1,380 research outputs found
Fuzzy Surfaces of Genus Zero
A fuzzy version of the ordinary round 2-sphere has been constructed with an
invariant curvature. We here consider linear connections on arbitrary fuzzy
surfaces of genus zero. We shall find as before that they are more or less
rigidly dependent on the differential calculus used but that a large number of
the latter can be constructed which are not covariant under the action of the
rotation group. For technical reasons we have been forced to limit our
considerations to fuzzy surfaces which are small perturbations of the fuzzy
sphere.Comment: 11 pages, Late
A Dynamical 2-dimensional Fuzzy Space
The noncommutative extension of a dynamical 2-dimensional space-time is given
and some of its properties discussed. Wick rotation to euclidean signature
yields a surface which has as commutative limit the doughnut but in a singular
limit in which the radius of the hole tends to zero.Comment: 13 pages, accepted for publication in Phys. Lett.
Topology at the Planck Length
A basic arbitrariness in the determination of the topology of a manifold at
the Planck length is discussed. An explicit example is given of a `smooth'
change in topology from the 2-sphere to the 2-torus through a sequence of
noncommuting geometries. Applications are considered to the theory of D-branes
within the context of the proposed (atrix) theory.Comment: Orsay Preprint 97/34, 17 pages, Late
Structure of the Three-dimensional Quantum Euclidean Space
As an example of a noncommutative space we discuss the quantum 3-dimensional
Euclidean space together with its symmetry structure in great detail.
The algebraic structure and the representation theory are clarified and
discrete spectra for the coordinates are found. The q-deformed Legendre
functions play a special role. A completeness relation is derived for these
functions.Comment: 22 pages, late
Matrix theory compactifications on twisted tori
We study compactifications of Matrix theory on twisted tori and
non-commutative versions of them. As a first step, we review the construction
of multidimensional twisted tori realized as nilmanifolds based on certain
nilpotent Lie algebras. Subsequently, matrix compactifications on tori are
revisited and the previously known results are supplemented with a background
of a non-commutative torus with non-constant non-commutativity and an
underlying non-associative structure on its phase space. Next we turn our
attention to 3- and 6-dimensional twisted tori and we describe consistent
backgrounds of Matrix theory on them by stating and solving the conditions
which describe the corresponding compactification. Both commutative and
non-commutative solutions are found in all cases. Finally, we comment on the
correspondence among the obtained solutions and flux compactifications of
11-dimensional supergravity, as well as on relations among themselves, such as
Seiberg-Witten maps and T-duality.Comment: 1+31 pages, v2: some comments and clarifications added, accepted for
publication in Physical Review
Unified Theories from Fuzzy Extra Dimensions
We combine and exploit ideas from Coset Space Dimensional Reduction (CSDR)
methods and Non-commutative Geometry. We consider the dimensional reduction of
gauge theories defined in high dimensions where the compact directions are a
fuzzy space (matrix manifold). In the CSDR one assumes that the form of
space-time is M^D=M^4 x S/R with S/R a homogeneous space. Then a gauge theory
with gauge group G defined on M^D can be dimensionally reduced to M^4 in an
elegant way using the symmetries of S/R, in particular the resulting four
dimensional gauge is a subgroup of G. In the present work we show that one can
apply the CSDR ideas in the case where the compact part of the space-time is a
finite approximation of the homogeneous space S/R, i.e. a fuzzy coset. In
particular we study the fuzzy sphere case.Comment: 6 pages, Invited talk given by G. Zoupanos at the 36th International
Symposium Ahrenshoop, Wernsdorf, Germany, 26-30 Aug 200
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