199 research outputs found
Geometry and observables in (2+1)-gravity
We review the geometrical properties of vacuum spacetimes in (2+1)-gravity
with vanishing cosmological constant. We explain how these spacetimes are
characterised as quotients of their universal cover by holonomies. We explain
how this description can be used to clarify the geometrical interpretation of
the fundamental physical variables of the theory, holonomies and Wilson loops.
In particular, we discuss the role of Wilson loop observables as the generators
of the two fundamental transformations that change the geometry of
(2+1)-spacetimes, grafting and earthquake. We explain how these variables can
be determined from realistic measurements by an observer in the spacetime.Comment: Talk given at 2nd School and Workshop on Quantum Gravity and Quantum
Geometry (Corfu, September 13-20 2009); 10 pages, 13 eps figure
Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry
We consider Chern-Simons theories for the Poincare, de Sitter and anti-de
Sitter groups in three dimensions which generalise the Chern-Simons formulation
of 3d gravity. We determine conditions under which kappa-Poincare symmetry and
its de Sitter and anti-de Sitter analogues can be associated to these theories
as quantised symmetries. Assuming the usual form of those symmetries, with a
timelike vector as deformation parameter, we find that such an association is
possible only in the de Sitter case, and that the associated Chern-Simons
action is not the gravitational one. Although the resulting theory and 3d
gravity have the same equations of motion for the gauge field, they are not
equivalent, even classically, since they differ in their symplectic structure
and the coupling to matter. We deduce that kappa-Poincare symmetry is not
associated to either classical or quantum gravity in three dimensions. Starting
from the (non-gravitational) Chern-Simons action we explain how to construct a
multi-particle model which is invariant under the classical analogue of
kappa-de Sitter symmetry, and carry out the first steps in that construction.Comment: 31 pages, minor corrections and additional comment
Impacts of anthropogenic and environmental factors on the occurrence of shallow landslides in an alpine catchment (Urseren Valley, Switzerland)
Changes in climate and land use pose a risk to stability of alpine soils, but the direction and magnitude of the impact is still discussed controversially with respect to the various alpine regions. In this study, we explicitly consider the influence of dynamic human-induced changes on the occurrence of landslides in addition to natural factors. Our hypothesis was that if changes in land use and climate have a significant influence on the occurrence of landslides we would see a trend in the incidence of landslides over time. We chose the Urseren Valley in the Central Swiss Alps as investigation site because the valley is dramatically affected by landslides and the land use history is well documented. Maps of several environmental factors were used to analyse the spatial landslide pattern. In order to explain the causation of the temporal variation, time-series (45 years) of precipitation characteristics, cattle stocking and pasture maps were compared to a series of seven landslide investigation maps between 1959 and 2004. We found that the area affected by landslides increased by 92% from 1959 to 2004. Even though catchment characteristics like geology and slope largely explain the spatial variation in landslide susceptibility (68%), this cannot explain the temporal trend in landslide activity. The increase in stocking numbers and the increased intensity of torrential rain events had most likely an influence on landslide incidence. In addition, our data and interviews with farmers pointed to the importance of management practice
Quantum double and -Poincar\'e symmetries in (2+1)-gravity and Chern-Simons theory
We review the role of Drinfeld doubles and kappa-Poincare symmetries in
quantised (2+1)-gravity and Chern-Simons theory. We discuss the conditions
under which a given Hopf algebra symmetry is compatible with a Chern-Simons
theory and determine this compatibility explicitly for the Drinfeld doubles and
kappa-Poincare symmetries associated with the isometry groups of (2+1)-gravity.
In particular, we explain that the usual kappa-Poincare symmetries with a
timelike deformation are not directly associated with (2+1)-gravity. These
kappa-Poincare symmetries are linked to Chern-Simons theory only in the de
Sitter case, and the relevant Chern-Simons theory is physically inequivalent to
(2+1)-gravity.Comment: 11 pages, no figures, expanded version of talk at the conference
Theory Canada 4, references and explanations added, typos correcte
Algebraic quantization of the closed bosonic string
The gauge invariant observables of the closed bosonic string are quantized
without anomalies in four space-time dimensions by constructing their quantum
algebra in a manifestly covariant approach. The quantum algebra is the kernel
of a derivation on the universal envelopping algebra of an infinite-dimensional
Lie algebra. The search for Hilbert space representations of this algebra is
separated from its construction, and postponed.Comment: 22 pages. Revised: minor changes as in the published version (CMP
Dual generators of the fundamental group and the moduli space of flat connections
We define the dual of a set of generators of the fundamental group of an
oriented two-surface of genus with punctures and the
associated surface with a disc removed. This dual is
another set of generators related to the original generators via an involution
and has the properties of a dual graph. In particular, it provides an algebraic
prescription for determining the intersection points of a curve representing a
general element of the fundamental group with the
representatives of the generators and the order in which these intersection
points occur on the generators.We apply this dual to the moduli space of flat
connections on and show that when expressed in terms both, the
holonomies along a set of generators and their duals, the Poisson structure on
the moduli space takes a particularly simple form. Using this description of
the Poisson structure, we derive explicit expressions for the Poisson brackets
of general Wilson loop observables associated to closed, embedded curves on the
surface and determine the associated flows on phase space. We demonstrate that
the observables constructed from the pairing in the Chern-Simons action
generate of infinitesimal Dehn twists and show that the mapping class group
acts by Poisson isomorphisms.Comment: 54 pages, 13 .eps figure
Geometrical (2+1)-gravity and the Chern-Simons formulation: Grafting, Dehn twists, Wilson loop observables and the cosmological constant
We relate the geometrical and the Chern-Simons description of
(2+1)-dimensional gravity for spacetimes of topology , where
is an oriented two-surface of genus , for Lorentzian signature and general
cosmological constant and the Euclidean case with negative cosmological
constant. We show how the variables parametrising the phase space in the
Chern-Simons formalism are obtained from the geometrical description and how
the geometrical construction of (2+1)-spacetimes via grafting along closed,
simple geodesics gives rise to transformations on the phase space. We
demonstrate that these transformations are generated via the Poisson bracket by
one of the two canonical Wilson loop observables associated to the geodesic,
while the other acts as the Hamiltonian for infinitesimal Dehn twists. For
spacetimes with Lorentzian signature, we discuss the role of the cosmological
constant as a deformation parameter in the geometrical and the Chern-Simons
formulation of the theory. In particular, we show that the Lie algebras of the
Chern-Simons gauge groups can be identified with the (2+1)-dimensional Lorentz
algebra over a commutative ring, characterised by a formal parameter
whose square is minus the cosmological constant. In this
framework, the Wilson loop observables that generate grafting and Dehn twists
are obtained as the real and the -component of a Wilson loop
observable with values in the ring, and the grafting transformations can be
viewed as infinitesimal Dehn twists with the parameter .Comment: 50 pages, 6 eps figure
Boundary conditions and symplectic structure in the Chern-Simons formulation of (2+1)-dimensional gravity
We propose a description of open universes in the Chern-Simons formulation of
(2+1)-dimensional gravity where spatial infinity is implemented as a puncture.
At this puncture, additional variables are introduced which lie in the
cotangent bundle of the Poincar\'e group, and coupled minimally to the
Chern-Simons gauge field. We apply this description of spatial infinity to open
universes of general genus and with an arbitrary number of massive spinning
particles. Using results of [9] we give a finite dimensional description of the
phase space and determine its symplectic structure. In the special case of a
genus zero universe with spinless particles, we compare our result to the
symplectic structure computed by Matschull in the metric formulation of
(2+1)-dimensional gravity. We comment on the quantisation of the phase space
and derive a quantisation condition for the total mass and spin of an open
universe.Comment: 44 pages, 3 eps figure
Grafting and Poisson structure in (2+1)-gravity with vanishing cosmological constant
We relate the geometrical construction of (2+1)-spacetimes via grafting to
phase space and Poisson structure in the Chern-Simons formulation of
(2+1)-dimensional gravity with vanishing cosmological constant on manifolds of
topology , where is an orientable two-surface of genus
. We show how grafting along simple closed geodesics \lambda is
implemented in the Chern-Simons formalism and derive explicit expressions for
its action on the holonomies of general closed curves on S_g. We prove that
this action is generated via the Poisson bracket by a gauge invariant
observable associated to the holonomy of . We deduce a symmetry
relation between the Poisson brackets of observables associated to the Lorentz
and translational components of the holonomies of general closed curves on S_g
and discuss its physical interpretation. Finally, we relate the action of
grafting on the phase space to the action of Dehn twists and show that grafting
can be viewed as a Dehn twist with a formal parameter satisfying
.Comment: 43 pages, 10 .eps figures; minor modifications: 2 figures added,
explanations added, typos correcte
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