368 research outputs found
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
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
L^2 torsion without the determinant class condition and extended L^2 cohomology
We associate determinant lines to objects of the extended abelian category
built out of a von Neumann category with a trace. Using this we suggest
constructions of the combinatorial and the analytic L^2 torsions which, unlike
the work of the previous authors, requires no additional assumptions; in
particular we do not impose the determinant class condition. The resulting
torsions are elements of the determinant line of the extended L^2 cohomology.
Under the determinant class assumption the L^2 torsions of this paper
specialize to the invariants studied in our previous work. Applying a recent
theorem of D. Burghelea, L. Friedlander and T. Kappeler we obtain a Cheeger -
Muller type theorem stating the equality between the combinatorial and the
analytic L^2 torsions.Comment: 39 page
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
Gravity a la Born-Infeld
A simple technique for the construction of gravity theories in Born-Infeld
style is presented, and the properties of some of these novel theories are
investigated. They regularize the positive energy Schwarzschild singularity,
and a large class of models allows for the cancellation of ghosts. The possible
correspondence to low energy string theory is discussed. By including curvature
corrections to all orders in alpha', the new theories nicely illustrate a
mechanism that string theory might use to regularize gravitational
singularities.Comment: 21 pages, 2 figures, new appendix B with corrigendum: Class. Quantum
Grav. 21 (2004) 529
Approximation of holomorphic mappings on strongly pseudoconvex domains
Let D be a relatively compact strongly pseudoconvex domain in a Stein
manifold, and let Y be a complex manifold. We prove that the set A(D,Y),
consisting of all continuous maps from the closure of D to Y which are
holomorphic in D, is a complex Banach manifold. When D is the unit disc in C
(or any other topologically trivial strongly pseudoconvex domain in a Stein
manifold), A(D,Y) is locally modeled on the Banach space A(D,C^n)=A(D)^n with
n=dim Y. Analogous results hold for maps which are holomorphic in D and of
class C^r up to the boundary for any positive integer r. We also establish the
Oka property for sections of continuous or smooth fiber bundles over the
closure of D which are holomorphic over D and whose fiber enjoys the Convex
approximation property. The main analytic technique used in the paper is a
method of gluing holomorphic sprays over Cartan pairs in Stein manifolds, with
control up to the boundary, which was developed in our paper "Holomorphic
curves in complex manifolds" (Duke Math. J. 139 (2007), no. 2, 203--253)
Stability of relative equilibria with singular momentum values in simple mechanical systems
A method for testing -stability of relative equilibria in Hamiltonian
systems of the form "kinetic + potential energy" is presented. This method
extends the Reduced Energy-Momentum Method of Simo et al. to the case of
non-free group actions and singular momentum values. A normal form for the
symplectic matrix at a relative equilibrium is also obtained.Comment: Partially rewritten. Some mistakes fixed. Exposition improve
Topological origin of the phase transition in a mean-field model
We argue that the phase transition in the mean-field XY model is related to a
particular change in the topology of its configuration space. The nature of
this topological transition can be discussed on the basis of elementary Morse
theory using the potential energy per particle V as a Morse function. The value
of V where such a topological transition occurs equals the thermodynamic value
of V at the phase transition and the number of (Morse) critical points grows
very fast with the number of particles N. Furthermore, as in statistical
mechanics, also in topology the way the thermodynamic limit is taken is
crucial.Comment: REVTeX, 5 pages, with 1 eps figure included. Some changes in the
text. To appear in Physical Review Letter
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