835 research outputs found
Linear Connections in Non-Commutative Geometry
A construction is proposed for linear connections on non-commutative
algebras. The construction relies on a generalisation of the Leibnitz rules of
commutative geometry and uses the bimodule structure of . A special
role is played by the extension to the framework of non-commutative geometry of
the permutation of two copies of . The construction of the linear
connection as well as the definition of torsion and curvature is first proposed
in the setting of the derivations based differential calculus of Dubois-
Violette and then a generalisation to the framework proposed by Connes as well
as other non-commutative differential calculi is suggested. The covariant
derivative obtained admits an extension to the tensor product of several copies
of . These constructions are illustrated with the example of the
algebra of matrices.Comment: 15 pages, LMPM ../94 (uses phyzzx
Linear Connections on Fuzzy Manifolds
Linear connections are introduced on a series of noncommutative geometries
which have commutative limits. Quasicommutative corrections are calculated.Comment: 10 pages PlainTex; LPTHE Orsay 95/42; ESI Vienna 23
Almost commutative Riemannian geometry: wave operators
Associated to any (pseudo)-Riemannian manifold of dimension is an
-dimensional noncommutative differential structure (\Omega^1,\extd) on
the manifold, with the extra dimension encoding the classical Laplacian as a
noncommutative `vector field'. We use the classical connection, Ricci tensor
and Hodge Laplacian to construct (\Omega^2,\extd) and a natural
noncommutative torsion free connection on . We show
that its generalised braiding \sigma:\Omega^1\tens\Omega^1\to
\Omega^1\tens\Omega^1 obeys the quantum Yang-Baxter or braid relations only
when the original is flat, i.e their failure is governed by the Riemann
curvature, and that \sigma^2=\id only when is Einstein. We show that if
has a conformal Killing vector field then the cross product algebra
viewed as a noncommutative analogue of has a
natural -dimensional calculus extending and a natural spacetime
Laplacian now directly defined by the extra dimension. The case
recovers the Majid-Ruegg bicrossproduct flat spacetime model and the
wave-operator used in its variable speed of light preduction, but now as an
example of a general construction. As an application we construct the wave
operator on a noncommutative Schwarzschild black hole and take a first look at
its features. It appears that the infinite classical redshift/time dilation
factor at the event horizon is made finite.Comment: 39 pages, 4 pdf images. Removed previous Sections 5.1-5.2 to a
separate paper (now ArXived) to meet referee length requirements.
Corresponding slight restructure but no change to remaining conten
Shadow of noncommutativity
We analyse the structure of the limit of a family of algebras
describing noncommutative versions of space-time, with a
parameter of noncommutativity. Assuming the Poincar\'e covariance of the
limit, we show that, besides the algebra of functions on Minkowski
space, must contain a nontrivial extra factor which is Lorentz
covariant and which does not commute with the functions whenever it is not
commutative. We give a general description of the possibilities and analyse
some representative examples.Comment: 19 pages, Latex2e, available at http://qcd.th.u-psud.f
Abundance of local actions for the vacuum Einstein equations
We exhibit large classes of local actions for the vacuum Einstein equations.
In presence of fermions, or more generally of matter which couple to the
connection, these actions lead to inequivalent equations revealing an arbitrary
number of parameters. Even in the pure gravitational sector, any corresponding
quantum theory would depend on these parameters.Comment: 10 pages. Final version to appear in Letters in Mathematical Physic
Higher Spin BRS Cohomology of Supersymmetric Chiral Matter in D=4
We examine the BRS cohomology of chiral matter in , supersymmetry
to determine a general form of composite superfield operators which can suffer
from supersymmetry anomalies. Composite superfield operators \Y_{(a,b)} are
products of the elementary chiral superfields and \ov S and the
derivative operators D_\a, \ov D_{\dot \b} and \pa_{\a \dot \b}. Such
superfields \Y_{(a,b)} can be chosen to have `' symmetrized undotted
indices \a_i and `' symmetrized dotted indices \dot \b_j. The result
derived here is that each composite superfield \Y_{(a,b)} is subject to
potential supersymmetry anomalies if is an odd number, which means that
\Y_{(a,b)} is a fermionic superfield.Comment: 15 pages, CPT-TAMU-20/9
Chirality and Dirac Operator on Noncommutative Sphere
We give a derivation of the Dirac operator on the noncommutative -sphere
within the framework of the bosonic fuzzy sphere and define Connes' triple. It
turns out that there are two different types of spectra of the Dirac operator
and correspondingly there are two classes of quantized algebras. As a result we
obtain a new restriction on the Planck constant in Berezin's quantization. The
map to the local frame in noncommutative geometry is also discussed.Comment: 24 pages, latex, no figure
Gauge Theories in Noncommutative Homogeneous K\"ahler Manifolds
We construct a gauge theory on a noncommutative homogeneous K\"ahler
manifold, where we employ the deformation quantization with separation of
variables for K\"ahler manifolds formulated by Karabegov. A key point in this
construction is to obtaining vector fields which act as inner derivations for
the deformation quantization. We show that these vector fields are the only
Killing vector fields. We give an explicit construction of this gauge theory on
noncommutative and noncommutative .Comment: 27 pages, typos correcte
Lectures on graded differential algebras and noncommutative geometry
These notes contain a survey of some aspects of the theory of graded
differential algebras and of noncommutative differential calculi as well as of
some applications connected with physics. They also give a description of
several new developments.Comment: 71 pages; minor typo correction
Z-graded differential geometry of quantum plane
In this work, the Z-graded differential geometry of the quantum plane is
constructed. The corresponding quantum Lie algebra and its Hopf algebra
structure are obtained. The dual algebra, i.e. universal enveloping algebra of
the quantum plane is explicitly constructed and an isomorphism between the
quantum Lie algebra and the dual algebra is given.Comment: 17 page
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