Natural Symplectic Structures Research Articles

Symplectic Geometry of Teichmüller Spaces for Surfaces with Ideal Boundary

A hyperbolic 0-metric on a surface with boundary is a hyperbolic metric on its interior, exhibiting the boundary behavior of the standard metric on the Poincaré disk. Consider the infinite-dimensional Teichmüller spaces of hyperbolic 0-metrics on oriented surfaces with boundary, up to diffeomorphisms fixing the boundary and homotopic to the identity. We show that these spaces have natural symplectic structures, depending only on the choice of an invariant metric on sl(2,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}(2,\\mathbb {R})$$\\end{document}. We prove that these Teichmüller spaces are Hamiltonian Virasoro spaces for the action of the universal cover of the group of diffeomorphisms of the boundary. We give an explicit formula for the Hill potential on the boundary defining the moment map. Furthermore, using Fenchel–Nielsen parameters we prove a Wolpert formula for the symplectic form, leading to global Darboux coordinates on the Teichmüller space.

Symplectic aspects of the tt*-Toda equations

We evaluate explicitly, in terms of the asymptotic data, the ratio of the constant pre-factors in the large and small x asymptotics of the tau functions for global solutions of the tt*-Toda equations. This constant problem for the sinh–Gordon equation, which is the case n = 1 of the tt*-Toda equations, was solved by Tracy (1991 Commun. Math. Phys. 142 297–311). We also introduce natural symplectic structures on the space of asymptotic data and on the space of monodromy data for a wider class of solutions, and show that these symplectic structures are preserved by the Riemann-Hilbert correspondence.

Toric Geometry ofSL2(ℂ) Free Group Character Varieties from Outer Space

AbstractCuller and Vogtmann defined a simplicial spaceO(g), calledouter space, to study the outer automorphism group of the free groupFg. Using representation theoretic methods, we give an embedding ofO(g) into the analytification of X(Fg,SL2(ℂ)), theSL2(ℂ) character variety ofFg, reproving a result of Morgan and Shalen. Then we show that every pointvcontained in a maximal cell ofO(g) defines a flat degeneration of X(Fg,SL2(ℂ)) to a toric varietyX(PΓ). We relate X(Fg,SL2(ℂ)) andX(v) topologically by showing that there is a surjective, continuous, proper map Ξv:X(Fg,SL2(ℂ)) →X(v). We then show that this map is a symplectomorphism on a dense open subset of X(Fg, SL2(ℂ)) with respect to natural symplectic structures on X(Fg, SL2(ℂ)) andX(v). In this way, we construct an integrable Hamiltonian system in X(Fg, SL2(ℂ)) for each point in a maximal cell ofO(g), and we show that eachvdefines a topological decomposition of X(Fg, SL2(ℂ)) derived from the decomposition ofX(PΓ) by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell inO(g) all arise as divisorial valuations built from an associated projective compactification of X(Fg, SL2(ℂ)).

On volumes of classical supermanifolds

We consider the volumes of classical supermanifolds (such as the supersphere, complex projective superspace, Stiefel and Grassmann supermanifolds) with respect to natural metrics or symplectic structures. We show that the formulae for the volumes of these supermanifolds can be obtained from the formulae for the volumes of the corresponding ordinary manifolds (under some universal normalization of the volume) by analytic continuation with respect to parameters. The volumes of nontrivial supermanifolds may be identically equal to zero. In the 1970s Berezin showed that the total Haar measure of the unitary supergroup vanishes except in the cases and , when the supergroup is the ordinary unitary group or . Some time ago Witten conjectured that the Liouville volume of a compact even symplectic supermanifold is always equal to zero (except for ordinary manifolds). We give counterexamples to this conjecture, present a simple explanation of Berezin's theorem, and generalize this theorem to the Stiefel supermanifold . We mention a connection with recent work of Mkrtchyan and Veselov on universal formulae in Lie algebra theory. Bibliography: 32 titles.

Quantum Cohomologies on Products of Cosymplectic Manifolds and Circles

In this paper we study the products of cosymplectic manifolds and a unit circle, which have natural symplectic structures. We have some relations on moduli spaces, Gromov–Witten invariants, and quantum cohomologies of cosymplectic manifolds and the products. As an example we examine the cosymplectic manifold \(M=S^2\times T\times S^1\) and the product \(M\times S^1\), and we also calculate their moduli spaces, Gromov–Witten invariants, and quantum cohomologies.

Hawking radiation and classical tunneling: A ray phase space approach

Acoustic waves in fluids undergoing the transition from sub- to supersonic flow satisfy governing equations similar to those for light waves in the immediate vicinity of a black hole event horizon. This acoustic analogy has been used by Unruh and others as a conceptual model for “Hawking radiation.” Here, we use variational methods, originally introduced by Brizard for the study of linearized MHD, and ray phase space methods, to analyze linearized acoustics in the presence of background flows. The variational formulation endows the evolution equations with natural Hermitian and symplectic structures that prove useful for later analysis. We derive a 2 × 2 normal form governing the wave evolution in the vicinity of the “event horizon.” This shows that the acoustic model can be reduced locally (in ray phase space) to a standard (scalar) tunneling process weakly coupled to a unidirectional non-dispersive wave (the “incoming wave”). Given the normal form, the Hawking “thermal spectrum” can be derived by invoking standard tunneling theory, but only by ignoring the coupling to the incoming wave. Deriving the normal form requires a novel extension of the modular ray-based theory used previously to study tunneling and mode conversion in plasmas. We also discuss how ray phase space methods can be used to change representation, which brings the problem into a form where the wave functions are less singular than in the usual formulation, a fact that might prove useful in numerical studies.

Duality constructions from quantum state manifolds

The formalism of quantum state space geometry on manifolds of generalised coherent states is proposed as a natural setting for the construction of geometric dual descriptions of non-relativistic quantum systems. These state manifolds are equipped with natural Riemannian and symplectic structures derived from the Hilbert space inner product. This approach allows for the systematic construction of geometries which reflect the dynamical symmetries of the quantum system under consideration. We analyse here in detail the two dimensional case and demonstrate how existing results in the AdS_2/CFT_1 context can be understood within this framework. We show how the radial/bulk coordinate emerges as an energy scale associated with a regularisation procedure and find that, under quite general conditions, these state manifolds are asymptotically anti-de Sitter solutions of a class of classical dilaton gravity models. For the model of conformal quantum mechanics proposed by de Alfaro et. al. the corresponding state manifold is seen to be exactly AdS_2 with a scalar curvature determined by the representation of the symmetry algebra. It is also shown that the dilaton field itself is given by the quantum mechanical expectation values of the dynamical symmetry generators and as a result exhibits dynamics equivalent to that of a conformal mechanical system.

Gravitational Gauge Theory and the Existence of Time

General relativity may be formulated as a gauge theory more than one way using the quotient manifold approach. We contrast the structures that arise in four gravitational gauge theories, three of which give satisfactory gauge theoris of general relativity. Of particular interest is the quotient of the conformal group of a flat space by its Weyl subgroup, which always has natural symplectic and metric structures in addition to the requisite manifold. This quotient space admits canonically conjugate, orthogonal, metric submanifolds distinct from the original space if and only if the original flat space has signature n, -n or 0. In the Euclidean cases, the resultant configuration space must be Lorentzian. This gives a 1-1 mapping between Euclidean and Lorentzian submanifolds, with induced Euclidean gravity or general relativity, respectively.

THE EXISTENCE OF TIME

Of those gauge theories of gravity known to be equivalent to general relativity, only the biconformal gauging introduces new structures — the quotient of the conformal group of any pseudo-Euclidean space by its Weyl subgroup always has natural symplectic and metric structures. Using this metric and symplectic form, we show that there exist canonically conjugate, orthogonal, metric submanifolds if and only if the original gauged space is Euclidean or signature 0. In the Euclidean cases, the resultant configuration space must be Lorentzian. Therefore, in this context, time may be viewed as a derived property of general relativity.

Pseudo-Riemannian geodesics and billiards

In pseudo-Riemannian geometry the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. We describe the geometry of these structures and their generalizations. We also introduce and study pseudo-Euclidean billiards, emphasizing their distinction from Euclidean ones. We present a pseudo-Euclidean version of the Clairaut theorem on geodesics on surfaces of revolution. We prove pseudo-Euclidean analogs of the Jacobi–Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo-Euclidean space.

Conjugate variables in quantum field theory and natural symplectic structures

Within standard quantum field theory we establish relations which operators conjugate to the energy–momentum operator of the theory would have. They thus can be understood as representing the effect of coordinate operators. The non-trivial commutation relations we derive constitute natural symplectic structures in the theory. The example which is based on the energy–momentum tensor of the theory is constructed to all orders of perturbation theory. The reference theory is massless ϕ 4 . The extension to other theories is indicated.

Quasi-Hamiltonian geometry of meromorphic connections

For each connected complex reductive group G, we find a family of new examples of complex quasi-Hamiltonian G-spaces with G-valued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on principal G-bundles over a disk, and they generalise the conjugacy class example of Alekseev, Malkin, and Meinrenken [3] (which appears in the simple pole case). Using the “fusion product” in the theory, this gives a finite-dimensional construction of the natural symplectic structures on the spaces of monodromy/Stokes data of meromorphic connections over arbitrary genus Riemann surfaces, together with a new proof of the symplectic nature of isomonodromic deformations of such connections

Higher dimensional knot spaces for manifolds with vector cross products

Vector cross product structures on manifolds include symplectic, volume, G 2 - and Spin ( 7 ) -structures. We show that the knot spaces of such manifolds have natural symplectic structures, and relate instantons and branes in these manifolds to holomorphic disks and Lagrangian submanifolds in their knot spaces. For the complex case, the holomorphic volume form on a Calabi–Yau manifold defines a complex vector cross product structure. We show that its isotropic knot space admits a natural holomorphic symplectic structure. We also relate the Calabi–Yau geometry of the manifold to the holomorphic symplectic geometry of its isotropic knot space.

Moduli of Stable Parabolic Connections, Riemann–Hilbert Correspondence and Geometry of Painlevé Equation of Type _VI_, Part I

In this paper, we will give a complete geometric background for the geometry of Painlevé VI and Garnier equations. By geometric invariant theory, we will construct a smooth fine moduli space M_n^{\boldsymbol{α}} (\mathbf t, \boldsymbol λ, L) of stable parabolic connections on \mathbf P^1 with logarithmic poles at D(\mathbf t) = t_1 +· · ·+ t_n as well as its natural compactification. Moreover the moduli space \mathcal R(\mathcal P_{n,\mathbf t})_{\mathbf a} of Jordan equivalence classes of \mathit{SL}_2(\mathbf C) -representations of the fundamental group π_1(\mathbf P^1 \backslash D(\mathbf t), ∗) are defined as the categorical quotient. We define the Riemann–Hilbert correspondence \mathbf{RH} : M_n^{\boldsymbol α} (\mathbf t, \boldsymbol λ,L) → \mathcal R(\mathcal P_{n,\mathbf t})_{\mathbf a} and prove that \mathbf{RH} is a bimeromorphic proper surjective analytic map. Painlevé and Garnier equations can be derived from the isomonodromic flows and Painlevé property of these equations are easily derived from the properties of \mathbf{RH} . We also prove that the smooth parts of both moduli spaces have natural symplectic structures and \mathbf{RH} is a symplectic resolution of singularities of \mathcal R(\mathcal P_{n,\mathbf t})_{\mathbf a} , from which one can give geometric backgrounds for other interesting phenomena, like Hamiltonian structures, Bäcklund transformations, special solutions of these equations.

Symplectic Manifolds and Isomonodromic Deformations

We study moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces together with the corresponding spaces of monodromy data (involving Stokes matrices). Natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint (generalising the Atiyah–Bott approach). This enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of Jimbo, Miwa and Ueno, thereby putting the existing results for the six Painlevé equations and Schlesinger's equations into a uniform framework.

The Wigner Function for General Lie Groups and the Wavelet Transform

We build Wigner maps, functions and operators on general phase spaces arising from a class of Lie groups, including non-unimodular groups (such as the affine group). The phase spaces are coadjoint orbits in the dual of the Lie algebra of these groups and they come equipped with natural symplectic structures and Liouville-type invariant measures. When the group admits square-integrable representations, we present a very general construction of a Wigner function which enjoys all the desirable properties, including full covariance and reconstruction formulae. We study in detail the case of the affine group on the line. In particular, we put into focus the close connection between the well-known wavelet transform and the Wigner function on such groups.

Schottky uniformization and the symplectic structure of the cotangent bundle of a Teichmüller space

Using the Schottky uniformization of a marked Riemann surface, the space of equivalence classes of complex projective structures on a compact oriented C ∞ surface gets identified, in an obvious fashion, with the total space of the holomorphic cotangent bundle of the corresponding Teichmüller space. This identification is proved to be compatible with the natural symplectic structures on these two spaces. If we consider the other identification of the same two spaces obtained using the Bers’ construction of universal Riemann surface, then the compatibility of the symplectic structures was established by Kawai [S. Kawai, Math. Ann. 305 (1996) 161–182].

Canonical co-ordinates and Hamiltonian framework for the relativistic two spinning particles and three-body systems

Starting from the natural symplectic structures determined by the Weyl group for the single massive spinning particle, we build the composed system of two spinning particles. The groupal analysis allows us to determine the centre-of-mass co-ordinates, to which we add a set of complementary ones in such a way as to get collective and relative canonical variables of momentum and position. Exploiting these results we are able to introduce an analogous description of the phase-space of the scalar three-body. The equations of motion are obtained by means of the full foliation on the state equations and we are able to get them in a purely Hamiltonian scheme on the reduced physical phase-space. For the three-body we get three first-class constraints with two independent potentials; however, the Hamiltonian scheme suggests a more flexible method for introducing interactions. We remark that the direct canonical quantization, allowed by the Hamiltonian framework, gives rise, in the two-scalar-particle case with the Coulomb potential, to a Lamb shift term and with the rising linear potential to very good results about the mesons properties. Indeed, we find a wave equation formally coinciding with the one obtained and studied in the domain of the anisotropic chromodynamics.

Phase-space approach to relativistic quantum mechanics. II. Geometrical aspects

The formalism introduced in Paper I [J. Math. Phys. 18, 952 (1977)] is made manifestly covariant by including as an admissible phase space any 2n-dimensional submanifold of the forward tube 𝒯={x−iy∈Cn+1‖ y0≳‖y‖} which is of the ’’product’’ form σ=S−iΩλ, where Ωλ ={y∈Rn+1‖ y0= (λ2+y2)1/2}, λ≳0, and S is any space-or-lightlike submanifold of space–time Rn+1. The σ’s have natural symplectic structures covariant with respect to the Poincaré group, and a norm ∥¨∥σ on the space 𝒦 of solutions is defined by integrating with respect to the Liouville measure on σ. This automatically gives ∥f∥2σ as the total flux of a conserved space–time vector field, implying that ∥f∥σ is independent of σ. Some inconsistencies encountered in the space–time theory of Klein–Gordon particles appear to be resolved in the phase-space framework.