8 research outputs found
The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere
Equivariance under the action of Uq(so(5)) is used to compute the left
regular and (chiral) spinorial representations of the algebra of the orthogonal
quantum 4-sphere S^4_q. These representations are the constituents of a
spectral triple on this sphere with a Dirac operator which is isospectral to
the canonical one on the round undeformed four-sphere and which gives metric
dimension four for the noncommutative geometry. Non-triviality of the geometry
is proved by pairing the associated Fredholm module with an `instanton'
projection. We also introduce a real structure which satisfies all required
properties modulo smoothing operators.Comment: 40 pages, no figures, Latex. v2: Title changed. Sect. 9 on real
structure completely rewritten and results strengthened. Additional minor
changes throughout the pape
Local Index Formula on the Equatorial Podles Sphere
We discuss spectral properties of the equatorial Podles sphere. As a
preparation we also study the `degenerate' (i.e. ) case (related to the
quantum disk). We consider two different spectral triples: one related to the
Fock representation of the Toeplitz algebra and the isopectral one. After the
identification of the smooth pre--algebra we compute the dimension
spectrum and residues. We check the nontriviality of the (noncommutative) Chern
character of the associated Fredholm modules by computing the pairing with the
fundamental projector of the -algebra (the nontrivial generator of the
-group) as well as the pairing with the -analogue of the Bott
projector. Finally, we show that the local index formula is trivially
satisfied.Comment: 18 pages, no figures; minor correction
The Behavior of Kasner Cosmologies with Induced Matter
We extend the induced matter model, previously applied to a variety of
isotropic cases, to a generalization of Bianchi type-I anisotropic cosmologies.
The induced matter model is a 5D Kaluza-Klein approach in which assumptions of
compactness are relaxed for the fifth coordinate, leading to extra geometric
terms. One interpretation of these extra terms is to identify them as an
``induced matter'' contribution to the stress-energy tensor. In similar spirit,
we construct a five dimensional metric in which the spatial slices possess
Bianchi type-I geometry. We find a set of solutions for the five dimensional
Einstein equations, and determine the pressure and density of induced matter.
We comment on the long-term dynamics of the model, showing that the assumption
of positive density leads to the contraction over time of the fifth scale
factor.Comment: 14 page