502 research outputs found
Superintegrable Hamiltonian systems with noncompact invariant submanifolds. Kepler system
The Mishchenko-Fomenko theorem on superintegrable Hamiltonian systems is
generalized to superintegrable Hamiltonian systems with noncompact invariant
submanifolds. It is formulated in the case of globally superintegrable
Hamiltonian systems which admit global generalized action-angle coordinates.
The well known Kepler system falls into two different globally superintegrable
systems with compact and noncompact invariant submanifolds.Comment: 23 page
The Picard group of the loop space of the Riemann sphere
The loop space of the Riemann sphere consisting of all C^k or Sobolev W^{k,p}
maps from the circle S^1 to the sphere is an infinite dimensional complex
manifold. We compute the Picard group of holomorphic line bundles on this loop
space as an infinite dimensional complex Lie group with Lie algebra the first
Dolbeault group. The group of Mobius transformations G and its loop group LG
act on this loop space. We prove that an element of the Picard group is
LG-fixed if it is G-fixed; thus completely answer the question by Millson and
Zombro about G-equivariant projective embedding of the loop space of the
Riemann sphere.Comment: International Journal of Mathematic
Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics
In this article we develop some new existence results for the Einstein
constraint equations using the Lichnerowicz-York conformal rescaling method.
The mean extrinsic curvature is taken to be an arbitrary smooth function
without restrictions on the size of its spatial derivatives, so that it can be
arbitrarily far from constant. The rescaled background metric belongs to the
positive Yamabe class, and the freely specifiable part of the data given by the
traceless-transverse part of the rescaled extrinsic curvature and the matter
fields are taken to be sufficiently small, with the matter energy density not
identically zero. Using topological fixed-point arguments and global barrier
constructions, we then establish existence of solutions to the constraints. Two
recent advances in the analysis of the Einstein constraint equations make this
result possible: A new type of topological fixed-point argument without
smallness conditions on spatial derivatives of the mean extrinsic curvature,
and a new construction of global super-solutions for the Hamiltonian constraint
that is similarly free of such conditions on the mean extrinsic curvature. For
clarity, we present our results only for strong solutions on closed manifolds.
However, our results also hold for weak solutions and for other cases such as
compact manifolds with boundary; these generalizations will appear elsewhere.
The existence results presented here for the Einstein constraints are
apparently the first such results that do not require smallness conditions on
spatial derivatives of the mean extrinsic curvature.Comment: 4 pages, no figures, accepted for publication in Physical Review
Letters. (Abstract shortenned and other minor changes reflecting v4 version
of arXiv:0712.0798
Polar Actions on Berger Spheres
The object of this article is to study a torus action on a so-called Berger sphere. We also make some comments on polar actions on naturally reductive homogeneous spaces. Finally, we prove a rigidity-type theorem for Riemannian manifolds carrying a polar action with a fix point
Pedestrian index theorem a la Aharonov-Casher for bulk threshold modes in corrugated multilayer graphene
Zero-modes, their topological degeneracy and relation to index theorems have
attracted attention in the study of single- and bilayer graphene. For
negligible scalar potentials, index theorems explain why the degeneracy of the
zero-energy Landau level of a Dirac hamiltonian is not lifted by gauge field
disorder, for example due to ripples, whereas other Landau levels become
broadened by the inhomogenous effective magnetic field. That also the bilayer
hamiltonian supports such protected bulk zero-modes was proved formally by
Katsnelson and Prokhorova to hold on a compact manifold by using the
Atiyah-Singer index theorem. Here we complement and generalize this result in a
pedestrian way by pointing out that the simple argument by Aharonov and Casher
for degenerate zero-modes of a Dirac hamiltonian in the infinite plane extends
naturally to the multilayer case. The degeneracy remains, though at nonzero
energy, also in the presence of a gap. These threshold modes make the spectrum
asymmetric. The rest of the spectrum, however, remains symmetric even in
arbitrary gauge fields, a fact related to supersymmetry. Possible benefits of
this connection are discussed.Comment: 6 pages, 2 figures. The second version states now also in words that
the conjugation symmetry that in the massive case gets replaced by
supersymmetry is the chiral symmetry. Changes in figure
Shortcuts to Spherically Symmetric Solutions: A Cautionary Note
Spherically symmetric solutions of generic gravitational models are
optimally, and legitimately, obtained by expressing the action in terms of the
two surviving metric components. This shortcut is not to be overdone, however:
a one-function ansatz invalidates it, as illustrated by the incorrect solutions
of [1].Comment: 2 pages. Amplified derivation, accepted for publication in Class
Quant Gra
Shortcuts to high symmetry solutions in gravitational theories
We apply the Weyl method, as sanctioned by Palais' symmetric criticality
theorems, to obtain those -highly symmetric -geometries amenable to explicit
solution, in generic gravitational models and dimension. The technique consists
of judiciously violating the rules of variational principles by inserting
highly symmetric, and seemingly gauge fixed, metrics into the action, then
varying it directly to arrive at a small number of transparent, indexless,
field equations. Illustrations include spherically and axially symmetric
solutions in a wide range of models beyond D=4 Einstein theory; already at D=4,
novel results emerge such as exclusion of Schwarzschild solutions in cubic
curvature models and restrictions on ``independent'' integration parameters in
quadratic ones. Another application of Weyl's method is an easy derivation of
Birkhoff's theorem in systems with only tensor modes. Other uses are also
suggested.Comment: 10 page
A Categorical Equivalence between Generalized Holonomy Maps on a Connected Manifold and Principal Connections on Bundles over that Manifold
A classic result in the foundations of Yang-Mills theory, due to J. W.
Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills
Theory." Int. J. Th. Phys. 30(9), (1991)], establishes that given a
"generalized" holonomy map from the space of piece-wise smooth, closed curves
based at some point of a manifold to a Lie group, there exists a principal
bundle with that group as structure group and a principal connection on that
bundle such that the holonomy map corresponds to the holonomies of that
connection. Barrett also provided one sense in which this "recovery theorem"
yields a unique bundle, up to isomorphism. Here we show that something stronger
is true: with an appropriate definition of isomorphism between generalized
holonomy maps, there is an equivalence of categories between the category whose
objects are generalized holonomy maps on a smooth, connected manifold and whose
arrows are holonomy isomorphisms, and the category whose objects are principal
connections on principal bundles over a smooth, connected manifold. This result
clarifies, and somewhat improves upon, the sense of "unique recovery" in
Barrett's theorems; it also makes precise a sense in which there is no loss of
structure involved in moving from a principal bundle formulation of Yang-Mills
theory to a holonomy, or "loop", formulation.Comment: 20 page
Experimental realization of three-color entanglement at optical fiber communication and atomic storage wavelengths
Multi-color entangled states of light including low-loss optical fiber
transmission and atomic resonance frequencies are essential resources for
future quantum information network. We present the experimental achievement on
the three-color entanglement generation at 852 nm, 1550 nm and 1440 nm
wavelengths for optical continuous variables. The entanglement generation
system consists of two cascaded non-degenerated optical parametric oscillators
(NOPOs). The flexible selectivity of nonlinear crystals in the two NOPOs and
the tunable property of NOPO provide large freedom for the frequency selection
of three entangled optical beams, so the present system is possible to be
developed as practical devices used for quantum information networks with
atomic storage units and long fiber transmission lines.Comment: 4pages, 4 figure
Gauge theory of Faddeev-Skyrme functionals
We study geometric variational problems for a class of nonlinear sigma-models
in quantum field theory. Mathematically, one needs to minimize an energy
functional on homotopy classes of maps from closed 3-manifolds into compact
homogeneous spaces G/H. The minimizers are known as Hopfions and exhibit
localized knot-like structure. Our main results include proving existence of
Hopfions as finite energy Sobolev maps in each (generalized) homotopy class
when the target space is a symmetric space. For more general spaces we obtain a
weaker result on existence of minimizers in each 2-homotopy class.
Our approach is based on representing maps into G/H by equivalence classes of
flat connections. The equivalence is given by gauge symmetry on pullbacks of
G-->G/H bundles. We work out a gauge calculus for connections under this
symmetry, and use it to eliminate non-compactness from the minimization problem
by fixing the gauge.Comment: 34 pages, no figure
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