Canonical Symplectic Structure Research Articles

Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

We consider the moduli space of vector bundles of rank n and degree ng over a fixed Riemann surface of genus g⩾2 with the explicit parametrization in terms of the Tyurin data. The ‘non-abelian’ theta divisor consists of bundles such that h1⩾1 . On the complement of this divisor we construct a non-abelian (i.e. matrix) Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which also depends holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the GLn character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by h 1.

Variational and Dirac structures for interconnected distributed-discrete systems

We present how the interconnection of distributed and discrete systems can be realized within the setting of variational principles and Dirac structures on the Lagrangian side. For the distributed system we focus on the case when the configuration space is an infinite-dimensional vector space and the system is subject to boundary energy flow. A key property of our approach is that it systematically extends the canonical variational and symplectic structures of finite and infinite-dimensional Lagrangian and Hamiltonian mechanics.

Quantum geodesics in quantum mechanics

We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus Ω1 depending on the Hamiltonian p2/2m + V(x), and a flat quantum connection ∇ with torsion such that a previous quantum-geometric formulation of flow along autoparallel curves (or “geodesics”) is exactly Schrödinger’s equation. The connection ∇ preserves a non-symmetric quantum metric given by the canonical symplectic structure lifted to a rank (0, 2) tensor on the extended phase space where we adjoin a time variable. We also apply the same approach to obtain a novel flow generated by the Klein–Gordon operator on Minkowski spacetime with a background electromagnetic field, by formulating quantum “geodesics” on the relativistic Heisenberg algebra with proper time for the external geodesic parameter. Examples include quantum geodesics that look like a relativistic free particle wave packet and a hydrogen-like atom.

A local normal form for Hamiltonian actions of compact semisimple Poisson–Lie groups

The main contribution of this manuscript is a local normal form for Hamiltonian actions of Poisson-Lie groups $K$ on a symplectic manifold equipped with an $AN$-valued moment map, where $AN$ is the dual Poisson-Lie group of $K$. Our proof uses the delinearization theorem of Alekseev which relates a classical Hamiltonian action of $K$ with $\mathfrak{k}^*$-valued moment map to a Hamiltonian action with an $AN$-valued moment map, via a deformation of symplectic structures. We obtain our main result by proving a ``delinearization commutes with symplectic quotients'' theorem which is also of independent interest, and then putting this together with the local normal form theorem for classical Hamiltonian actions wtih $\mathfrak{k}^*$-valued moment maps. A key ingredient for our main result is the delinearization $\mathcal{D}(\omega_{can})$ of the canonical symplectic structure on $T^*K$, so we additionally take some steps toward explicit computations of $\mathcal{D}(\omega_{can})$. In particular, in the case $K=SU(2)$, we obtain explicit formulas for the matrix coefficients of $\mathcal{D}(\omega_{can})$ with respect to a natural choice of coordinates on $T^*SU(2)$.

Rank-adaptive structure-preserving model order reduction of Hamiltonian systems

This work proposes an adaptive structure-preserving model order reduction method for finite-dimensional parametrized Hamiltonian systems modeling non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width typical of transport problems, the full model is approximated on local reduced spaces that are adapted in time using dynamical low-rank approximation techniques. The reduced dynamics is prescribed by approximating the symplectic projection of the Hamiltonian vector field in the tangent space to the local reduced space. This ensures that the canonical symplectic structure of the Hamiltonian dynamics is preserved during the reduction. In addition, accurate approximations with low-rank reduced solutions are obtained by allowing the dimension of the reduced space to change during the time evolution. Whenever the quality of the reduced solution, assessed via an error indicator, is not satisfactory, the reduced basis is augmented in the parameter direction that is worst approximated by the current basis. Extensive numerical tests involving wave interactions, nonlinear transport problems, and the Vlasov equation demonstrate the superior stability properties and considerable runtime speedups of the proposed method as compared to global and traditional reduced basis approaches.

On Extensions of Canonical Symplectic Structure from Coadjoint Orbit of Complex General Linear Group

On Extensions of Canonical Symplectic Structure from Coadjoint Orbit of Complex General Linear Group

Bi-Hamiltonian Structure of Spin Sutherland Models: The Holomorphic Case

We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of mathrm{GL}(n,mathbb {C}), which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard mathrm{GL}(n,mathbb {C}) Poisson–Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.

Deformation Quantization and Homological Reduction of a Lattice Gauge Model

For a compact Lie group G we consider a lattice gauge model given by the G-Hamiltonian system which consists of the cotangent bundle of a power of G with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi–Civita connection of the Killing metric on G. We then explain and refine quantized homological reduction for the construction of a star product on the symplectically reduced space in the singular case. Afterwards we show that for $$G = {\mathrm {SU}}(2)$$ the main hypotheses ensuring the method of quantized homological reduction to be applicable hold in the case of our lattice gauge model. For that case, this implies that the—in general singular—symplectically reduced phase space of the corresponding lattice gauge model carries a star product.

On Geodesic Flow and Energy Functional on Riemannian Manifolds

In this paper, we study the geodesic flow and the energy functional on a Riemannian manifold and show that the geodesics have minimal energy, in other words, are the minimizers of the energy functional, from the new perspective of involution, and that the geodesic flow is a Hamiltonian flow which has a close connection with the canonical symplectic structure on the tangent bundle of a Riemannian manifold.

The geometry of graded cotangent bundles

Given a vector bundle A→M we study the geometry of the graded manifolds T∗[k]A[1], including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical structures, such as higher Courant algebroids on A⊕⋀k−1A∗ and higher Dirac structures therein, semi-direct products of Lie algebroid structures on A with their coadjoint representations up to homotopy, and branes on certain AKSZ σ-models.

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants.

We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented.

Pre-symplectic structures on the space of connections

Let X be a four-manifold with boundary three-manifold M. We shall describe (i) a pre-symplectic structure on the space A(X) of connections on the bundle X×SU(n) that comes from the canonical symplectic structure on the cotangent space T⁎A(X), and (ii) a pre-symplectic structure on the space A0♭(M) of flat connections on M×SU(n) that have null charge. These two structures are related by the boundary restriction map. We discuss also the Hamiltonian features of the space of connections A(X) together with the action of the group of gauge transformations.

Stein and Weinstein structures on disk cotangent bundles of surfaces

Gompf (Ann Math (2) 148(2):619–693, 1998) describes a Stein domain structure on the disk cotangent bundle of any closed surface S by a Legendrian handlebody diagram. We prove that Gompf’s Stein domain is symplectomorphic to the disk cotangent bundle equipped with its canonical symplectic structure and the boundary of this domain is contactomorphic to the unit cotangent bundle of S equipped with its canonical contact structure. As a corollary, we obtain a surgery diagram for the canonical contact structure on the unit cotangent bundle of S.

Bayesian Inference from Symplectic Geometric Viewpoint

The purpose of this article is to formulate Bayesian updating from dynamical viewpoint. We prove that Bayesian updating for population mean vectors of multivariate normal distributions can be expressed as an affine symplectic transformation on a phase space with the canonical symplectic structure.

Symplectic Wick rotations between moduli spaces of 3-manifolds

Given a closed hyperbolic surface $S$, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on $S\times\R$. We describe natural maps between (parts of) $\cQF$ and $\cGH_{-1}$, called "Wick rotations", defined in terms of special surfaces (e.g. minimal/maximal surfaces, CMC surfaces, pleated surfaces) and prove that these maps are at least $C^1$ smooth and symplectic with respect to the canonical symplectic structures on both $\cQF$ and $\cGH_{-1}$. Similar results involving the spaces of globally hyperbolic de Sitter and Minkowski metrics are also described. These 3-dimensional results are shown to be equivalent to purely 2-dimensional ones. Namely, consider the double harmonic map $\cH:T^*\cT\to\cTT$, sending a conformal structure $c$ and a holomorphic quadratic differential $q$ on $S$ to the pair of hyperbolic metrics $(m_L,m_R)$ such that the harmonic maps isotopic to the identity from $(S,c)$ to $(S,m_L)$ and to $(S,m_R)$ have, respectively, Hopf differentials equal to $i q$ and $-i q$, and the double earthquake map $\cE:\cT\times\cML\to\cTT$, sending a hyperbolic metric $m$ and a measured lamination $l$ on $S$ to the pair $(E_L(m,l), E_R(m,l))$, where $E_L$ and $E_R$ denote the left and right earthquakes. We describe how such 2-dimensional double maps are related to 3-dimensional Wick rotations and prove that they are also $C^1$ smooth and symplectic.

Canonical symplectic structure and structure-preserving geometric algorithms for Schrödinger–Maxwell systems

An infinite dimensional canonical symplectic structure and structure-preserving geometric algorithms are developed for the photon–matter interactions described by the Schrödinger–Maxwell equations. The algorithms preserve the symplectic structure of the system and the unitary nature of the wavefunctions, and bound the energy error of the simulation for all time-steps. This new numerical capability enables us to carry out first-principle based simulation study of important photon–matter interactions, such as the high harmonic generation and stabilization of ionization, with long-term accuracy and fidelity.

On geometric quantization of the Dirac magnetic monopole

We give a simple derivation of the spectrum of the Dirac magnetic monopole on a unit sphere S2 based on geometric quantization and the Frobenius reciprocity formula. The starting point is the calculation by Novikov and Schmelzer of the canonical symplectic structure on the coadjoint orbits of the isometry group of 3-dimensional Euclidean space E(3), which showed the appearance of the Dirac magnetic term.

Symplectic Structures on the Cotangent Bundles of Open 4-Manifolds

We show that, for any two orientable smooth open 4-manifolds $X_0,X_1$ which are homeomorphic, their cotangent bundles $T^*X_0,T^*X_1$ are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold $R$ homeomorphic to $\mathbb{R}^4$, the standard Stein structure on $T^*R$ is Stein homotopic to the standard Stein structure on $T^*\mathbb{R}^4 = \mathbb{R}^8$. We use this to show that any exotic $\mathbb{R}^4$ embeds in the standard symplectic $\mathbb{R}^8$ as a Lagrangian submanifold. As a corollary, we show that $\mathbb{R}^8$ has uncountably many smoothly distinct foliations by Lagrangian $\mathbb{R}^4$s with their standard smooth structure.

Special Kähler manifolds and generalized geometry

Given a special Kähler manifold (M,ω,J,∇) we construct a subbundle of the generalized tangent bundle of M endowed with a natural special Kähler structure. Precisely we consider E=T(M)⊕T⁎(M) and the subbundle Lω=graph(ω); we prove that Lω is invariant with respect to the calibrated complex structure Jg=(O−g−1gO) of E defined by the Riemannian metric g=−ωJ on M and we define a special connection ∇˜ on E by using a natural contravariant connection on T⁎(M) defined by ω. We prove that (Lω,(,)|Lω,J|Lωg,∇˜|Lω) is special Kähler, where (,) is the canonical symplectic structure on E. Moreover, by using the identification of T(M)⊕T⁎(M) with T(T⁎(M)) defined by the symplectic connection ∇, we describe the corresponding special Kähler subbundle of T(T⁎(M)). Also we prove that the construction is invariant with respect to the class of connections {∇θ} introduced in Alekseevsky et al. [1].

SYMPLECTIC STRUCTURES ON STATISTICAL MANIFOLDS

AbstractA relationship between symplectic geometry and information geometry is studied. The square of a dually flat space admits a natural symplectic structure that is the pullback of the canonical symplectic structure on the cotangent bundle of the dually flat space via the canonical divergence. With respect to the symplectic structure, there exists a moment map whose image is the dually flat space. As an example, we obtain a duality relation between the Fubini–Study metric on a projective space and the Fisher metric on a statistical model on a finite set. Conversely, a dually flat space admitting a symplectic structure is locally symplectically isomorphic to the cotangent bundle with the canonical symplectic structure of some dually flat space. We also discuss nonparametric cases.