluedness of the graph of the Hamiltonian flow past the caustics.

This paper aims at explaining some incarnations of the idea of topological recursion: in two-dimensional quantum field theories (2d TQFTs), in cohomological field theories (CohFT), and in the computation of volumes of the moduli space of curves. It gives an introduction to the formalism of quantum Airy structures on which the topological recursion is based, which is seen at work in the above topics.

Generalizing local Gromov–Witten theory, in this paper we define a local version of symplectic field theory. When the symplectic manifold with cylindrical ends is four-dimensional and the underlying simple curve is regular by automatic transversality, we establish a transversality result for all its multiple covers and discuss the resulting algebraic structures.

The unimodality conjecture posed by Tolman in [L. Jeffrey, T. Holm, Y. Karshon, E. Lerman and E. Meinrenken, Moment maps in various geometries, http://www.birs.ca/workshops/2005/05w5072/report05w5072.pdf] states that if $(M,\omega )$ is a $2n$-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, then the sequence of Betti numbers $\{b_0(M),b_2(M),\dots ,b_{2n}(M)\}$ is unimodal, i.e. $b_i(M)\le b_{i+2}(M)$ for every $i<n$. Recently, the author and Kim [Y. Cho and M. Kim, Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points, Math. Res. Lett.21(4) (2014) 691–696] proved that the unimodality holds in eight-dimensional case by using equivariant cohomology theory. In this paper, we generalize the idea in [Y. Cho and M. Kim, Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points, Math. Res. Lett.21(4) (2014) 691–696] to an arbitrary dimensional case. We prove the conjecture in arbitrary dimension under the assumption that the moment map $H:M\to ℝ$ is index-increasing, which means that $\mathrm{\text{ind}}(p)<\mathrm{\text{ind}}(q)$ implies $H(p)<H(q)$ for every pair of critical points $p$ and $q$ of $H$, where $\mathrm{\text{ind}}(p)$ is the Morse index of $p$ with respect to $H$.

A two-dimensional integrable system being a deformation of the rational Calogero–Moser system is constructed via the symplectic reduction, performed with respect to the Sklyanin algebra action. We explicitly resolve the respective classical equations of motion via the projection method and quantize the system.

Using a covariant and gauge invariant geometric structure constructed on the Witten covariant phase space for Dirac–Nambu–Goto bosonic p-branes propagating in a curved background, we find the canonically conjugate variables, and the relevant commutation relations are considered. We also find the canonical variables for the Gauss–Bonnet topological term in string theory.

In this paper, we find that the asymptotic nonlinear Maslov index defined on the universal cover of the group of all contact Hamiltonian diffeomorphisms of the standard (2n - 1)-dimensional contact sphere is a quasimorphism. Then we show our main result: Let M be standard (n - 1)-dimensional complex projective space. We prove that the value of the pullback of the asymptotic nonlinear Maslov index to the universal cover of the group of Hamiltonian diffeomorphisms of M, when evaluated on a diffeomorphism supported in a sufficiently small open subset of M, equals times the Calabi invariant of this diffeomorphism.

In this paper we discuss various geometric aspects related to the Schrödinger and the Pauli equations. First we resume the Madelung–Bohm hydrodynamical approach to quantum mechanics and recall the Hamiltonian structure of the Schrödinger equation. The probability current provides an equivariant moment map for the group $G=\mathrm{\text{sDiff}}(R^3)$ of volume-preserving diffeomorphisms of $R^3$ (rapidly approaching the identity at infinity) and leads to a current algebra of Rasetti–Regge type. The moment map picture is then extended, mutatis mutandis, to the Pauli equation and to generalized Schrödinger equations of the Pauli–Thomas type. A gauge theoretical reinterpretation of all equations is obtained via the introduction of suitable Maurer–Cartan gauge fields and it is then related to Weyl geometric and pilot wave ideas. A general framework accommodating Aharonov–Bohm and Aharonov–Casher effects is presented within the gauge approach. Furthermore, a kind of holomorphic geometric quantization can be performed and yields natural “coherent state” representations of $G$. The relationship with the covariant phase space and density manifold approaches is then outlined. Comments on possible extensions to nonlinear Schrödinger equations, on Fisher-information theoretic aspects and on stochastic mechanics are finally made.

In this paper, using the elegant language of differential forms, we provide a covariant formulation of the equilibrium statistical mechanics of non-Hamiltonian systems. The key idea of the paper is to focus on the structure of phase space and its kinematical and dynamical roles. While in the case of Hamiltonian systems, the structure of the phase space is a symplectic structure (a nondegenerate closed two-form), we consider an almost symplectic structure for the more general case of non-Hamiltonian systems. An almost symplectic structure is a nondegenerate but not necessarily closed two-form structure. Consequently, the dynamics becomes non-Hamiltonian and based on the fact that the structure is nondegenerate, we can also define a volume element. With a well-defined volume in hand, we derive the Liouville equation and find an invariant statistical state. Recasting non-Hamiltonian systems in terms of the almost symplectic geometry has at least two advantages: the formalism is covariant and therefore does not depend on coordinates and there is no confusion in the determination of the natural volume element of the system. For clarification, we investigate the application of the formalism in two examples in which the underlying geometry of the phase space is locally conformal symplectic geometry.

In this paper, we analyze in detail a collection of motivating examples to consider $b^m$-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every $b^m$-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to $b$-symplectic manifolds: $b$-contact manifolds.

Coherently with the principle of analogy suggested by Dirac, we describe a general setting for reducing a classical dynamics, and the role of the Noether theorem — connecting symmetries with constants of the motion — within a reduction. This is the first of two papers, and it focuses on the reduction within the Poisson and the symplectic formalism.

Dirac stated that the Cartesian coordinates are uniquely suited for expressing the canonical commutation relations in a simple form. By contrast, expressing these commutation relations in any other coordinate system is more complicated and less obvious. The question that we address in this work is the reason why this is true. We claim that this unique role of the Cartesian coordinates is a result of the existence and uniqueness of the Hofer metric on the space of canonical transformations of the phase space of the system getting quantized.

The contact geometric structure of the thermodynamic phase space is used to introduce a novel symplectic structure on the tangent bundle of the equilibrium space. Moreover, it turns out that the equilibrium space can be interpreted as a Lagrange submanifold of the corresponding tangent bundle, if the fundamental equation is known explicitly. As a consequence, Hamiltonians can be defined that describe thermodynamic processes.

The influence of a curved spacetime $M$ on the physical behavior of an ideal gas of $N$ particles is analyzed by considering the phase space of the system as a region of the cotangent bundle $T^{\ast }{M}^N$ and using Souriau’s Lie group thermodynamics to define the corresponding probability distribution function. While the construction of the phase space respects the separability of the system, by forcing each particle to satisfy the so-called mass-shell constraint, the probability distribution is constructed by mixing the natural symplectic structure of the cotangent bundle with a Hamiltonian description of the system. In this way, the spacetime is introduced into the statistics and its isometries turn out to be of special interest because the distributions are parametrized by the elements of the Lie algebra of the isometry group, through the momentum map of the action of the isometries in $T^{\ast }{M}^N$. We find the Gibbs distribution that, in the simplest case of a flat spacetime, reduces to the so-called modified Jüttner distribution, used to describe ideal gases in the regime of special relativity. We also define a temperature-like function using the norm of a Killing vector, which allows us to recover the so-called Tolman–Ehrenfest effect. As a particular example, we study the outer region of a Schwarzschild black hole, for which a power series expansion of the Schwarzschild radius allows us to represent the partition function and the Gibbs distribution in terms of the corresponding quantities of the Minkowski spacetime.

In this paper, we present the coisotropic embedding theorem as a tool to provide a solution for the inverse problem of the calculus of variations for a particular class of implicit differential equations, namely the equations of motion of free Electrodynamics.

We carry on the approach used in [L. Schiavone, The inverse problem within free Electrodynamics and the coisotropic embedding theorem, Inter. J. Geom. Meth. Mod. Phys.21(7) (2024) 2450131] to provide a solution for the inverse problem of the calculus of variations for Maxwell equations in vacuum and we provide an abstract theory including all implicit differential equations that can be formulated in terms of vector fields over pre-symplectic manifolds.

The author first proves the existences of J-holomorphic curves in the symplectizations of Legendre fibrations and then as an application confirms the Weinstein conjectures on contact manifolds of Legendre fibrations. As a corollary, a new proof on the theorem due to Hofer, Viterbo, Gluck, Ziller, Weinstein and Ljusternik-Fet Theorem is provided, which is quite different from their original proofs.

We study the Aubry set which appears in Mather theory of convex Hamiltonian systems from the point of view of symplectic geometry.

Let $M$ be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian $H:T^{\ast }M\to ℝ$ and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if $M$ is not aspherical, then contractible periodic orbits exist for almost all energies above the maximum critical value of $H$.