115 research outputs found
Localized Exotic Smoothness
Gompf's end-sum techniques are used to establish the existence of an infinity
of non-diffeomorphic manifolds, all having the same trivial
topology, but for which the exotic differentiable structure is confined to a
region which is spatially limited. Thus, the smoothness is standard outside of
a region which is topologically (but not smoothly) ,
where is the compact three ball. The exterior of this region is
diffeomorphic to standard . In a
space-time diagram, the confined exoticness sweeps out a world tube which, it
is conjectured, might act as a source for certain non-standard solutions to the
Einstein equations. It is shown that smooth Lorentz signature metrics can be
globally continued from ones given on appropriately defined regions, including
the exterior (standard) region. Similar constructs are provided for the
topology, of the Kruskal form of the Schwarzschild
solution. This leads to conjectures on the existence of Einstein metrics which
are externally identical to standard black hole ones, but none of which can be
globally diffeomorphic to such standard objects. Certain aspects of the Cauchy
problem are also discussed in terms of \models which are
``half-standard'', say for all but for which cannot be globally
smooth.Comment: 8 pages plus 6 figures, available on request, IASSNS-HEP-94/2
Wilson Line Picture of Levin-Wen Partition Functions
Levin and Wen [Phys. Rev. B 71, 045110 (2005)] have recently given a lattice
Hamiltonian description of doubled Chern-Simons theories. We relate the
partition function of these theories to an expectation of Wilson loops that
form a link in 2+1 dimensional spacetime known in the mathematical literature
as Chain-Mail. This geometric construction gives physical interpretation of the
Levin-Wen Hilbert space and Hamiltonian, its topological invariance, exactness
under coarse-graining, and how two opposite chirality sectors of the doubled
theory arise.Comment: Final published version; Appendix adde
Exotic Smoothness and Physics
The essential role played by differentiable structures in physics is reviewed
in light of recent mathematical discoveries that topologically trivial
space-time models, especially the simplest one, , possess a rich
multiplicity of such structures, no two of which are diffeomorphic to each
other and thus to the standard one. This means that physics has available to it
a new panoply of structures available for space-time models. These can be
thought of as source of new global, but not properly topological, features.
This paper reviews some background differential topology together with a
discussion of the role which a differentiable structure necessarily plays in
the statement of any physical theory, recalling that diffeomorphisms are at the
heart of the principle of general relativity. Some of the history of the
discovery of exotic, i.e., non-standard, differentiable structures is reviewed.
Some new results suggesting the spatial localization of such exotic structures
are described and speculations are made on the possible opportunities that such
structures present for the further development of physical theories.Comment: 13 pages, LaTe
A cocycle on the group of symplectic diffeomorphisms
We define a cocycle on the group of symplectic diffeomorphisms of a
symplectic manifold and investigate its properties. The main applications are
concerned with symplectic actions of discrete groups. For example, we give an
alternative proof of the Polterovich theorem about the distortion of cyclic
subgroups in finitely generated groups of Hamiltonian diffeomorphisms.Comment: 19 pages, no figures, corrected versio
Exotic Smoothness and Quantum Gravity
Since the first work on exotic smoothness in physics, it was folklore to
assume a direct influence of exotic smoothness to quantum gravity. Thus, the
negative result of Duston (arXiv:0911.4068) was a surprise. A closer look into
the semi-classical approach uncovered the implicit assumption of a close
connection between geometry and smoothness structure. But both structures,
geometry and smoothness, are independent of each other. In this paper we
calculate the "smoothness structure" part of the path integral in quantum
gravity assuming that the "sum over geometries" is already given. For that
purpose we use the knot surgery of Fintushel and Stern applied to the class
E(n) of elliptic surfaces. We mainly focus our attention to the K3 surfaces
E(2). Then we assume that every exotic smoothness structure of the K3 surface
can be generated by knot or link surgery a la Fintushel and Stern. The results
are applied to the calculation of expectation values. Here we discuss the two
observables, volume and Wilson loop, for the construction of an exotic
4-manifold using the knot and the Whitehead link . By using Mostow
rigidity, we obtain a topological contribution to the expectation value of the
volume. Furthermore we obtain a justification of area quantization.Comment: 16 pages, 1 Figure, 1 Table subm. Class. Quant. Grav
Exotic Smooth Structures on Small 4-Manifolds
Let M be either CP^2#3CP^2bar or 3CP^2#5CP^2bar. We construct the first
example of a simply-connected symplectic 4-manifold that is homeomorphic but
not diffeomorphic to M.Comment: 11 page
Exotic smooth structures on 4-manifolds with zero signature
For every integer , we construct infinite families of mutually
nondiffeomorphic irreducible smooth structures on the topological -manifolds
and (2k-1)(\CP#\CPb), the connected sums of
copies of and \CP#\CPb.Comment: 6 page
Paper Session II-B - Performance Status of the Mars Environmental Compatibility Assessment Electrometer
The Mars Environmental Compatibility Assessment electrometer is an instrument intended to fly on a future Mars lander mission. The electrometer was designed primarily to investigate (1) the electrostatic interaction between the Martian soil and five different types of insulators attached to the electrometer, which are to be rubbed over the Martian soil. The MECA Electrometer is also capable of measuring (2) the presence of charged particles in the Martian atmosphere, (3) the local electric field strength, and (4) the local temperature. We have tested and evaluated the measurement capabilities of the MECA Electrometer under simulated Martian surface conditions using facilities located in the Electromagnetic Physics Testbed at KSC. The results of the study have demonstrated that rubbing an insulator over the Martian soil simulant does triboelectrically charge up the insulator\u27s surface. However, the charge buildup on an insulator was found to be as low as 1% of the current maximum range of the electrometer when it is rubbed through Martian soil. This indicates that the overall gain of the MECA Electrometer could be increased by a factor of 50, if measurements at the 50% level of full-range sensitivity are desired. The ion gauge, which detects the presence of charged particles, was also evaluated over the pressure range 13 - 533 mbar, and results will be presented
On the geometrization of matter by exotic smoothness
In this paper we discuss the question how matter may emerge from space. For
that purpose we consider the smoothness structure of spacetime as underlying
structure for a geometrical model of matter. For a large class of compact
4-manifolds, the elliptic surfaces, one is able to apply the knot surgery of
Fintushel and Stern to change the smoothness structure. The influence of this
surgery to the Einstein-Hilbert action is discussed. Using the Weierstrass
representation, we are able to show that the knotted torus used in knot surgery
is represented by a spinor fulfilling the Dirac equation and leading to a
mass-less Dirac term in the Einstein-Hilbert action. For sufficient complicated
links and knots, there are "connecting tubes" (graph manifolds, torus bundles)
which introduce an action term of a gauge field. Both terms are genuinely
geometrical and characterized by the mean curvature of the components. We also
discuss the gauge group of the theory to be U(1)xSU(2)xSU(3).Comment: 30 pages, 3 figures, svjour style, complete reworking now using
Fintushel-Stern knot surgery of elliptic surfaces, discussion of Lorentz
metric and global hyperbolicity for exotic 4-manifolds added, final version
for publication in Gen. Rel. Grav, small typos errors fixe
Fake R^4's, Einstein Spaces and Seiberg-Witten Monopole Equations
We discuss the possible relevance of some recent mathematical results and
techniques on four-manifolds to physics. We first suggest that the existence of
uncountably many R^4's with non-equivalent smooth structures, a mathematical
phenomenon unique to four dimensions, may be responsible for the observed
four-dimensionality of spacetime. We then point out the remarkable fact that
self-dual gauge fields and Weyl spinors can live on a manifold of Euclidean
signature without affecting the metric. As a specific example, we consider
solutions of the Seiberg-Witten Monopole Equations in which the U(1) fields are
covariantly constant, the monopole Weyl spinor has only a single constant
component, and the 4-manifold M_4 is a product of two Riemann surfaces
Sigma_{p_1} and Sigma_{p_2}. There are p_{1}-1(p_{2}-1) magnetic(electric)
vortices on \Sigma_{p_1}(\Sigma_{p_2}), with p_1 + p_2 \geq 2 (p_1=p_2= 1 being
excluded). When the two genuses are equal, the electromagnetic fields are
self-dual and one obtains the Einstein space \Sigma_p x \Sigma_p, the monopole
condensate serving as the cosmological constant.Comment: 9 pages, Talk at the Second Gursey Memorial Conference, June 2000,
Istanbu
- …