38,936 research outputs found
Consistent Digital Curved Rays and Pseudoline Arrangements
Representing a family of geometric objects in the digital world where each object is represented by a set of pixels is a basic problem in graphics and computational geometry. One important criterion is the consistency, where the intersection pattern of the objects should be consistent with axioms of the Euclidean geometry, e.g., the intersection of two lines should be a single connected component. Previously, the set of linear rays and segments has been considered. In this paper, we extended this theory to families of curved rays going through the origin. We further consider some psudoline arrangements obtained as unions of such families of rays
Charged sectors, spin and statistics in quantum field theory on curved spacetimes
The first part of this paper extends the Doplicher-Haag-Roberts theory of
superselection sectors to quantum field theory on arbitrary globally hyperbolic
spacetimes. The statistics of a superselection sector may be defined as in flat
spacetime and each charge has a conjugate charge when the spacetime possesses
non-compact Cauchy surfaces. In this case, the field net and the gauge group
can be constructed as in Minkowski spacetime.
The second part of this paper derives spin-statistics theorems on spacetimes
with appropriate symmetries. Two situations are considered: First, if the
spacetime has a bifurcate Killing horizon, as is the case in the presence of
black holes, then restricting the observables to the Killing horizon together
with "modular covariance" for the Killing flow yields a conformally covariant
quantum field theory on the circle and a conformal spin-statistics theorem for
charged sectors localizable on the Killing horizon. Secondly, if the spacetime
has a rotation and PT symmetry like the Schwarzschild-Kruskal black holes,
"geometric modular action" of the rotational symmetry leads to a
spin-statistics theorem for charged covariant sectors where the spin is defined
via the SU(2)-covering of the spatial rotation group SO(3).Comment: latex2e, 73 page
The Geometry of Almost Einstein (2,3,5) Distributions
We analyze the classic problem of existence of Einstein metrics in a given
conformal structure for the class of conformal structures inducedf Nurowski's
construction by (oriented) (2,3,5) distributions. We characterize in two ways
such conformal structures that admit an almost Einstein scale: First, they are
precisely the oriented conformal structures that are induced by at
least two distinct oriented (2,3,5) distributions; in this case there is a
1-parameter family of such distributions that induce . Second, they
are characterized by the existence of a holonomy reduction to ,
, or a particular semidirect product , according to the sign of the Einstein constant of the
corresponding metric. Via the curved orbit decomposition formalism such a
reduction partitions the underlying manifold into several submanifolds and
endows each ith a geometric structure. This establishes novel links between
(2,3,5) distributions and many other geometries - several classical geometries
among them - including: Sasaki-Einstein geometry and its paracomplex and
null-complex analogues in dimension 5; K\"ahler-Einstein geometry and its
paracomplex and null-complex analogues, Fefferman Lorentzian conformal
structures, and para-Fefferman neutral conformal structures in dimension 4; CR
geometry and the point geometry of second-order ordinary differential equations
in dimension 3; and projective geometry in dimension 2. We describe a
generalized Fefferman construction that builds from a 4-dimensional
K\"ahler-Einstein or para-K\"ahler-Einstein structure a family of (2,3,5)
distributions that induce the same (Einstein) conformal structure. We exploit
some of these links to construct new examples, establishing the existence of
nonflat almost Einstein (2,3,5) conformal structures for which the Einstein
constant is positive and negative
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