244 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
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
Homotopy types of stabilizers and orbits of Morse functions on surfaces
Let be a smooth compact surface, orientable or not, with boundary or
without it, either the real line or the circle , and
the group of diffeomorphisms of acting on by the rule
, where and .
Let be a Morse function and be the orbit of under this
action. We prove that for , and
except for few cases. In particular, is aspherical, provided so is .
Moreover, is an extension of a finitely generated free abelian
group with a (finite) subgroup of the group of automorphisms of the Reeb graph
of .
We also give a complete proof of the fact that the orbit is tame
Frechet submanifold of of finite codimension, and that the
projection is a principal locally trivial -fibration.Comment: 49 pages, 8 figures. This version includes the proof of the fact that
the orbits of a finite codimension of tame action of tame Lie group on tame
Frechet manifold is a tame Frechet manifold itsel
On the Strong Coupling Limit of the Faddeev-Hopf Model
The variational calculus for the Faddeev-Hopf model on a general Riemannian
domain, with general Kaehler target space, is studied in the strong coupling
limit. In this limit, the model has key similarities with pure Yang-Mills
theory, namely conformal invariance in dimension 4 and an infinite dimensional
symmetry group. The first and second variation formulae are calculated and
several examples of stable solutions are obtained. In particular, it is proved
that all immersive solutions are stable. Topological lower energy bounds are
found in dimensions 2 and 4. An explicit description of the spectral behaviour
of the Hopf map S^3 -> S^2 is given, and a conjecture of Ward concerning the
stability of this map in the full Faddeev-Hopf model is proved.Comment: 21 pages, 0 figure
An infinite genus mapping class group and stable cohomology
We exhibit a finitely generated group \M whose rational homology is
isomorphic to the rational stable homology of the mapping class group. It is
defined as a mapping class group associated to a surface \su of infinite
genus, and contains all the pure mapping class groups of compact surfaces of
genus with boundary components, for any and . We
construct a representation of \M into the restricted symplectic group of the real Hilbert space generated by the homology
classes of non-separating circles on \su, which generalizes the classical
symplectic representation of the mapping class groups. Moreover, we show that
the first universal Chern class in H^2(\M,\Z) is the pull-back of the
Pressley-Segal class on the restricted linear group
via the inclusion .Comment: 14p., 8 figures, to appear in Commun.Math.Phy
Hamiltonian reductions of free particles under polar actions of compact Lie groups
Classical and quantum Hamiltonian reductions of free geodesic systems of
complete Riemannian manifolds are investigated. The reduced systems are
described under the assumption that the underlying compact symmetry group acts
in a polar manner in the sense that there exist regularly embedded, closed,
connected submanifolds meeting all orbits orthogonally in the configuration
space. Hyperpolar actions on Lie groups and on symmetric spaces lead to
families of integrable systems of spin Calogero-Sutherland type.Comment: 15 pages, minor correction and updated references in v
Homothetic perfect fluid space-times
A brief summary of results on homotheties in General Relativity is given,
including general information about space-times admitting an r-parameter group
of homothetic transformations for r>2, as well as some specific results on
perfect fluids. Attention is then focussed on inhomogeneous models, in
particular on those with a homothetic group (acting multiply
transitively) and . A classification of all possible Lie algebra
structures along with (local) coordinate expressions for the metric and
homothetic vectors is then provided (irrespectively of the matter content), and
some new perfect fluid solutions are given and briefly discussed.Comment: 27 pages, Latex file, Submitted to Class. Quantum Gra
Egorov's theorem for transversally elliptic operators on foliated manifolds and noncommutative geodesic flow
The main result of the paper is Egorov's theorem for transversally elliptic
operators on compact foliated manifolds. This theorem is applied to describe
the noncommutative geodesic flow in noncommutative geometry of Riemannian
foliations.Comment: 23 pages, no figures. Completely revised and improved version of
dg-ga/970301
A handlebody calculus for topology change
We consider certain interesting processes in quantum gravity which involve a
change of spatial topology. We use Morse theory and the machinery of
handlebodies to characterise topology changes as suggested by Sorkin. Our
results support the view that that the pair production of Kaluza-Klein
monopoles and the nucleation of various higher dimensional objects are allowed
transitions with non-zero amplitude.Comment: Latex, 32 pages, 7 figure
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