On the geometry of reduced cotangent bundles at zero momentum

We consider the problem of cotangent bundle reduction for proper non-free group actions at zero momentum. We show that in this context the symplectic stratification obtained by Sjamaar and Lerman refines in two ways: (i) each symplectic stratum admits a stratification which we call the secondary stratification with two distinct types of pieces, one of which is open and dense and symplectomorphic to a cotangent bundle; (ii) the reduced space at zero momentum admits a finer stratification than the symplectic one into pieces that are coisotropic in their respective symplectic strata.     

Symplectic and hyperkähler structures in a dimensional reduction of the Seiberg-Witten equations with a Higgs field

In this paper we show that the dimensionally reduced Seiberg-Witten equations lead to a Higgs field and we study the resulting moduli spaces. The moduli space arising out of a subset of the equations, shown to be non-empty for a compact Riemann surface of genus g ≥ 1, gives rise to a family of moduli spaces carrying a hyperkähler structure. For the full set of equations the corresponding moduli space does not have the aforementioned hyperkähler structure but has a natural symplectic structure. For the case of the torus, g = 1, we show that the full set of equations has a solution, different from the “vortex solutions”.     

Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle

We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in a toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states) and some non-monotone Lagrangian torus fibers.     

Submanifolds in the thermodynamic phase space

The paper deals with submanifolds (constraints) in the so-called thermodynamic phase space (TPS). It has been shown that in each regular submanifold one can define a reduced contact space and that under some conditions properties of TPS may be projected onto the reduced space. It is also proved that Dirac's concept of a class of constraints can be determined algebraically by means of Cartan's bracket. Only contact manifolds are considered, without referring to symplectic ones.     

A variational principle for symplectic connections

We introduce a variational principle for symplectic connections and study the corresponding field equations. For two-dimensional compact symplectic manifolds we determine all solutions of the field equations. For two-dimensional non-compact simply connected symplectic manifolds we give an essentially exhaustive list of solutions of the field equations. Finally we indicate how to construct from solutions of the field equations on (M, ω) solutions of the field equations on the cotangent bundle to M with its standard symplectic structure.     

A Tolman-like Compact Model with Conformal Geometry

In this investigation, we study a model of a charged anisotropic compact star by assuming a relationship between the metric functions arising from a conformal symmetry. This mechanism leads to a first-order differential equation containing pressure anisotropy and the electric field. Particular forms of the electric field intensity, combined with the Tolman VII metric, are used to solve the Einstein–Maxwell field equations. New classes of exact solutions generated are expressed in terms of elementary functions. For specific parameter values based on the physical requirements, it is shown that the model satisfies the causality, stability and energy conditions. Numerical values generated for masses, radii, central densities, surface redshifts and compactness factors are consistent with compact objects such as PSR J1614-2230 and SMC X-1.     

Particle on a torus knot: Constrained dynamics and semi-classical quantization in a magnetic field

Kinematics and dynamics of a particle moving on a torus knot poses an interesting problem as a constrained system. In the first part of the paper we have derived the modified symplectic structure or Dirac brackets of the above model in Dirac’s Hamiltonian framework, both in toroidal and Cartesian coordinate systems. This algebra has been used to study the dynamics, in particular small fluctuations in motion around a specific torus. The spatial symmetries of the system have also been studied.     

The structure of the space of solutions of Einstein's equations II: several Killing fields and the Einstein-Yang-Mills equations

The space of solutions of Einstein's vacuum equations is shown to have conical singularities at each spacetime possessing a compact Cauchy surface of constant mean curvature and a nontrivial set of Killing fields. Similar results are shown for the coupled Einstein-Yang-Mills system. Combined with an appropriate slice theorem, the results show that the space of geometrically equivalent solutions is a stratified manifold with each stratum being a symplectic manifold characterized by the symmetry type of its members.     

Quantization of Multiply Connected Manifolds

The standard (Berezin-Toeplitz) geometric quantization of a compact Kähler manifold is restricted by integrality conditions. These restrictions can be circumvented by passing to the universal covering space, provided that the lift of the symplectic form is exact. I relate this construction to the Baum-Connes assembly map and prove that it gives a strict quantization of the original manifold. I also propose a further generalization, classify the required structure, and provide a means of computing the resulting algebras. These constructions involve twisted group C*-algebras of the fundamental group which are determined by a group cocycle constructed from the cohomology class of the symplectic form. This provides an algebraic counterpart to the Morita equivalence of a symplectic manifold with its fundamental group.     

Relation between Tolman length and isothermal compressibility for simple liquids

The Tolman length δ 0 of a liquid with a plane surface has attracted increasing theoretical attention in recent years,but the expression of Tolman length in terms of observable quantities is still not very clear.In 2001,Bartell gave a simple expression of Tolman length δ 0 in terms of isothermal compressibility.However,this expression predicts that Tolman length is always negative,which is contrary to the results of molecular dynamics simulations(MDS) for simple liquids.In this paper,this contradiction is analyzed and the reason for the discrepancy in the sign is found.In addition,we introduce a new expression of Tolman length in terms of isothermal compressibility for simple fluids not near the critical points under some weak restrictions.The Tolman length of simple liquids calculated by using this formula is consistent with that obtained using MDS regarding the sign.     

Yang-Mills on Surfaces with Boundary: Quantum Theory and Symplectic Limit

The quantum field measure for gauge fields over a compact surface with boundary, with holonomy around the boundary components specified, is constructed. Loop expectation values for general loop configurations are computed. For a compact oriented surface with one boundary component, let be the moduli space of flat connections with boundary holonomy lying in a conjugacy class in the gauge group G. We prove that a certain natural closed 2-form on , introduced in an earlier work by C. King and the author, is a symplectic structure on the generic stratum of for generic . We then prove that the quantum Yang-Mills measure, with the boundary holonomy constrained to lie in , converges in a natural sense to the corresponding symplectic volume measure in the classical limit. We conclude with a detailed treatment of the case , and determine the symplectic volume of this moduli space. Received: 30 June 1996 / Accepted: 22 July 1996     

Oscillatory Modules

Developing the ideas of Bressler and Soibelman and of Karabegov, we introduce a notion of an oscillatory module on a symplectic manifold which is a sheaf of modules over the sheaf of deformation quantization algebras with an additional structure. We compare the category of oscillatory modules on a torus to the Fukaya category as computed by Polishchuk and Zaslow.     

Coherent states in geometric quantization

In this paper we study overcomplete systems of coherent states associated to compact integral symplectic manifolds by geometric quantization. Our main goals are to give a systematic treatment of the construction of such systems and to collect some recent results. We begin by recalling the basic constructions of geometric quantization in both the Kähler and non-Kähler cases. We then study the reproducing kernels associated to the quantum Hilbert spaces and use them to define symplectic coherent states. The rest of the paper is dedicated to the properties of symplectic coherent states and the corresponding Berezin–Toeplitz quantization. Specifically, we study overcompleteness, symplectic analogues of the basic properties of Bargmann’s weighted analytic function spaces, and the ‘maximally classical’ behavior of symplectic coherent states. We also find explicit formulas for symplectic coherent states on compact Riemann surfaces.     

Rainbow’s stars

In recent years, a growing interest in the equilibrium of compact astrophysical objects like white dwarf and neutron stars has been manifested. In particular, various modifications due to Planck-scale energy effects have been considered. In this paper we analyze the modification induced by gravity’s rainbow on the equilibrium configurations described by the Tolman–Oppenheimer–Volkoff (TOV) equation. Our purpose is to explore the possibility that the rainbow Planck-scale deformation of space-time could support the existence of different compact stars.     

Categories of Holomorphic Vector Bundles on Noncommutative Two-Tori

 In this paper we study the category of standard holomorphic vector bundles on a noncommutative two-torus. We construct a functor from the derived category of such bundles to the derived category of coherent sheaves on an elliptic curve and prove that it induces an equivalence with the subcategory of stable objects. By the homological mirror symmetry for elliptic curves this implies an equivalence between the derived category of holomorphic bundles on a noncommutative two-torus and the Fukaya category of the corresponding symplectic (commutative) torus. Received: 24 November 2002 / Accepted: 25 November 2002 Published online: 28 February 2003 RID="⋆" ID="⋆" The work of both authors was partially supported by NSF grants. Communicated by A. Connes     

Symplectic fixed points and holomorphic spheres

LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the symplectic action function perturbed by a Hamiltonian term. In particular, we associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading. It is shown to have properties similar to the homology of the Conley index in locally compact spaces. As an application, we show that if the fixed point set of an exact diffeomorphism onP is nondegenerate, then it satisfies the Morse inequalities onP. We also discuss fixed point estimates for general exact diffeomorphisms.     

Can one define a preferred symplectic connection?

We define a variational principle for symplectic connections of the Yang-Mills type. When the symplectic manifold is a compact surface we show that the moduli space of the connections which are extremals of the functional coincides with the Teichmüller space of the surface. We indicate that the noncompact situation is very different.     

Hamiltonian finite-dimensional models of baroclinic instability

A hierarchy of N-dimensional systems is constructed starting from the standard continuous two-layer quasi-geostrophic model of the geophysical fluid dynamics. These models (“truncations”) preserve the Hamiltonian structure of the parent model and tend to it in the limit N → ∞. The construction is based on the known correspondence SU(N) → SDiff(T²) when N → ∞ between the finite-dimensional group of unitary unimodular N × N matrices and the group of symplectic diffeomorphisms of the torus and the fact that the above-mentioned continuous model has an intrinsic geometric structure related to SDiff(T²) in the case of periodic boundary conditions. A fast symplectic solver for these truncations is proposed and used to study the baroclinic instability. 7     

Geometric quantization of symplectic manifolds with respect to reducible non-negative polarizations

The leafwise complex of a reducible non-negative polarization with values in the prequantum bundle on a prequantizable symplectic manifold is studied. The cohomology groups of this complex is shown to vanish in rank less than the rank of the real part of the non-negative polarization. The Bohr-Sommerfeld set for a reducible non-negative polarization is defined. A factorization theorem is proved for these reducible non-negative polarizations. For compact symplectic manifolds, it is shown that the above complex has finite dimensional cohomology groups, more-over a Lefschetz fixed point theorem and an index theorem for these non-elliptic complexes is proved. As a corollary of the index theorem, we deduce that the cardinality of the Bohr-Sommerfeld set for any reducible real polarization on a compact symplectic manifold is determined by the volume and the dimension of the manifold. Supported in part by NSF grant DMS-93-09653, while the author was visiting University of California Berkeley.